Introduction
Bose–Einstein condensation means that a single one-particle state carries a macroscopic part of a many-particle system. For a homogeneous gas the natural candidate is the constant orbital. We prove that this orbital contains a fixed positive fraction of every ground state of a sufficiently dilute three-dimensional hard-sphere Bose gas, in the thermodynamic limit at fixed density and fixed exclusion distance. The conclusion permits complex wave functions and a degenerate ground space.
The result
Let Λ L = ( R / L Z ) 3 \Lambda_L=(\mathbb{R}/L\mathbb{Z})^3 Λ L = ( R / L Z ) 3 d L d_L d L N N N a > 0 a>0 a > 0
Ω N , L , a = { X ∈ Λ L N : d L ( x i , x j ) > a whenever i < j } . \Omega_{N,L,a}=\{X\in\Lambda_L^N:d_L(x_i,x_j)>a\text{ whenever }i<j\}. Ω N , L , a = { X ∈ Λ L N : d L ( x i , x j ) > a whenever i < j } . The hard-sphere Hamiltonian is the operator associated with the Dirichlet form
q N , L , a [ f ] = ∑ i = 1 N ∫ Ω N , L , a ∣ ∇ i f ∣ 2 , f ∈ H 0 1 ( Ω N , L , a ) , q_{N,L,a}[f]=\sum_{i=1}^{N}\int_{\Omega_{N,L,a}}|\nabla_i f|^2,\qquad f\in H_0^1(\Omega_{N,L,a}), q N , L , a [ f ] = i = 1 ∑ N ∫ Ω N , L , a ∣ ∇ i f ∣ 2 , f ∈ H 0 1 ( Ω N , L , a ) , restricted to symmetric functions. Thus its kinetic energy is − ∑ i Δ i -\sum_i \Delta_i − ∑ i Δ i a a a 1 − a / ∣ x ∣ 1-a/\lvert x\rvert 1 − a / ∣ x ∣ Λ L N \Lambda_L^N Λ L N
For a normalized bosonic vector Ψ \Psi Ψ
Γ Ψ ( 1 ) ( x ; y ) = N ∫ Λ L N − 1 Ψ ( x , Y ) Ψ ( y , Y ) ‾ d Y , φ 0 , L = L − 3 / 2 . \Gamma^{(1)}_\Psi(x;y)=N\int_{\Lambda_L^{N-1}}\Psi(x,Y)\overline{\Psi(y,Y)}\,\mathrm{d}Y,\qquad\varphi_{0,L}=L^{-3/2}. Γ Ψ ( 1 ) ( x ; y ) = N ∫ Λ L N − 1 Ψ ( x , Y ) Ψ ( y , Y ) d Y , φ 0 , L = L − 3/2 . In particular, Tr Γ Ψ ( 1 ) = N \operatorname{Tr}\Gamma^{(1)}_\Psi=N Tr Γ Ψ ( 1 ) = N
B ( Ψ ) = ⟨ φ 0 , L , Γ Ψ ( 1 ) φ 0 , L ⟩ N = 1 L 3 ∫ ∣ ∫ Ψ ( x , Y ) d x ∣ 2 d Y . (1) B(\Psi)=\frac{\langle\varphi_{0,L},\Gamma^{(1)}_\Psi\varphi_{0,L}\rangle}{N}
=\frac{1}{L^3}\int\left|\int\Psi(x,Y)\,\mathrm{d}x\right|^2\mathrm{d}Y. \tag*{(1)} B ( Ψ ) = N ⟨ φ 0 , L , Γ Ψ ( 1 ) φ 0 , L ⟩ = L 3 1 ∫ ∫ Ψ ( x , Y ) d x 2 d Y . (1) This quantity lies in [ 0 , 1 ] [0,1] [ 0 , 1 ]
Theorem 1.1 (Condensation in every ground state). There are absolute constants ε 0 , c 0 > 0 \varepsilon_0,c_0>0 ε 0 , c 0 > 0 with the following property. Fix a , ρ > 0 a,\rho>0 a , ρ > 0 satisfying ρ a 3 < ε 0 \rho a^3<\varepsilon_0 ρ a 3 < ε 0 For every sequence of particle numbers N k N_k N k and lengths L k L_k L k such that
N k ⟶ ∞ , L k ⟶ ∞ , N k L k 3 ⟶ ρ , N_k\longrightarrow\infty,\qquad L_k\longrightarrow\infty,\qquad\frac{N_k}{L_k^3}\longrightarrow\rho, N k ⟶ ∞ , L k ⟶ ∞ , L k 3 N k ⟶ ρ , and every choice of normalized, possibly complex, bosonic ground eigenvectors Ψ k \Psi_k Ψ k of the hard-sphere Hamiltonians,
lim inf k → ∞ B ( Ψ k ) ≥ c 0 . \liminf_{k\to\infty}B(\Psi_k)\ge c_0. k → ∞ lim inf B ( Ψ k ) ≥ c 0 . The same constant c 0 c_0 c 0 ρ a 3 \rho a^3 ρ a 3 a a a ρ \rho ρ ρ a 3 \rho a^3 ρ a 3
Corollary 1.2 (Ground-space states). Under the hypotheses of Theorem 1.1, the same lower bound holds for every sequence of density operators supported on the bosonic ground eigenspaces. In particular, it holds for the normalized ground-space projections Π k / Tr Π k \Pi_k/\operatorname{Tr}\Pi_k Π k / Tr Π k and for every sequence of normalized nonnegative ground functions.
Here the one-particle density matrix of a density operator is defined by linearity. The proof below is uniform over its pure ground vectors; convexity then proves the corollary. No choice of a preferred ground vector or a basis of positive ground functions is required.
Context
The ideal-gas prediction grew from Bose’s counting of light quanta and Einstein’s extension to a gas of material particles [8 , 13 , 14 ] . For an interacting gas, the density-matrix criterion of Penrose and Onsager [30 ] identifies condensation through a macroscopic one-particle occupation. Bogoliubov’s theory assumes a macroscopically occupied zero mode to derive the weakly interacting gas’s excitation picture [7 ] . For hard spheres, Lee, Huang and Yang’s pseudopotential calculation predicts a depleted fraction of order ρ a 3 \sqrt{\rho a^3} ρ a 3 [23 ] . The rigorous occupation problem is to justify macroscopic condensation itself before seeking this sharper depletion law.
The hard-sphere energy theory runs from Dyson’s bounds [12 ] and the leading dilute asymptotics of Lieb and Yngvason [26 ] to the sharp Lee–Huang–Yang lower bound of Fournais and Solovej, which includes hard cores [17 ] . Basti, Brooks, Cenatiempo, Olgiati and Schlein proved the matching upper coefficient for hard spheres and subsequently for a class of general nonnegative potentials [3 , 4 ] . These energy asymptotics do not determine single-orbital occupation as the one-particle spectral gap vanishes in large volume. Lieb and Yngvason explicitly separated that question from their energy theorem [26 ] .
Continuum condensation results allowing scaled hard cores include the trapped Gross–Pitaevskii theorem of Lieb and Seiringer [25 ] and the homogeneous canonical positive-temperature theorem of Deuchert and Seiringer [11 ] . Low-energy results of Fournais and of Chong, Liang and Nam [10 , 16 ] , and Junge’s canonical low-temperature result in Neumann boxes [22 ] Corollary 6, treat joint dilute and increasing-volume limits. These regimes permit the gas parameter to decrease as the volume grows. Here a a a ρ \rho ρ
Sütő’s thermodynamic condensation theorem, above a temperature-dependent density threshold, assumes nonnegative potentials with nonnegative Fourier transform and additional regularity and integrability [33 ] Theorem 1.2; these hypotheses exclude a literal hard core. Galanda and Pinamonti construct algebraic equilibrium states for smooth interactions, using coupled cutoff-removal and weak-interaction limits in one setting and Gross–Pitaevskii scaling in another [19 , 20 ] . Their state formulations and limit orders differ from the present hard-sphere problem. Solovej’s 2025 survey identifies fixed-density ground-state condensation as a major open problem [32 ] Section 5. Theorem 1.1 establishes a positive lower limiting occupation fraction for sufficiently dilute three-dimensional hard spheres, with the fixed-parameter order of limits in Equation (1.2) .
The path method has separate antecedents. Benjamini, Pemantle and Peres constructed unpredictable paths with controlled intersections [5 ] Theorem 1.3 and Lemma 3.1. Abbe, Massoulié, Montanari, Sly and Srivastava averaged noisy group differences along paths joining prescribed endpoints [1 ] ; Garban and Spencer adapted this averaging method to symmetry breaking in classical disordered spin systems [21 ] . Averaging path likelihoods before taking their second moments also underlies trail detection [2 , 6 ] and the point-reassignment argument of Peres and Sly [31 ] , Proposition 2.1. Our exterior-shell detours use Kesten’s boundary-connectivity theorem in Timár’s formulation [34 ] , arXiv version 2, Theorem 4.
Our sole companion input is the explicit random lattice skeleton and both encounter moments in A density-uniform condensate bound for dilute Bose gases [29 ] , Section 8.1, pp. 22–24; Lemma 8.1 and Equations (24)–(25), p. 23]. These statements contain no physical parameters. The hard-core energy bounds, obstacle adaptation, conditional-normalizer calculation and phase argument are proved here.
How the proof works
Write Φ = ∣ Ψ ∣ \Phi=|\Psi| Φ = ∣Ψ∣ B ( Φ ) B(\Phi) B ( Φ )
B ( Φ ) = 1 L 3 ∫ Φ ( x , Y ) Φ ( y , Y ) d x d y d Y . B(\Phi)=\frac{1}{L^3}\int\Phi(x,Y)\Phi(y,Y)\,dx\,dy\,dY. B ( Φ ) = L 3 1 ∫ Φ ( x , Y ) Φ ( y , Y ) d x d y d Y . The same bath Y Y Y x x x y y y B ( Ψ ) B(\Psi) B ( Ψ )
Local trajectory changes provide the comparisons between neighboring cells. Multiplying their costs along a long route would lose a factor at every step. Instead we average changes over random routes and bound the second moment of the averaged likelihood. Exact cancellation will leave a cost determined by mutual encounters of two routes, with a bound independent of their lengths.
Local trajectories and the exposed environment. Change length units so that each unit cell contains on average D D D r = α / D r = \alpha/D r = α / D α \alpha α D D D (6) records the invariant gas parameter. Local kinetic inequalities and normalized particle insertion show that cells with too few particles or too many close neighbors have small joint probabilities. An exact stationary killed-Brownian law, whose position marginal is Φ 2 \Phi^2 Φ 2
At each cell we allocate the movable labels to many disjoint groups. For one group at a time, expose the outer endpoints and all trajectories outside the group. A cell is usable when enough of its remaining labels admit successful replacements against these frozen trajectories. The unexposed law is a product of bridge laws that already impose individual oscillation and confinement cutoffs and avoidance of the frozen paths, conditioned further on mutual hard avoidance. This exact conditional law, and an unconditional joint bound on unusable cells, are the two outputs of Section 6. We keep the two probability statements separate throughout the comparison.
Distant cells and a common bath. For each required pair of distant cells, the joint obstacle bounds retain a set of exposed environments of fixed positive probability. On this set, draw a random lattice route from the companion’s skeleton law and reroute it around encountered finite unusable clusters along their good exterior shells. The resulting route uses nearest-neighbor steps, so every step admits the local replacements. Averaging the retained distribution of environments, with two independent route choices drawn in each common environment, gives a bounded exponential moment for their mutual encounters. One imported estimate controls encounters of the nominal skeletons; the other controls the additional encounters caused by detours, after the local obstacle probabilities have been summed.
Along a route, choose one group label from each departure cell. Construct two copies of the selected trajectories, preserving each label’s two outer endpoints. Its midpoint lies in its departure cell in the first copy and in the next cell in the second. At every intermediate cell the two different labels share one midpoint. Removing the initial tag from the first copy and the terminal tag from the second leaves the same bath of N − 1 N - 1 N − 1 Figure 1 depicts these assignments. This produces a common finite measure for two position measures that place a tagged particle in the respective endpoint cells and retain the same observed bath. Their square-root overlap contributes to B ( Φ ) B(\Phi) B ( Φ ) (68) .
Figure 1. The two midpoint assignments. Each departure label moves one cell along the path in the second copy; the drawing suppresses the intermediate cells before U d U_d U d
To lower-bound this overlap, average the change of measure over routes and label choices, and bound its second moment. The interaction normalizers are retained exactly. In the product of two likelihoods, the normalizers from portions of the routes far from one another cancel; only nearby encounters and shared labels contribute a cost. The preceding encounter moment controls that cost uniformly in the endpoint separation. Section 8 constructs the two-copy measure before proving and applying this cancellation. Finally, summing the overlap contributions of a number of disjoint groups proportional to D D D B ( Φ ) B(\Phi) B ( Φ )
Recovering the phases. Equality of the kinetic energies of Ψ \Psi Ψ Φ \Phi Φ δ g e o m \delta_{\mathrm{geom}} δ geom
B ( Ψ ) ≥ B ( Φ ) − 4 δ g e o m B(\Psi) \ge B(\Phi) - 4\sqrt{\delta_{\mathrm{geom}}} B ( Ψ ) ≥ B ( Φ ) − 4 δ geom transfers the modulus bound to the original vector. We first fix the positive modulus fraction and only then strengthen the dilution threshold to make this phase loss smaller.
Section 2 gives the form and phase facts. Sections 3–5 supply the spatial, path and bridge estimates for the local replacements. Section 6 organizes the groups and their conditional law, Section 7 constructs the routes, and Section 8 proves the modulus bound. Section 9 controls the filled regions. Section 10 fixes the constants and returns to arbitrary thermodynamic sequences in the original units.
Ground states and phase transfer
The probabilistic part of the proof uses the squared modulus of a ground state. This section justifies that reduction and identifies the additional estimate needed to recover the occupation of an arbitrary complex state. The relevant phase information is local: moving one particle with the bath fixed preserves its phase whenever the motion stays in the allowed region.
Length units and the occupation fraction
We work on the scaled torus Λ K = ( R / K Z ) 3 \Lambda_K = (\mathbb{R}/K\mathbb{Z})^3 Λ K = ( R / K Z ) 3 K K K V = K 3 V = K^3 V = K 3 r > 0 r > 0 r > 0 d d d n ≥ 0 n \ge0 n ≥ 0
Ω n = { ( x 1 , … , x n ) ∈ Λ K n : d ( x i , x j ) > r for i < j } , \Omega_n = \{(x_1,\ldots,x_n) \in\Lambda_K^n : d(x_i,x_j) > r\ \text{for } i < j\}, Ω n = {( x 1 , … , x n ) ∈ Λ K n : d ( x i , x j ) > r for i < j } , with Ω 0 \Omega_0 Ω 0
q n [ f ] = ∑ i = 1 n ∫ Ω n ∣ ∇ i f ∣ 2 , D ( q n ) = H 0 1 ( Ω n ) . q_n[f] = \sum_{i=1}^{n} \int_{\Omega_n} |\nabla_i f|^2,\qquad D(q_n) = H_0^1(\Omega_n). q n [ f ] = i = 1 ∑ n ∫ Ω n ∣ ∇ i f ∣ 2 , D ( q n ) = H 0 1 ( Ω n ) . Here H 0 1 H_0^1 H 0 1 E n E_n E n Ω n \Omega_n Ω n E n = + ∞ E_n = +\infty E n = + ∞ E 0 = 0 E_0 = 0 E 0 = 0 Ω N \Omega_N Ω N
The mean particle count per unit cell and its product with the exclusion distance are
D = N K 3 , α = D r , α 0 2 ≤ α ≤ 2 α 0 , D ≥ D 0 . D = \frac{N}{K^3},\qquad\alpha= Dr,\qquad\frac{\alpha_0}{2} \le\alpha\le2\alpha_0,\qquad D \ge D_0. D = K 3 N , α = D r , 2 α 0 ≤ α ≤ 2 α 0 , D ≥ D 0 . The numerical constant α 0 \alpha_0 α 0 D 0 D_0 D 0 B v = v + [ − 1 / 2 , 1 / 2 ) 3 B_v = v + [-1/2,1/2)^3 B v = v + [ − 1/2 , 1/2 ) 3 L = ( Z / K Z ) 3 \mathcal{L} = (\mathbb{Z}/K\mathbb{Z})^3 L = ( Z / K Z ) 3 K K K α \alpha α N N N K K K D D D [ D 0 , ∞ ) [D_0,\infty) [ D 0 , ∞ )
To relate these units to the original torus of side L L L a a a s = L / K s=L/K s = L / K r = a / s r=a/s r = a / s
Ψ ~ ( z 1 , … , z N ) = s 3 N / 2 Ψ ( s z 1 , … , s z N ) \widetilde{\Psi}(z_1,\ldots,z_N)=s^{3N/2}\Psi(sz_1,\ldots,sz_N) Ψ ( z 1 , … , z N ) = s 3 N /2 Ψ ( s z 1 , … , s z N ) is unitary and multiplies the quadratic form by s 2 s^2 s 2
D r 3 = N L 3 a 3 = a 3 D 2 . (2) Dr^3=\frac{N}{L^3}a^3=\frac{a^3}{D^2}. \tag*{(2)} D r 3 = L 3 N a 3 = D 2 a 3 . (2) Thus, with α \alpha α D D D K K K
Let Ψ \Psi Ψ Λ K \Lambda_K Λ K Y = ( x 2 , … , x N ) Y=(x_2,\ldots,x_N) Y = ( x 2 , … , x N )
Γ Ψ ( 1 ) ( x ; y ) = N ∫ Ψ ( x , Y ) Ψ ( y , Y ) ‾ d Y , φ 0 ( x ) = V − 1 / 2 . \Gamma_{\Psi}^{(1)}(x;y)=N\int\Psi(x,Y)\overline{\Psi(y,Y)}\,\mathrm{d}Y,\qquad\varphi_0(x)=V^{-1/2}. Γ Ψ ( 1 ) ( x ; y ) = N ∫ Ψ ( x , Y ) Ψ ( y , Y ) d Y , φ 0 ( x ) = V − 1/2 . The operator with kernel Ψ ( x , Y ) \Psi(x,Y) Ψ ( x , Y ) Γ Ψ ( 1 ) \Gamma_{\Psi}^{(1)} Γ Ψ ( 1 ) N N N
B ( Ψ ) = ⟨ φ 0 , Γ Ψ ( 1 ) φ 0 ⟩ N = 1 V ∫ ∣ ∫ Ψ ( x , Y ) d x ∣ 2 d Y . (3) B(\Psi)=\frac{\langle\varphi_0,\Gamma_{\Psi}^{(1)}\varphi_0\rangle}{N}
=\frac{1}{V}\int\left|\int\Psi(x,Y)\,\mathrm{d}x\right|^2\mathrm{d}Y. \tag*{(3)} B ( Ψ ) = N ⟨ φ 0 , Γ Ψ ( 1 ) φ 0 ⟩ = V 1 ∫ ∫ Ψ ( x , Y ) d x 2 d Y . (3) Changing variables in this identity proves its invariance under the unitary rescaling above. For the rest of the proof put
Φ = ∣ Ψ ∣ , μ ( d X ) = Φ ( X ) 2 d X , m ( Y ) = ∫ Φ ( x , Y ) 2 d x . (4) \Phi=|\Psi|,\qquad\mu(\mathrm{d}X)=\Phi(X)^2\,\mathrm{d}X,\qquad m(Y)=\int\Phi(x,Y)^2\,\mathrm{d}x. \tag*{(4)} Φ = ∣Ψ∣ , μ ( d X ) = Φ ( X ) 2 d X , m ( Y ) = ∫ Φ ( x , Y ) 2 d x . (4) Thus m ( Y ) d Y m(Y)\,\mathrm{d}Y m ( Y ) d Y μ \mu μ
The full ground equation and its equality case
The componentwise positivity mechanism is classical; compare Faris and Simon [15 ] 1 [18 ] , Section 2.3. Its direct proof here supplies the required form-domain statement.
Lemma 2.1 (Ground states and component phases). For every integer n ≥ 1 n\ge1 n ≥ 1 Ω n \Omega_n Ω n E n E_n E n Ψ \Psi Ψ Φ = ∣ Ψ ∣ \Phi=|\Psi| Φ = ∣Ψ∣ E N E_N E N Ω N \Omega_N Ω N C C C Ω N \Omega_N Ω N
Φ > 0 , Ψ = e i ϑ C Φ \Phi>0,\qquad\Psi=e^{i\vartheta_C}\Phi Φ > 0 , Ψ = e i ϑ C Φ for one constant ϑ C ∈ R \vartheta_C\in\mathbb{R} ϑ C ∈ R u u u Λ K N \Lambda_K^N Λ K N
q N [ u Φ ] − E N ∥ u Φ ∥ 2 2 = ∫ Φ 2 ∣ ∇ u ∣ 2 . (5) q_N[u\Phi]-E_N\lVert u\Phi\rVert_2^2=\int\Phi^2|\nabla u|^2. \tag*{(5)} q N [ u Φ ] − E N ∥ u Φ ∥ 2 2 = ∫ Φ 2 ∣∇ u ∣ 2 . (5) The gradient on the right includes all particle coordinates.
Ψ ~ ( z 1 , … , z N ) = s 3 N / 2 Ψ ( s z 1 , … , s z N ) \widetilde{\Psi}(z_1,\ldots,z_N)=s^{3N/2}\Psi(sz_1,\ldots,sz_N) Ψ ( z 1 , … , z N ) = s 3 N /2 Ψ ( s z 1 , … , s z N ) is unitary and multiplies the quadratic form by s 2 s^2 s 2
D r 3 = N L 3 a 3 = a 3 D 2 . (6) Dr^3=\frac{N}{L^3}a^3=\frac{a^3}{D^2}. \tag*{(6)} D r 3 = L 3 N a 3 = D 2 a 3 . (6) Thus, with α \alpha α D D D K K K
Let Ψ \Psi Ψ Λ K \Lambda_K Λ K Y = ( x 2 , … , x N ) Y=(x_2,\ldots,x_N) Y = ( x 2 , … , x N )
Γ Ψ ( 1 ) ( x ; y ) = N ∫ Ψ ( x , Y ) Ψ ( y , Y ) ‾ d Y , φ 0 ( x ) = V − 1 / 2 . \Gamma_{\Psi}^{(1)}(x;y)=N\int\Psi(x,Y)\overline{\Psi(y,Y)}\,\mathrm{d}Y,\qquad\varphi_0(x)=V^{-1/2}. Γ Ψ ( 1 ) ( x ; y ) = N ∫ Ψ ( x , Y ) Ψ ( y , Y ) d Y , φ 0 ( x ) = V − 1/2 . The operator with kernel Ψ ( x , Y ) \Psi(x,Y) Ψ ( x , Y ) Γ Ψ ( 1 ) \Gamma_{\Psi}^{(1)} Γ Ψ ( 1 ) N N N
B ( Ψ ) = ⟨ φ 0 , Γ Ψ ( 1 ) φ 0 ⟩ N = 1 V ∫ ∣ ∫ Ψ ( x , Y ) d x ∣ 2 d Y . (7) B(\Psi)=\frac{\langle\varphi_0,\Gamma_{\Psi}^{(1)}\varphi_0\rangle}{N}
=\frac{1}{V}\int\left|\int\Psi(x,Y)\,\mathrm{d}x\right|^2\mathrm{d}Y. \tag*{(7)} B ( Ψ ) = N ⟨ φ 0 , Γ Ψ ( 1 ) φ 0 ⟩ = V 1 ∫ ∫ Ψ ( x , Y ) d x 2 d Y . (7) Changing variables in this identity proves its invariance under the unitary rescaling above. For the rest of the proof put
Φ = ∣ Ψ ∣ , μ ( d X ) = Φ ( X ) 2 d X , m ( Y ) = ∫ Φ ( x , Y ) 2 d x . (8) \Phi=|\Psi|,\qquad\mu(\mathrm{d}X)=\Phi(X)^2\,\mathrm{d}X,\qquad m(Y)=\int\Phi(x,Y)^2\,\mathrm{d}x. \tag*{(8)} Φ = ∣Ψ∣ , μ ( d X ) = Φ ( X ) 2 d X , m ( Y ) = ∫ Φ ( x , Y ) 2 d x . (8) Thus m ( Y ) d Y m(Y)\,\mathrm{d}Y m ( Y ) d Y μ \mu μ
The full ground equation and its equality case
The componentwise positivity mechanism is classical; compare Faris and Simon [15 ] , Lemmas 1 and 3. We give the form argument to include the symmetry restriction and the possibly disconnected torus domain. The multiplier identity below is the classical ground-state representation; see [18 ] , Section 2.3. Its direct proof here supplies the required form-domain statement.
Lemma 2.1 (Ground states and component phases). For every integer n ≥ 1 n\ge1 n ≥ 1 Ω n \Omega_n Ω n E n E_n E n Ψ \Psi Ψ Φ = ∣ Ψ ∣ \Phi=|\Psi| Φ = ∣Ψ∣ E N E_N E N Ω N \Omega_N Ω N C C C Ω N \Omega_N Ω N
Φ > 0 , Ψ = e i ϑ C Φ \Phi>0,\qquad\Psi=e^{i\vartheta_C}\Phi Φ > 0 , Ψ = e i ϑ C Φ for one constant ϑ C ∈ R \vartheta_C\in\mathbb{R} ϑ C ∈ R u u u Λ K N \Lambda_K^N Λ K N
q N [ u Φ ] − E N ∥ u Φ ∥ 2 2 = ∫ Φ 2 ∣ ∇ u ∣ 2 . (9) q_N[u\Phi]-E_N\lVert u\Phi\rVert_2^2=\int\Phi^2|\nabla u|^2. \tag*{(9)} q N [ u Φ ] − E N ∥ u Φ ∥ 2 2 = ∫ Φ 2 ∣∇ u ∣ 2 . (9) The gradient on the right includes all particle coordinates.
Proof. Zero extension maps H 0 1 ( Ω n ) H^1_0(\Omega_n) H 0 1 ( Ω n ) H 1 ( Λ K n ) H^1(\Lambda^n_K) H 1 ( Λ K n ) L 2 L^2 L 2 R R R C R − 2 CR^{-2} C R − 2
For f ∈ H 0 1 ( Ω n ) f \in H^1_0(\Omega_n) f ∈ H 0 1 ( Ω n )
F ( X ) = ( 1 n ! ∑ π ∈ S n ∣ f ( π X ) ∣ 2 ) 1 / 2 . F(X)=\left(\frac{1}{n!}\sum_{\pi\in S_n}\lvert f(\pi X)\rvert^2\right)^{1/2}. F ( X ) = ( n ! 1 π ∈ S n ∑ ∣ f ( π X ) ∣ 2 ) 1/2 . The gradient inequality for the Euclidean norm of a vector gives ∥ F ∥ 2 = ∥ f ∥ 2 \lVert F\rVert_2=\lVert f\rVert_2 ∥ F ∥ 2 = ∥ f ∥ 2 q n [ F ] ≤ q n [ f ] q_n[F]\leq q_n[f] q n [ F ] ≤ q n [ f ] f f f Ω n \Omega_n Ω n F F F L 2 L^2 L 2 H 1 H^1 H 1 H 0 1 H^1_0 H 0 1 f f f F F F
The bosonic minimizer Ψ \Psi Ψ
E N ≤ q N [ Φ ] ≤ q N [ Ψ ] = E N , ∥ Φ ∥ 2 = 1. E_N\leq q_N[\Phi]\leq q_N[\Psi]=E_N,\qquad\lVert\Phi\rVert_2=1. E N ≤ q N [ Φ ] ≤ q N [ Ψ ] = E N , ∥ Φ ∥ 2 = 1. Thus Φ \Phi Φ Φ = ∣ Ψ ∣ \Phi=\lvert\Psi\rvert Φ = ∣ Ψ ∣ Δ Φ = − E N Φ ≤ 0 \Delta\Phi=-E_N\Phi\leq0 ΔΦ = − E N Φ ≤ 0 Φ \Phi Φ
On a positive component set v = Ψ / Φ v=\Psi/\Phi v = Ψ/Φ ∣ v ∣ = 1 \lvert v\rvert=1 ∣ v ∣ = 1 Re ( v ∇ v ) = 0 \operatorname{Re}(v\nabla v)=0 Re ( v ∇ v ) = 0
∣ ∇ Ψ ∣ 2 = ∣ ∇ Φ ∣ 2 + Φ 2 ∣ ∇ v ∣ 2 . \lvert\nabla\Psi\rvert^2=\lvert\nabla\Phi\rvert^2+\Phi^2\lvert\nabla v\rvert^2. ∣ ∇Ψ ∣ 2 = ∣ ∇Φ ∣ 2 + Φ 2 ∣ ∇ v ∣ 2 . There are at most countably many open components. Summing their energy identities and using q N [ Ψ ] = q N [ Φ ] q_N[\Psi]=q_N[\Phi] q N [ Ψ ] = q N [ Φ ] ∇ v = 0 \nabla v=0 ∇ v = 0 v v v Φ = 0 \Phi=0 Φ = 0 Ψ \Psi Ψ
Finally, multiplication by a bounded Lipschitz function preserves the form domain. Test the weak equation for Φ \Phi Φ u 2 Φ u^2\Phi u 2 Φ q N [ u Φ ] q_N[u\Phi] q N [ u Φ ] (9) . □
Lemma 2.2 (Allowed slices). For almost every fixed bath Y Y Y f ∈ H 0 1 ( Ω N ) f\in H^1_0(\Omega_N) f ∈ H 0 1 ( Ω N ) H 0 1 ( S Y ) H^1_0(S_Y) H 0 1 ( S Y )
S Y = { x ∈ Λ K : ( x , Y ) ∈ Ω N } . S_Y=\{x\in\Lambda_K:(x,Y)\in\Omega_N\}. S Y = { x ∈ Λ K : ( x , Y ) ∈ Ω N } . The same assertion holds when any nonempty subset of coordinates is left unfixed. For the smooth ground representatives, every connected subset of S Y S_Y S Y
Proof. Choose f j ∈ C c ∞ ( Ω N ) f_j\in C^\infty_c(\Omega_N) f j ∈ C c ∞ ( Ω N ) f f f H 1 H^1 H 1 j j j f j f_j f j Ω N \Omega_N Ω N H 0 1 H^1_0 H 0 1
The continuous map x ↦ ( x , Y ) x \mapsto(x,Y) x ↦ ( x , Y ) S Y S_Y S Y Ω N \Omega_N Ω N S Y S_Y S Y
The slicing statement uses compactly supported approximation rather than a boundary trace theorem. It therefore applies even when the allowed slice has several components or an irregular boundary. Moreover m ( Y ) m(Y) m ( Y ) m ( Y ) = 0 m(Y)=0 m ( Y ) = 0
The price of removing a part of each slice
The triangle inequality in (7) gives B ( Ψ ) ≤ B ( Φ ) B(\Psi) \le B(\Phi) B ( Ψ ) ≤ B ( Φ )
Lemma 2.3 (Phase transfer). Let { ( x , Y ) : x ∈ U Y } \{(x,Y):x \in U_Y\} {( x , Y ) : x ∈ U Y } m ( Y ) > 0 m(Y)>0 m ( Y ) > 0 U Y U_Y U Y S Y S_Y S Y
A Y = Λ K ∖ U Y , δ g e o m = ∫ m ( Y ) ∣ A Y ∣ V d Y . A_Y=\Lambda_K\setminus U_Y,\qquad\delta_{\mathrm{geom}}=\int m(Y)\frac{|A_Y|}{V}\,dY. A Y = Λ K ∖ U Y , δ geom = ∫ m ( Y ) V ∣ A Y ∣ d Y . Then
B ( Ψ ) ≥ B ( Φ ) − 4 δ g e o m . (10) B(\Psi)\ge B(\Phi)-4\sqrt{\delta_{\mathrm{geom}}}. \tag*{(10)} B ( Ψ ) ≥ B ( Φ ) − 4 δ geom . (10) In particular, if B ( Φ ) ≥ c ∗ > 0 B(\Phi)\ge c_* >0 B ( Φ ) ≥ c ∗ > 0 δ g e o m ≤ c ∗ 2 / 64 \delta_{\mathrm{geom}}\le c_*^2/64 δ geom ≤ c ∗ 2 /64 B ( Ψ ) ≥ c ∗ / 2 B(\Psi)\ge c_*/2 B ( Ψ ) ≥ c ∗ /2
Proof. By Lemma 2.2, the real part of Ψ ( x , Y ) Ψ ( y , Y ) ‾ \Psi(x,Y)\overline{\Psi(y,Y)} Ψ ( x , Y ) Ψ ( y , Y ) Φ ( x , Y ) Φ ( y , Y ) \Phi(x,Y)\Phi(y,Y) Φ ( x , Y ) Φ ( y , Y ) x , y ∈ U Y x,y\in U_Y x , y ∈ U Y 2 Φ ( x , Y ) Φ ( y , Y ) 2\Phi(x,Y)\Phi(y,Y) 2Φ ( x , Y ) Φ ( y , Y )
B ( Φ ) − B ( Ψ ) ≤ 2 V ∭ ( 1 A Y ( x ) + 1 A Y ( y ) ) Φ ( x , Y ) Φ ( y , Y ) d x d y d Y . B(\Phi)-B(\Psi)\le\frac{2}{V}\iiint\left(1_{A_Y}(x)+1_{A_Y}(y)\right)\Phi(x,Y)\Phi(y,Y)\,dx\,dy\,dY. B ( Φ ) − B ( Ψ ) ≤ V 2 ∭ ( 1 A Y ( x ) + 1 A Y ( y ) ) Φ ( x , Y ) Φ ( y , Y ) d x d y d Y . For the term containing 1 A Y ( x ) 1_{A_Y}(x) 1 A Y ( x )
V − 1 / 2 Φ ( x , Y ) , V − 1 / 2 1 A Y ( x ) Φ ( y , Y ) . V^{-1/2}\Phi(x,Y),\qquad V^{-1/2}1_{A_Y}(x)\Phi(y,Y). V − 1/2 Φ ( x , Y ) , V − 1/2 1 A Y ( x ) Φ ( y , Y ) . Their squared norms, with respect to d x d y d Y dx\,dy\,dY d x d y d Y 1 1 1 δ g e o m \delta_{\mathrm{geom}} δ geom δ g e o m \sqrt{\delta_{\mathrm{geom}}} δ geom x x x y y y (10) . The final assertion follows by substitution. This argument chooses no phase as a function of the bath; all quantities integrated are already measurable.
The rest of the proof must now supply two estimates with compatible constants: a lower bound B ( Φ ) ≥ c ∗ > 0 B(\Phi)\ge c_* >0 B ( Φ ) ≥ c ∗ > 0 D D D δ g e o m \delta_{\mathrm{geom}} δ geom
Energy comparisons and joint spatial bounds
We need a supply of particles in most unit cells, and a small probability that many prescribed cells contain too many close neighbors. The estimates must hold jointly for arbitrary sets of cells. They will follow from local energy penalties, tested against products of count cutoffs.
Throughout this section the torus is Λ K = ( R / K Z ) 3 \Lambda_K = (\mathbb{R}/K\mathbb{Z})^3 Λ K = ( R / K Z ) 3 K ≥ 100 K \ge100 K ≥ 100
V = K 3 , D = N / V , r = α / D , α 0 / 2 ≤ α ≤ 2 α 0 . V = K^3,\qquad D = N/V,\qquad r = \alpha/D,\qquad\alpha_0/2 \le\alpha\le2\alpha_0. V = K 3 , D = N / V , r = α / D , α 0 /2 ≤ α ≤ 2 α 0 . The fixed constant α 0 \alpha_0 α 0 D D D L = ( Z / K Z ) 3 \mathcal{L} = (\mathbb{Z}/K\mathbb{Z})^3 L = ( Z / K Z ) 3 B v = v + [ − 1 / 2 , 1 / 2 ) 3 B_v = v + [-1/2,1/2)^3 B v = v + [ − 1/2 , 1/2 ) 3 E n E_n E n n n n r r r E 0 = 0 E_0 = 0 E 0 = 0 Φ \Phi Φ μ ( d X ) = Φ ( X ) 2 d X \mu(\mathrm{d}X) = \Phi(X)^2\mathrm{d}X μ ( d X ) = Φ ( X ) 2 d X
For a subset I I I
q I [ F ] = ∑ i ∈ I ∫ ∣ ∇ i F ∣ 2 , n v I ( X ) = # { i ∈ I : x i ∈ B v } , n v = n v { 1 , … , N } . q_I[F] = \sum_{i\in I}\int|\nabla_i F|^2,\qquad n_v^I(X) = \#\{i\in I:x_i\in B_v\},\qquad n_v=n_v^{\{1,\ldots,N\}}. q I [ F ] = i ∈ I ∑ ∫ ∣ ∇ i F ∣ 2 , n v I ( X ) = # { i ∈ I : x i ∈ B v } , n v = n v { 1 , … , N } . Integrals of form functions use their zero extensions. Constants c , C c,C c , C α \alpha α K , N , D K,N,D K , N , D
Two local kinetic inequalities
We use the hard-core two-particle coercivity underlying local exclusion bounds [28 ] and the Neumann subdivision method [26 ] . The following ray and subdivision argument proves the occupation estimate needed here.
Lemma 3.1 (Cube energy). Let Q Q Q l l l F ∈ H 1 ( Q n ) F \in H^1(Q^n) F ∈ H 1 ( Q n ) ∣ x i − x j ∣ ≤ r |x_i-x_j| \le r ∣ x i − x j ∣ ≤ r i ≠ j i \ne j i = j
∑ i = 1 n ∫ Q n ∣ ∇ i F ∣ 2 ≥ c c u b e r l 3 n ( n − 1 ) ∫ Q n ∣ F ∣ 2 , c c u b e = 1 64 3 . (11) \sum_{i=1}^n \int_{Q^n} |\nabla_i F|^2 \ge c_{\mathrm{cube}}\frac{r}{l^3}n(n-1)\int_{Q^n}|F|^2,\qquad c_{\mathrm{cube}}=\frac{1}{64\sqrt{3}}. \tag*{(11)} i = 1 ∑ n ∫ Q n ∣ ∇ i F ∣ 2 ≥ c cube l 3 r n ( n − 1 ) ∫ Q n ∣ F ∣ 2 , c cube = 64 3 1 . (11) Consequently, for every torus form function F ∈ H 1 ( Ω N ) F \in H^1(\Omega_N) F ∈ H 1 ( Ω N ) I ⊂ { 1 , … , N } I\subset\{1,\ldots,N\} I ⊂ { 1 , … , N }
q I [ F ] ≥ c c u b e r ∫ ∑ v ∈ L n v I ( n v I − 1 ) ∣ F ∣ 2 . (12) q_I[F] \ge c_{\mathrm{cube}}r\int\sum_{v\in\mathcal{L}}n_v^I(n_v^I-1)|F|^2. \tag*{(12)} q I [ F ] ≥ c cube r ∫ v ∈ L ∑ n v I ( n v I − 1 ) ∣ F ∣ 2 . (12) For D ≥ 2 D\ge2 D ≥ 2
E N N ≥ c E α , c E = c c u b e / 2. (13) \frac{E_N}{N}\ge c_E\alpha,\qquad c_E=c_{\mathrm{cube}}/2. \tag*{(13)} N E N ≥ c E α , c E = c cube /2. (13) Proof. Fix a partner y ∈ Q y\in Q y ∈ Q f f f ∣ x − y ∣ ≤ r |x-y|\le r ∣ x − y ∣ ≤ r x = y + s ω x=y+s\omega x = y + s ω R ( ω ) R(\omega) R ( ω ) R ( ω ) ≤ 3 l R(\omega)\le\sqrt{3}l R ( ω ) ≤ 3 l r < s < R ( ω ) r<s<R(\omega) r < s < R ( ω )
∣ f ( y + s ω ) ∣ 2 ≤ ( ∫ r s t − 2 d t ) ( ∫ r s ∣ ∂ t f ( y + t ω ) ∣ 2 t 2 d t ) ≤ 1 r ∫ r R ( ω ) ∣ ∂ t f ∣ 2 t 2 d t . |f(y+s\omega)|^2\le\left(\int_r^s t^{-2}\,\mathrm{d}t\right)\left(\int_r^s|\partial_t f(y+t\omega)|^2t^2\,\mathrm{d}t\right)\le\frac{1}{r}\int_r^{R(\omega)}|\partial_t f|^2t^2\,\mathrm{d}t. ∣ f ( y + s ω ) ∣ 2 ≤ ( ∫ r s t − 2 d t ) ( ∫ r s ∣ ∂ t f ( y + t ω ) ∣ 2 t 2 d t ) ≤ r 1 ∫ r R ( ω ) ∣ ∂ t f ∣ 2 t 2 d t . Integrating in s s s ω \omega ω
∫ Q ∣ f ∣ 2 ≤ 3 l 3 r ∫ Q ∣ ∇ f ∣ 2 . (14) \int_Q |f|^2\le\frac{\sqrt{3}l^3}{r}\int_Q|\nabla f|^2. \tag*{(14)} ∫ Q ∣ f ∣ 2 ≤ r 3 l 3 ∫ Q ∣∇ f ∣ 2 . (14) Rays ending before r r r r r r r r r
For n ≥ 2 n \ge2 n ≥ 2 Q Q Q j 3 j^3 j 3 j = ⌊ ( n / 2 ) 1 / 3 ⌋ j=\lfloor(n/2)^{1/3}\rfloor j = ⌊( n /2 ) 1/3 ⌋ n / 16 ≤ j 3 ≤ n / 2 n/16 \le j^3 \le n/2 n /16 ≤ j 3 ≤ n /2 n n n j 3 j^3 j 3 n / 2 n/2 n /2 (14) to each such coordinate gives at least
r j 3 3 l 3 n 2 ≥ r n 2 32 3 l 3 \frac{rj^3}{\sqrt{3}l^3}\frac{n}{2} \ge\frac{rn^2}{32\sqrt{3}l^3} 3 l 3 r j 3 2 n ≥ 32 3 l 3 r n 2 times the sector norm. We only restrict the coordinate integrals to sectors; no derivative of a sector indicator is taken. Sum these inequalities over sectors. The cases n = 0 , 1 n=0,1 n = 0 , 1 (11) , with the stated smaller constant.
Apply the cube bound to all unit-cell assignment sectors of the labels in I I I (12) . Finally, ∑ ν n ν ( n ν − 1 ) ≥ N 2 / V − N \sum_\nu n_\nu(n_\nu-1) \ge N^2/V-N ∑ ν n ν ( n ν − 1 ) ≥ N 2 / V − N E N / N ≥ c c u b e r ( D − 1 ) ≥ c E α E_N/N \ge c_{\mathrm{cube}}r(D-1) \ge c_E\alpha E N / N ≥ c cube r ( D − 1 ) ≥ c E α
Lemma 3.2 (Close-neighbor energy). Let 0 < r < s < 1 0<r<s<1 0 < r < s < 1 F ∈ H 0 1 ( Ω N ) F\in H^1_0(\Omega_N) F ∈ H 0 1 ( Ω N ) I ⊂ { 1 , … , N } I\subset\{1,\ldots,N\} I ⊂ { 1 , … , N } d i I = min j ∈ I ∖ { i } d ( x i , x j ) d_i^I=\min_{j\in I\setminus\{i\}}d(x_i,x_j) d i I = min j ∈ I ∖ { i } d ( x i , x j ) + ∞ +\infty + ∞
q I [ F ] ≥ c n e a r r s − 3 ∫ # { i ∈ I : d i I ≤ 2 s } ∣ F ∣ 2 , c n e a r = 27 262144 3 . (15) q_I[F]\ge c_{\mathrm{near}}rs^{-3}\int\#\{i\in I:d_i^I\le2s\}|F|^2,\qquad c_{\mathrm{near}}=\frac{27}{262144\sqrt{3}}. \tag*{(15)} q I [ F ] ≥ c near r s − 3 ∫ # { i ∈ I : d i I ≤ 2 s } ∣ F ∣ 2 , c near = 262144 3 27 . (15) Proof. Partition each coordinate circle into ⌊ K / ( 8 s ) ⌋ \lfloor K/(8s)\rfloor ⌊ K / ( 8 s )⌋ l l l [ 8 s , 16 s ] [8s,16s] [ 8 s , 16 s ] 2 s 2s 2 s ( 1 − 2 s / l ) 3 ≥ 27 / 64 (1-2s/l)^3\ge27/64 ( 1 − 2 s / l ) 3 ≥ 27/64 (14) gives kinetic energy at least r / ( 3 l 3 ) r/(\sqrt{3}l^3) r / ( 3 l 3 ) i i i d i I ≤ 2 s d_i^I\le2s d i I ≤ 2 s 27 / 64 27/64 27/64 l 3 ≤ 4096 s 3 l^3\le4096s^3 l 3 ≤ 4096 s 3
Normalized insertion and energy differences
We next compare energies at different particle numbers. Normalizing the inserted orbital separately for each old configuration prevents the old kinetic energy from acquiring a multiplicative error. The trial follows the nearest-neighbor construction of Lieb, Seiringer, and Yngvason [27 ] ; the configurationwise normalization below preserves the old marginal for arbitrary old form functions.
Lemma 3.3 (Normalized insertion). Let w w w F F F n n n X X X F F F k k k 2 r 2r 2 r supp w \operatorname{supp}w supp w
C ∥ w ∥ ∞ 2 k r 3 ≤ 1 4 , (16) C\lVert w\rVert_\infty^2kr^3\le\frac{1}{4}, \tag*{(16)} C ∥ w ∥ ∞ 2 k r 3 ≤ 4 1 , (16) there is an admissible function F ~ \widetilde F F F F F
q n + 1 [ F ~ ] ≤ q n [ F ] + C ( ∥ ∇ w ∥ 2 2 + ∥ w ∥ ∞ 2 k r ) ∥ F ∥ 2 2 . (17) q_{n+1}[\widetilde F]\le q_n[F]+C\left(\lVert\nabla w\rVert_2^2+\lVert w\rVert_\infty^2kr\right)\lVert F\rVert_2^2. \tag*{(17)} q n + 1 [ F ] ≤ q n [ F ] + C ( ∥ ∇ w ∥ 2 2 + ∥ w ∥ ∞ 2 k r ) ∥ F ∥ 2 2 . (17) The constants depend only on a fixed radial cutoff profile.
Proof. Choose a nondecreasing Lipschitz function h r h_r h r 1.1 r 1.1r 1.1 r 2 r 2r 2 r C / r C/r C / r X X X
f X ( y ) = w ( y ) min 1 ≤ i ≤ n h r ( d ( y , x i ) ) , Z ( X ) = ∫ f X ( y ) 2 d y . f_X(y)=w(y)\min_{1\le i\le n}h_r(d(y,x_i)),\qquad Z(X)=\int f_X(y)^2\,dy. f X ( y ) = w ( y ) 1 ≤ i ≤ n min h r ( d ( y , x i )) , Z ( X ) = ∫ f X ( y ) 2 d y . The empty minimum is one. The union of the relevant radius-2 r 2r 2 r C k r 3 Ckr^3 C k r 3 Z ≥ 3 / 4 Z\ge3/4 Z ≥ 3/4 (16) . Where a nearest minimizer is unique, only that old coordinate and y y y
∫ ( ∣ ∇ y f X ∣ 2 + ∑ i = 1 n ∣ ∇ i f X ∣ 2 ) d y ≤ C ( ∥ ∇ w ∥ 2 2 + ∥ w ∥ ∞ 2 k r ) . (18) \int\left(|\nabla_y f_X|^2+\sum_{i=1}^n|\nabla_i f_X|^2\right)\,dy\le C\left(\|\nabla w\|_2^2+\|w\|_\infty^2kr\right). \tag*{(18)} ∫ ( ∣ ∇ y f X ∣ 2 + i = 1 ∑ n ∣ ∇ i f X ∣ 2 ) d y ≤ C ( ∥∇ w ∥ 2 2 + ∥ w ∥ ∞ 2 k r ) . (18) Indeed cutoff derivatives have squared size at most C / r 2 C/r^2 C / r 2 a X = f X / Z a_X=f_X/\sqrt{Z} a X = f X / Z Z > 1 / 4 Z>1/4 Z > 1/4
∂ a X = Z − 1 / 2 ( ∂ f X − a X ⟨ a X , ∂ f X ⟩ L 2 ( d y ) ) . \partial a_X=Z^{-1/2}\left(\partial f_X-a_X\langle a_X,\partial f_X\rangle_{L^2(dy)}\right). ∂ a X = Z − 1/2 ( ∂ f X − a X ⟨ a X , ∂ f X ⟩ L 2 ( d y ) ) . It follows that its squared L 2 ( d y ) L^2(dy) L 2 ( d y ) Z − 1 ∥ ∂ f X ∥ 2 2 Z^{-1}\|\partial f_X\|_2^2 Z − 1 ∥ ∂ f X ∥ 2 2 Z Z Z y y y
∫ a X 2 d y = 1 , ∫ a X ∇ i a X d y = 0. \int a_X^2\,dy=1,\qquad\int a_X\nabla_i a_X\,dy=0. ∫ a X 2 d y = 1 , ∫ a X ∇ i a X d y = 0. Thus the cross term vanishes on differentiating F ( X ) a X ( y ) F(X)a_X(y) F ( X ) a X ( y ) y y y (18) proves (17) .
For a globally defined multiplier replace Z \sqrt{Z} Z max ( Z , 1 / 2 ) \max(\sqrt{Z},1/2) max ( Z , 1/2 ) y y y 1.1 r 1.1r 1.1 r F ~ = F a X \widetilde F=F a_X F = F a X
Corollary 3.4 (Energy differences). For all sufficiently large D D D α \alpha α
E N − E N − m ≤ C α m ( 0 ≤ m ≤ N ) . E_N-E_{N-m}\le C\alpha m\qquad(0\le m\le N). E N − E N − m ≤ C α m ( 0 ≤ m ≤ N ) . Moreover E n / n E_n/n E n / n K , r K,r K , r
Proof. Use w = V − 1 / 2 w=V^{-1/2} w = V − 1/2 N N N C D r 3 = C α 3 / D 2 CDr^3=C\alpha^3/D^2 C D r 3 = C α 3 / D 2 C D r = C α CDr=C\alpha C D r = C α N N N (3.9) .
For the last assertion, fix k < n k<n k < n k k k n n n k / n k/n k / n E k E_k E k ( k / n ) q n [ F ] ≥ E k (k/n)q_n[F]\ge E_k ( k / n ) q n [ F ] ≥ E k E n / n ≥ E k / k E_n/n\ge E_k/k E n / n ≥ E k / k
A joint bound for deficient cells
We apply the ground-state multiplier identity, (9) , to products of local count cutoffs.
Proposition 3.5 (Joint low-count bound). One can fix α 0 \alpha_0 α 0 ℓ > 0 \ell> 0 ℓ > 0 A ⊂ L A \subset\mathcal{L} A ⊂ L D D D
μ { n v < ℓ D for every v ∈ A } ≤ ( C / D 2 ) ∣ A ∣ . \mu\{n_v < \ell D \text{ for every } v \in A\} \le(C/D^2)^{|A|}. μ { n v < ℓ D for every v ∈ A } ≤ ( C / D 2 ) ∣ A ∣ . These choices and the constant C C C K K K
Proof. Fix translates w v w_v w v B v B_v B v χ v ∈ [ 0 , 1 ] \chi_v \in[0,1] χ v ∈ [ 0 , 1 ] B v B_v B v supp w v \operatorname{supp} w_v supp w v ∣ ∇ χ v ∣ 2 ≤ C χ v |\nabla\chi_v|^2 \le C\chi_v ∣∇ χ v ∣ 2 ≤ C χ v m v = ∑ i χ v ( x i ) m_v = \sum_i \chi_v(x_i) m v = ∑ i χ v ( x i ) r r r r r r supp w v \operatorname{supp} w_v supp w v χ v = 1 \chi_v = 1 χ v = 1
Suppose F F F m v ≤ 2 ℓ D m_v \le2\ell D m v ≤ 2 ℓ D v ∈ A v \in A v ∈ A F = 0 F=0 F = 0 ∥ F ∥ 2 > 0 \lVert F\rVert_2 > 0 ∥ F ∥ 2 > 0 ⌊ ℓ D ⌋ \lfloor\ell D\rfloor ⌊ ℓ D ⌋ w v w_v w v 3 ℓ D 3\ell D 3 ℓ D D D D C ( 1 + 3 ℓ α ) C(1+3\ell\alpha) C ( 1 + 3 ℓ α )
First choose α 0 \alpha_0 α 0 ℓ \ell ℓ c E α / 2 c_E\alpha/2 c E α /2 α \alpha α k = ⌊ ℓ D ⌋ ∣ A ∣ k=\lfloor\ell D\rfloor|A| k = ⌊ ℓ D ⌋ ∣ A ∣ ∥ F ∥ 2 \lVert F\rVert_2 ∥ F ∥ 2 q N [ F ] + ( c E α / 2 ) k ∥ F ∥ 2 2 q_N[F]+(c_E\alpha/2)k\lVert F\rVert_2^2 q N [ F ] + ( c E α /2 ) k ∥ F ∥ 2 2 E N + k < ∞ E_{N+k}<\infty E N + k < ∞ (13) give E N + k ≥ E N + k E N / N ≥ E N + c E α k E_{N+k}\ge E_N+kE_N/N\ge E_N+c_E\alpha k E N + k ≥ E N + k E N / N ≥ E N + c E α k
q N [ F ] − E N ∥ F ∥ 2 2 ≥ c α D ∣ A ∣ ∥ F ∥ 2 2 , (19) q_N[F]-E_N\lVert F\rVert_2^2 \ge c\alpha D|A|\lVert F\rVert_2^2, \tag*{(19)} q N [ F ] − E N ∥ F ∥ 2 2 ≥ c α D ∣ A ∣ ∥ F ∥ 2 2 , (19) where c > 0 c>0 c > 0 ℓ \ell ℓ D D D ⌊ ℓ D ⌋ ≥ ℓ D / 2 \lfloor\ell D\rfloor\ge\ell D/2 ⌊ ℓ D ⌋ ≥ ℓ D /2
Let ζ : [ 0 , ∞ ) → [ 0 , 1 ] \zeta:[0,\infty)\to[0,1] ζ : [ 0 , ∞ ) → [ 0 , 1 ] u A = ∏ v ∈ A ζ ( m v / ( ℓ D ) ) u_A=\prod_{v\in A}\zeta(m_v/(\ell D)) u A = ∏ v ∈ A ζ ( m v / ( ℓ D )) χ v \chi_v χ v
∣ ∇ m v ∣ 2 = ∑ i ∣ ∇ χ v ( x i ) ∣ 2 ≤ C m v , ∣ ∇ u A ∣ 2 ≤ C D ∑ v ∈ A u A ∖ { v } 2 . |\nabla m_v|^2=\sum_i|\nabla\chi_v(x_i)|^2\le Cm_v,\qquad|\nabla u_A|^2\le\frac{C}{D}\sum_{v\in A}u_{A\setminus\{v\}}^2. ∣∇ m v ∣ 2 = i ∑ ∣∇ χ v ( x i ) ∣ 2 ≤ C m v , ∣∇ u A ∣ 2 ≤ D C v ∈ A ∑ u A ∖ { v } 2 . The second estimate uses m v ≤ 2 ℓ D m_v\le2\ell D m v ≤ 2 ℓ D (19) and (9) to u A Φ u_A\Phi u A Φ a ( A ) = ∫ u A 2 d μ a(A)=\int u_A^2\,d\mu a ( A ) = ∫ u A 2 d μ
a ( A ) ≤ C D 2 ∣ A ∣ ∑ v ∈ A a ( A ∖ { v } ) ( A ≠ ∅ ) . a(A)\le\frac{C}{D^2|A|}\sum_{v\in A}a(A\setminus\{v\})\qquad(A\ne\varnothing). a ( A ) ≤ D 2 ∣ A ∣ C v ∈ A ∑ a ( A ∖ { v }) ( A = ∅ ) . Starting from a ( ∅ ) = 1 a(\varnothing)=1 a ( ∅ ) = 1 ∣ A ∣ |A| ∣ A ∣ a ( A ) ≤ ( C / D 2 ) ∣ A ∣ a(A)\le(C/D^2)^{|A|} a ( A ) ≤ ( C / D 2 ) ∣ A ∣ m v ≤ n v m_v\le n_v m v ≤ n v (3.10) has u A = 1 u_A=1 u A = 1
The same constants work for arbitrarily many prescribed cells. We now obtain an upper count bound and a close-neighbor bound with that same joint scope. Their common input is a localized energy inequality.
Exponential spatial localization
For nonempty A ⊂ L A \subset\mathcal{L} A ⊂ L
f A ( X ) = ∑ i = 1 N e − d ( x i , A ) , (20) f_A(X)=\sum_{i=1}^{N}e^{-d(x_i,A)}, \tag*{(20)} f A ( X ) = i = 1 ∑ N e − d ( x i , A ) , (20) k A ( X ) = # { i : d ( x i , A ) ≤ 6 , d i ≤ 2 s 0 } , d i = min j ≠ i d ( x i , x j ) , (21) k_A(X)=\#\{i:d(x_i,A)\le6,\ d_i\le2s_0\},\qquad d_i=\min_{j\ne i}d(x_i,x_j), \tag*{(21)} k A ( X ) = # { i : d ( x i , A ) ≤ 6 , d i ≤ 2 s 0 } , d i = j = i min d ( x i , x j ) , (21) s 0 = b D − 1 / 3 , 0 < b < 1. (22) s_0=bD^{-1/3},\qquad0<b<1. \tag*{(22)} s 0 = b D − 1/3 , 0 < b < 1. (22) The parameter b b b D D D r < s 0 < 1 / 2 r<s_0<1/2 r < s 0 < 1/2
The particle-number localization uses the many-body IMS construction; compare [24 ] , Section 3. We prove the product partition and its weighted penalty explicitly. We separate particles near A A A
Lemma 3.6 (Weighted local penalty). There are constants C , c > 0 C,c>0 C , c > 0 b b b N N N F F F
q N [ F ] − E N ∥ F ∥ 2 2 ≥ α ∫ ( f A − C D ∣ A ∣ + c b − 3 k A ) ∣ F ∣ 2 . (23) q_N[F]-E_N\lVert F\rVert_2^2\ge\alpha\int\left(f_A-CD|A|+cb^{-3}k_A\right)|F|^2. \tag*{(23)} q N [ F ] − E N ∥ F ∥ 2 2 ≥ α ∫ ( f A − C D ∣ A ∣ + c b − 3 k A ) ∣ F ∣ 2 . (23) Proof. Partitioning the particles. Let H ≥ 0 H\ge0 H ≥ 0 P ( H ≥ h ) = e − h \mathbb{P}(H\ge h)=e^{-h} P ( H ≥ h ) = e − h ϑ \vartheta ϑ π / 2 \pi/2 π /2 ( − ∞ , 0 ] (-\infty,0] ( − ∞ , 0 ] [ 1 , ∞ ) [1,\infty) [ 1 , ∞ )
a H ( x ) = sin ϑ ( d ( x , A ) − H ) , c H ( x ) = cos ϑ ( d ( x , A ) − H ) . a_H(x)=\sin\vartheta(d(x,A)-H),\qquad c_H(x)=\cos\vartheta(d(x,A)-H). a H ( x ) = sin ϑ ( d ( x , A ) − H ) , c H ( x ) = cos ϑ ( d ( x , A ) − H ) . Thus a H 2 + c H 2 = 1 a_H^2+c_H^2=1 a H 2 + c H 2 = 1 a H = 1 a_H=1 a H = 1 H H H H + 1 H+1 H + 1 C 1 { H < d ( x , A ) < H + 1 } C1_{\{H<d(x,A)<H+1\}} C 1 { H < d ( x , A ) < H + 1 } I ⊂ { 1 , … , N } I\subset\{1,\ldots,N\} I ⊂ { 1 , … , N }
F I , H ( X ) = F ( X ) ∏ i ∈ I a H ( x i ) ∏ j ∉ I c H ( x j ) . F_{I,H}(X)=F(X)\prod_{i\in I}a_H(x_i)\prod_{j\notin I}c_H(x_j). F I , H ( X ) = F ( X ) i ∈ I ∏ a H ( x i ) j ∈ / I ∏ c H ( x j ) . Expanding the gradients, using a H ∇ a H + c H ∇ c H = 0 a_H\nabla a_H+c_H\nabla c_H=0 a H ∇ a H + c H ∇ c H = 0
∑ I q N [ F I , H ] = q N [ F ] + ∫ ∑ i ( ∣ ∇ a H ( x i ) ∣ 2 + ∣ ∇ c H ( x i ) ∣ 2 ) ∣ F ∣ 2 . (24) \sum_I q_N[F_{I,H}]=q_N[F]+\int\sum_i\left(|\nabla a_H(x_i)|^2+|\nabla c_H(x_i)|^2\right)|F|^2. \tag*{(24)} I ∑ q N [ F I , H ] = q N [ F ] + ∫ i ∑ ( ∣∇ a H ( x i ) ∣ 2 + ∣∇ c H ( x i ) ∣ 2 ) ∣ F ∣ 2 . (24) The squared norms sum to ∥ F ∥ 2 2 \lVert F\rVert_2^2 ∥ F ∥ 2 2
The energy retained outside the localized set. The index set I I I A A A I I I I I I
q I c [ F I , H ] ≥ E N − ∣ I ∣ ∥ F I , H ∥ 2 2 ≥ ( E N − C α ∣ I ∣ ) ∥ F I , H ∥ 2 2 . q_{\mathcal{I}^c}[F_{I,H}]\ge E_{N-|I|}\lVert F_{I,H}\rVert_2^2\ge(E_N-C\alpha|I|)\lVert F_{I,H}\rVert_2^2. q I c [ F I , H ] ≥ E N − ∣ I ∣ ∥ F I , H ∥ 2 2 ≥ ( E N − C α ∣ I ∣ ) ∥ F I , H ∥ 2 2 . To sum these removal costs, regard the squared product factors at fixed X , H X,H X , H I I I a H ( x i ) 2 a_H(x_i)^2 a H ( x i ) 2
∑ I ∣ I ∣ ∏ i ∈ I a H ( x i ) 2 ∏ j ∉ I c H ( x j ) 2 = ∑ i a H ( x i ) 2 . \sum_I |I|\prod_{i\in I}a_H(x_i)^2\prod_{j\notin I}c_H(x_j)^2=\sum_i a_H(x_i)^2. I ∑ ∣ I ∣ i ∈ I ∏ a H ( x i ) 2 j ∈ / I ∏ c H ( x j ) 2 = i ∑ a H ( x i ) 2 . Since
E H a H ( x ) 2 ≤ e e − d ( x , A ) , E H 1 { H < d ( x , A ) < H + 1 } ≤ e e − d ( x , A ) , \mathbb{E}H a_H(x)^2 \le e e^{-d(x,A)}, \qquad\mathbb{E}H\mathbf{1}_{\{H<d(x,A)<H+1\}} \le e e^{-d(x,A)}, E H a H ( x ) 2 ≤ e e − d ( x , A ) , E H 1 { H < d ( x , A ) < H + 1 } ≤ e e − d ( x , A ) , the mean removal cost and the mean localization error together are at most C α ∫ f A ∣ F ∣ 2 C\alpha\int f_A|F|^2 C α ∫ f A ∣ F ∣ 2 α \alpha α α \alpha α
The local energy that pays for removal. The kinetic energy in the coordinates of I I I (12) and half of (15) to q I [ F I , H ] q_I[F_{I,H}] q I [ F I , H ] p v = e − d ( v , A ) p_v=e^{-d(v,A)} p v = e − d ( v , A ) H ≥ d ( v , A ) + 1 H\ge d(v,A)+1 H ≥ d ( v , A ) + 1 B v B_v B v a H = 1 a_H=1 a H = 1 I I I H H H
c r ∫ ∑ v p v n v ( n v − 1 ) ∣ F ∣ 2 . cr\int\sum_v p_v n_v(n_v-1)|F|^2. cr ∫ v ∑ p v n v ( n v − 1 ) ∣ F ∣ 2 . For a coordinate counted by k A k_A k A 2 s 0 < 1 2s_0<1 2 s 0 < 1 A A A H ≥ 8 H\ge8 H ≥ 8 I I I e − 8 e^{-8} e − 8
c r s 0 − 3 ∫ k A ∣ F ∣ 2 = c α b − 3 ∫ k A ∣ F ∣ 2 . crs_0^{-3}\int k_A|F|^2=c\alpha b^{-3}\int k_A|F|^2. cr s 0 − 3 ∫ k A ∣ F ∣ 2 = c α b − 3 ∫ k A ∣ F ∣ 2 . In particular the partner used here belongs to the removed set before the subset-label inequality is applied.
Combining these estimates with (24) gives
q N [ F ] − E N ∥ F ∥ 2 2 ≥ ∫ ( c r ∑ v p v n v ( n v − 1 ) − C α f A + c α b − 3 k A ) ∣ F ∣ 2 . (25) q_N[F]-E_N\|F\|_2^2\ge\int\left(cr\sum_v p_vn_v(n_v-1)-C\alpha f_A+c\alpha b^{-3}k_A\right)|F|^2. \tag*{(25)} q N [ F ] − E N ∥ F ∥ 2 2 ≥ ∫ ( cr v ∑ p v n v ( n v − 1 ) − C α f A + c α b − 3 k A ) ∣ F ∣ 2 . (25) Leaving a positive count penalty. Within each unit cell its distance to A A A
f A ≤ e ∑ v p v n v , ∑ v p v ≤ ∑ w ∈ A ∑ v e − d ( v , w ) ≤ C ∣ A ∣ . f_A\le e\sum_v p_vn_v,\qquad\sum_v p_v\le\sum_{w\in A}\sum_v e^{-d(v,w)}\le C|A|. f A ≤ e v ∑ p v n v , v ∑ p v ≤ w ∈ A ∑ v ∑ e − d ( v , w ) ≤ C ∣ A ∣. The last lattice exponential sum is uniformly bounded on every periodic unit grid. Since r = α / D r=\alpha/D r = α / D n ≥ 0 n\ge0 n ≥ 0 D ≥ 1 D\ge1 D ≥ 1
c D n ( n − 1 ) − ( C + 1 ) e n ≥ − C ′ D . \frac{c}{D}n(n-1)-(C+1)en\ge-C'D. D c n ( n − 1 ) − ( C + 1 ) e n ≥ − C ′ D . Multiply by α p v \alpha p_v α p v (25) this leaves the positive term α f A \alpha f_A α f A (23) . None of the constants in this calculation depends on b b b
Joint count and close-neighbor tails
Proposition 3.7 (Joint spatial bounds). For every A ⊂ L A\subset\mathcal{L} A ⊂ L
log ∫ e z f A d μ ≤ C z D ∣ A ∣ ( 0 ≤ z ≤ 1 ) , (26) \log\int e^{zf_A}\,d\mu\le CzD|A|\qquad(0\le z\le1), \tag*{(26)} log ∫ e z f A d μ ≤ C z D ∣ A ∣ ( 0 ≤ z ≤ 1 ) , (26) where f ∅ = 0 f_{\varnothing}=0 f ∅ = 0 θ > 0 \theta>0 θ > 0 b > 0 b>0 b > 0 θ \theta θ D D D
Q ( X ) = { v ∈ L : # { i : d ( x i , v ) ≤ 5 , d i ≤ s 0 } ≥ θ D } (27) Q(X)=\{v\in\mathcal{L}:\#\{i:d(x_i,v)\le5,\ d_i\le s_0\}\ge\theta D\} \tag*{(27)} Q ( X ) = { v ∈ L : # { i : d ( x i , v ) ≤ 5 , d i ≤ s 0 } ≥ θ D } (27) satisfies
μ { A ⊂ Q } ≤ p D ∣ A ∣ , p D = ( C θ b − 2 D − 4 / 3 ) 1 / Q ⟶ 0. (28) \mu\{A\subset Q\}\le p_D^{|A|},\qquad p_D=\left(C\theta b^{-2}D^{-4/3}\right)^{1/Q}\longrightarrow0. \tag*{(28)} μ { A ⊂ Q } ≤ p D ∣ A ∣ , p D = ( C θ b − 2 D − 4/3 ) 1/ Q ⟶ 0. (28) Here Q Q Q D D D α ∈ [ α 0 / 2 , 2 α 0 ] \alpha\in[\alpha_0/2,2\alpha_0] α ∈ [ α 0 /2 , 2 α 0 ] K ≥ 100 K\ge100 K ≥ 100
Proof. Distance is 1-Lipschitz and e − d ≤ 1 e^{-d} \le1 e − d ≤ 1 ∣ ∇ f A ∣ 2 ≤ f A |\nabla f_A|^2 \le f_A ∣∇ f A ∣ 2 ≤ f A M ( z ) = ∫ e z f A d μ M(z) = \int e^{z f_A}\,d\mu M ( z ) = ∫ e z f A d μ (9) and (23) with u = e z f A / 2 u=e^{z f_A/2} u = e z f A /2
α ( M ′ ( z ) − C D ∣ A ∣ M ( z ) ) ≤ z 2 4 M ′ ( z ) . \alpha\bigl(M'(z)-CD|A|M(z)\bigr) \le\frac{z^2}{4}M'(z). α ( M ′ ( z ) − C D ∣ A ∣ M ( z ) ) ≤ 4 z 2 M ′ ( z ) . Our choice of α 0 \alpha_0 α 0 α ≥ 1 \alpha\ge1 α ≥ 1 0 ≤ z ≤ 1 0 \le z \le1 0 ≤ z ≤ 1 M ′ ( z ) / M ( z ) ≤ C D ∣ A ∣ M'(z)/M(z) \le CD|A| M ′ ( z ) / M ( z ) ≤ C D ∣ A ∣ M ( 0 ) = 1 M(0)=1 M ( 0 ) = 1 (26) . These differentiations are ordinary finite-volume ones, since 0 ≤ f A ≤ N 0 \le f_A \le N 0 ≤ f A ≤ N
For the second statement first assume that distinct sites of A A A a v ∈ [ 0 , 1 ] a_v \in[0,1] a v ∈ [ 0 , 1 ] ∣ ∇ a v ∣ 2 ≤ C a v |\nabla a_v|^2 \le Ca_v ∣∇ a v ∣ 2 ≤ C a v χ ∈ [ 0 , 1 ] \chi\in[0,1] χ ∈ [ 0 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] [ 2 , ∞ ) [2,\infty) [ 2 , ∞ ) ∣ χ ′ ∣ 2 ≤ C χ |\chi'|^2 \le C\chi ∣ χ ′ ∣ 2 ≤ C χ
m ~ v ( X ) = ∑ i a v ( x i ) χ ( d i / s 0 ) . \widetilde m_v(X)=\sum_i a_v(x_i)\chi(d_i/s_0). m v ( X ) = i ∑ a v ( x i ) χ ( d i / s 0 ) . Then ∑ v ∈ A m ~ v ≤ k A \sum_{v\in A}\widetilde m_v \le k_A ∑ v ∈ A m v ≤ k A (27) is at most m ~ v \widetilde m_v m v
∣ ∇ m ~ v ∣ 2 ≤ C s 0 − 2 m ~ v . (29) |\nabla\widetilde m_v|^2 \le Cs_0^{-2}\widetilde m_v. \tag*{(29)} ∣∇ m v ∣ 2 ≤ C s 0 − 2 m v . (29) Away from a null set, every nearest neighbor is unique. If x i , x j x_i,x_j x i , x j x k x_k x k ∣ x i − x j ∣ ≥ ∣ x i − x k ∣ |x_i-x_j| \ge|x_i-x_k| ∣ x i − x j ∣ ≥ ∣ x i − x k ∣ x k x_k x k π / 3 \pi/3 π /3 2 s 0 2s_0 2 s 0 s 0 < 1 / 2 s_0 < 1/2 s 0 < 1/2 K ≥ 100 K \ge100 K ≥ 100 m ~ v \widetilde m_v m v a v , χ a_v,\chi a v , χ C s 0 − 2 a v ( x i ) χ ( d i / s 0 ) Cs_0^{-2}a_v(x_i)\chi(d_i/s_0) C s 0 − 2 a v ( x i ) χ ( d i / s 0 ) (29) . All coordinates in those derivatives are within 6 + 2 s 0 < 7 6+2s_0<7 6 + 2 s 0 < 7 v v v
Choose a Lipschitz increasing function η \eta η [ 0 , 1 / 2 ] [0,1/2] [ 0 , 1/2 ] [ 1 , ∞ ) [1,\infty) [ 1 , ∞ )
u A = ∏ v ∈ A η ( m ~ v / ( θ D ) ) , a ( A ) = ∫ u A 2 d μ . u_A=\prod_{v\in A}\eta\bigl(\widetilde m_v/(\theta D)\bigr),\qquad a(A)=\int u_A^2\,d\mu. u A = v ∈ A ∏ η ( m v / ( θ D ) ) , a ( A ) = ∫ u A 2 d μ . On the nonzero set of u A u_A u A k A ≥ ( θ D / 2 ) ∣ A ∣ k_A \ge(\theta D/2)|A| k A ≥ ( θ D /2 ) ∣ A ∣ b b b (23) satisfy c b − 3 θ / 2 − C ≥ 1 cb^{-3}\theta/2-C \ge1 c b − 3 θ /2 − C ≥ 1
q N [ u A Φ ] − E N ∥ u A Φ ∥ 2 ≥ α D ∣ A ∣ a ( A ) . q_N[u_A\Phi]-E_N\|u_A\Phi\|^2 \ge\alpha D|A|a(A). q N [ u A Φ ] − E N ∥ u A Φ ∥ 2 ≥ α D ∣ A ∣ a ( A ) . At almost every configuration, gradients belonging to distinct sites involve disjoint particle coordinates, by the separation of the sites. Using (29) on the transition region m ~ v ≤ θ D \widetilde m_v \le\theta D m v ≤ θ D
∣ ∇ u A ∣ 2 ≤ C θ D s 0 2 ∑ v ∈ A u A ∖ { v } 2 . |\nabla u_A|^2 \le\frac{C}{\theta D s_0^2}\sum_{v\in A}u_{A\setminus\{v\}}^2. ∣∇ u A ∣ 2 ≤ θ D s 0 2 C v ∈ A ∑ u A ∖ { v } 2 . Equation (9) and induction from a ( ∅ ) = 1 a(\varnothing)=1 a ( ∅ ) = 1
a ( A ) ≤ ( C α θ D 2 s 0 2 ) ∣ A ∣ ≤ ( C 0 b − 2 D − 4 / 3 ) ∣ A ∣ . a(A)\le\left(\frac{C}{\alpha\theta D^2s_0^2}\right)^{|A|}
\le\left(C_0b^{-2}D^{-4/3}\right)^{|A|}. a ( A ) ≤ ( α θ D 2 s 0 2 C ) ∣ A ∣ ≤ ( C 0 b − 2 D − 4/3 ) ∣ A ∣ . On A ⊂ Q A\subset Q A ⊂ Q A A A
Finally the graph joining distinct lattice sites at distance at most 20 has uniformly bounded degree, also on the torus. It admits a coloring with an absolute number Q Q Q A A A A ′ A' A ′ ∣ A ∣ / Q |A|/Q ∣ A ∣/ Q A ⊂ Q A \subset Q A ⊂ Q A ′ ⊂ Q A' \subset Q A ′ ⊂ Q (28) for all A A A
These estimates concern the actual law μ \mu μ (28) under the bath marginal by integrating out the first coordinate. This inclusion will be used for the one-coordinate geometry.
The choices have a fixed order: α 0 \alpha_0 α 0 ℓ \ell ℓ θ \theta θ b b b D D D D D D K ≥ 100 K \ge100 K ≥ 100 D D D
A stationary sample of hard-core paths
The spatial estimates will supply many particles whose short trajectories can later be changed. We first represent the position law by an exact killed-Brownian sample, and prove that prescribed cells simultaneously lack the required trajectories with very small probability. All probabilities in this section refer either to that sample or to explicitly specified free Brownian paths; the interacting trajectories are never assumed independent.
We use the scaled torus, cells, and parameters of Equation (2.1) . In particular,
V = K 3 , D = N / V , r = α / D , α ∈ [ α 0 / 2 , 2 α 0 ] , μ ( d X ) = Φ ( X ) 2 d X . V = K^3,\qquad D = N/V,\qquad r = \alpha/D,\qquad\alpha\in[\alpha_0/2,2\alpha_0],\qquad\mu(\mathrm{d}X) = \Phi(X)^2\mathrm{d}X. V = K 3 , D = N / V , r = α / D , α ∈ [ α 0 /2 , 2 α 0 ] , μ ( d X ) = Φ ( X ) 2 d X . The constants α 0 , ℓ , θ , b \alpha_0,\ell,\theta,b α 0 , ℓ , θ , b s 0 = b D − 1 / 3 s_0 = bD^{-1/3} s 0 = b D − 1/3 θ \theta θ 3 θ < ℓ / 2 3\theta< \ell/2 3 θ < ℓ /2 b b b t ∈ ( 0 , 1 ] t \in(0,1] t ∈ ( 0 , 1 ] D D D
The endpoint-weighted law
For the killed Dirichlet semigroup on arbitrary open Euclidean domains, see [9 ] , arXiv version 2, Proposition 2.9 and Theorem 6.1. That reference uses Brownian generator Δ / 2 \Delta/2 Δ/2 Δ \Delta Δ
Let ν N \nu_N ν N N N N Δ \Delta Δ [ 0 , 2 t ] [0,2t] [ 0 , 2 t ] ω = ( ω 1 , … , ω N ) \omega= (\omega_1,\ldots,\omega_N) ω = ( ω 1 , … , ω N ) X ( u ) = ( ω 1 ( u ) , … , ω N ( u ) ) X(u) = (\omega_1(u),\ldots,\omega_N(u)) X ( u ) = ( ω 1 ( u ) , … , ω N ( u )) Y = X ( 0 ) Y = X(0) Y = X ( 0 ) Z = X ( 2 t ) Z = X(2t) Z = X ( 2 t )
d R ( ω ) = e 2 t E N Φ ( Y ) Φ ( Z ) 1 { d ( ω i ( u ) , ω j ( u ) ) > r ∀ u , i < j } d ν N ( ω ) . (30) \mathrm{d}R(\omega) = e^{2tE_N}\Phi(Y)\Phi(Z)\mathbf{1}_{\{d(\omega_i(u),\omega_j(u))>r\ \forall u,\ i<j\}}\,\mathrm{d}\nu_N(\omega). \tag*{(30)} d R ( ω ) = e 2 t E N Φ ( Y ) Φ ( Z ) 1 { d ( ω i ( u ) , ω j ( u )) > r ∀ u , i < j } d ν N ( ω ) . (30) Here and below d d d
Lemma 4.1 (Stationary killed paths). Equation (30) defines a probability measure. Every time marginal is μ \mu μ s s s e s E N Φ ( X − ) Φ ( X + ) e^{sE_N}\Phi(X_-)\Phi(X_+) e s E N Φ ( X − ) Φ ( X + ) Proof. For the Dirichlet Laplacian on the open allowed configuration set Ω \Omega Ω e − s H N e^{-sH_N} e − s H N H 0 1 ( Ω ) H^1_0(\Omega) H 0 1 ( Ω ) Ω \Omega Ω H 0 1 ( Ω ) H^1_0(\Omega) H 0 1 ( Ω ) Ω \Omega Ω Ω \Omega Ω
The integral in (30) is e 2 t E N ⟨ Φ , e − 2 t H N Φ ⟩ = 1 e^{2tE_N}\langle\Phi,e^{-2tH_N}\Phi\rangle=1 e 2 t E N ⟨ Φ , e − 2 t H N Φ ⟩ = 1 u u u
e 2 t E N ( e − u H N Φ ) ( x ) ( e − ( 2 t − u ) H N Φ ) ( x ) = Φ ( x ) 2 . e^{2tE_N}(e^{-uH_N}\Phi)(x)(e^{-(2t-u)H_N}\Phi)(x)=\Phi(x)^2. e 2 t E N ( e − u H N Φ ) ( x ) ( e − ( 2 t − u ) H N Φ ) ( x ) = Φ ( x ) 2 . Integrating an added initial or terminal interval proves the restriction statement. Symmetry of the killed kernel proves reversal. At fixed endpoints the two factors Φ \Phi Φ
The next inequality transfers a rare event for one free trajectory to an estimate for any specified set of labelled trajectories. It uses only the removal-energy estimate, not independence under R \mathcal{R} R
Lemma 4.2 (Dropping selected interactions). Let I ⊂ { 1 , … , N } I\subset\{1,\ldots,N\} I ⊂ { 1 , … , N } m m m h h h
E R ∏ i ∈ I h ( ω i ) ≤ e C t α m ∫ μ ( d Y ) ∏ i ∈ I E f r e e Y i h . (31) \mathbb{E}_{\mathcal{R}}\prod_{i\in I}h(\omega_i)\le e^{Ct\alpha m}\int\mu(\mathrm{d}Y)\prod_{i\in I}\mathbb{E}_{\mathrm{free}}^{Y_i}h. \tag*{(31)} E R i ∈ I ∏ h ( ω i ) ≤ e C t α m ∫ μ ( d Y ) i ∈ I ∏ E free Y i h . (31) The free expectation uses a Brownian path of length 2 t 2t 2 t
Proof. First take bounded h h h I I I y , z y,z y , z Φ \Phi Φ
e − 2 t E N − m a ( y ) a ( z ) , a ( y ) 2 = ∫ Φ ( y , x ′ ) 2 d x ′ . e^{-2tE_{N-m}}a(y)a(z),\qquad a(y)^2=\int\Phi(y,x')^2\,\mathrm{d}x'. e − 2 t E N − m a ( y ) a ( z ) , a ( y ) 2 = ∫ Φ ( y , x ′ ) 2 d x ′ . The free selected paths, weighted by ∏ h \prod h ∏ h ν I \nu_I ν I y = ω I ( 0 ) y=\omega_I(0) y = ω I ( 0 ) z = ω I ( 2 t ) z=\omega_I(2t) z = ω I ( 2 t )
∫ a ( y ) a ( z ) ∏ i ∈ I h ( ω i ) d ν I ≤ 1 2 ∫ ( a ( y ) 2 + a ( z ) 2 ) ∏ i ∈ I h ( ω i ) d ν I = ∫ a ( y ) 2 ∏ i ∈ I E f r e e y i h d y . \begin{aligned}
\int a(y)a(z)\prod_{i\in I}h(\omega_i)\,\mathrm{d}\nu_I
\le\frac{1}{2}\int\left(a(y)^2+a(z)^2\right)\prod_{i\in I}h(\omega_i)\,\mathrm{d}\nu_I \\
&=\int a(y)^2\prod_{i\in I}\mathbb{E}_{\mathrm{free}}^{y_i}h\,\mathrm{d}y.
\end{aligned} ∫ a ( y ) a ( z ) i ∈ I ∏ h ( ω i ) d ν I ≤ 2 1 ∫ ( a ( y ) 2 + a ( z ) 2 ) i ∈ I ∏ h ( ω i ) d ν I = ∫ a ( y ) 2 i ∈ I ∏ E free y i h d y . Here the equality reverses the paths in the term containing a ( z ) 2 a(z)^2 a ( z ) 2 a a a ∫ μ ( d Y ) ∏ i ∈ I E f r e e Y i h \int\mu(\mathrm{d}Y)\prod_{i\in I}\mathbb{E}_{\mathrm{free}}^{Y_i}h ∫ μ ( d Y ) ∏ i ∈ I E free Y i h e 2 t ( E N − E N − m ) e^{2t(E_N-E_{N-m})} e 2 t ( E N − E N − m ) (31) . Applying this to h ∧ M h\wedge M h ∧ M M M M
Put
γ = 1 12 , h 0 = D − 1 , m t = ⌈ t r 2 ⌉ , τ = t m t . (32) \gamma= \frac{1}{12}, \qquad h_0 = D^{-1}, \qquad m_t = \left\lceil\frac{t}{r^2} \right\rceil, \qquad\tau= \frac{t}{m_t}. \tag*{(32)} γ = 12 1 , h 0 = D − 1 , m t = ⌈ r 2 t ⌉ , τ = m t t . (32) We henceforth take D D D h 0 < t h_0 < t h 0 < t r 2 / 2 ≤ τ ≤ r 2 r^2/2 \le\tau\le r^2 r 2 /2 ≤ τ ≤ r 2 [ 0 , 2 t ] [0,2t] [ 0 , 2 t ] m t m_t m t I 1 , … , I 2 m t I_1,\ldots,I_{2m_t} I 1 , … , I 2 m t
A j ( ω ) = r − 1 diam ( ω ~ ( I j ) ) , A_j(\omega) = r^{-1}\operatorname{diam}(\widetilde{\omega}(I_j)), A j ( ω ) = r − 1 diam ( ω ( I j )) , where ω ~ \widetilde{\omega} ω
Definition 4.3. Fix a sufficiently large numerical constant C r e g C_{\mathrm{reg}} C reg regular if
max j A j ≤ D γ , 1 2 m t ∑ j = 1 2 m t ( 1 + A j ) 3 ≤ C r e g , (33) \max_j A_j \le D^\gamma, \qquad\frac{1}{2m_t}\sum_{j=1}^{2m_t}(1+A_j)^3 \le C_{\mathrm{reg}}, \tag*{(33)} j max A j ≤ D γ , 2 m t 1 j = 1 ∑ 2 m t ( 1 + A j ) 3 ≤ C reg , (33) and its lifted diameter on each of [ 0 , h 0 ] [0,h_0] [ 0 , h 0 ] [ 2 t − h 0 , 2 t ] [2t-h_0,2t] [ 2 t − h 0 , 2 t ] s 0 / 8 s_0/8 s 0 /8
For two regular paths whose corresponding outer endpoints are more than s 0 s_0 s 0 3 s 0 / 4 3s_0/4 3 s 0 /4
Lemma 4.4 (Free cutoff bounds). For every fixed q > 0 q>0 q > 0 t > 0 t>0 t > 0 C q , t D − q C_{q,t}D^{-q} C q , t D − q K K K α \alpha α 2 t 2t 2 t t t t D D D t t t C r e g C_{\mathrm{reg}} C reg
If all three prescribed points lie within distance 3 3 3 v v v 10 10 10 v v v C e − c / t Ce^{-c/t} C e − c / t c , C > 0 c,C>0 c , C > 0
Proof. For Brownian motion, the variables A j A_j A j τ / r 2 ∈ [ 1 / 2 , 1 ] \tau/r^2 \in[1/2,1] τ / r 2 ∈ [ 1/2 , 1 ] C m t e − c D 2 γ C m_t e^{-cD^{2\gamma}} C m t e − c D 2 γ U j = ( 1 + A j ) 3 U_j=(1+A_j)^3 U j = ( 1 + A j ) 3 2 k 2k 2 k ∑ j ( U j − E U j ) \sum_j(U_j-\mathbb{E}U_j) ∑ j ( U j − E U j ) C k m t k C_km_t^k C k m t k
E ∣ 1 2 m t ∑ j ( U j − E U j ) ∣ 2 k ≤ C k m t − k . \mathbb{E}\left|\frac{1}{2m_t}\sum_j(U_j-\mathbb{E}U_j)\right|^{2k} \le C_km_t^{-k}. E 2 m t 1 j ∑ ( U j − E U j ) 2 k ≤ C k m t − k . Choose C r e g C_{\mathrm{reg}} C reg 1 + sup E U j 1+\sup\mathbb{E}U_j 1 + sup E U j m t m_t m t D 2 D^2 D 2 t t t
s 0 2 h 0 = b 2 D 1 / 3 , \frac{s_0^2}{h_0}=b^2D^{1/3}, h 0 s 0 2 = b 2 D 1/3 , which proves the remaining free estimate.
For a bridge leg of duration t t t W ( u ) − ( u / t ) W ( t ) W(u) - (u/t)W(t) W ( u ) − ( u / t ) W ( t ) W W W r r r
r t ( ∣ W ( t ) ∣ + C ) . \frac{r}{t}\left(\lvert W(t)\rvert+ C\right). t r ( ∣ W ( t )∣ + C ) . Outside an event of arbitrarily small polynomial probability this tends to zero, uniformly for bounded endpoint displacement. The inequality ( x + y ) 3 ≤ 4 x 3 + 4 y 3 (x+y)^3 \le4x^3 + 4y^3 ( x + y ) 3 ≤ 4 x 3 + 4 y 3 C r e g C_{\mathrm{reg}} C reg h 0 ( ∣ W ( t ) ∣ + C ) / t = o ( s 0 ) h_0(\lvert W(t)\rvert+ C)/t = o(s_0) h 0 (∣ W ( t )∣ + C ) / t = o ( s 0 ) 2 t 2t 2 t 2 t 2t 2 t t t t
Finally the endpoint lines of the two legs remain in the radius-3 ball. Leaving the radius-10 ball requires a centered fluctuation of size at least seven on one leg. The Brownian representation and the maximal Gaussian bound give the stated C e − c / t Ce^{-c/t} C e − c / t
Many paths visiting a prescribed set
For a set A A A J A ( ω ) J_A(\omega) J A ( ω ) v ∈ A v \in A v ∈ A D \mathcal{D} D ε D = sup y P f r e e y ( D ) \varepsilon_{\mathcal{D}} = \sup_y \mathbb{P}_{\mathrm{free}}^y(\mathcal{D}) ε D = sup y P free y ( D ) t ≤ 1 t \le1 t ≤ 1 K K K
Lemma 4.5 (A free visit bound). For every λ > 0 \lambda> 0 λ > 0
E f r e e y [ e λ J A 1 D − 1 ] ≤ ζ e − d ( y , A ) , ζ = C λ λ ε D 1 / 3 . (34) \mathbb{E}_{\mathrm{free}}^y\left[e^{\lambda J_A}\mathbf{1}_{\mathcal{D}} - 1\right] \le\zeta e^{-d(y,A)}, \qquad\zeta= C_\lambda\lambda\varepsilon_{\mathcal{D}}^{1/3}. \tag*{(34)} E free y [ e λ J A 1 D − 1 ] ≤ ζ e − d ( y , A ) , ζ = C λ λ ε D 1/3 . (34) Here d ( y , A ) d(y,A) d ( y , A ) y y y C λ C_\lambda C λ λ \lambda λ
Proof. Mark the lifted Brownian path at its initial point and at each first exit from the unit ball about its last mark. If Q Q Q 2 t 2t 2 t J A ≤ C ( Q + 1 ) J_A \le C(Q+1) J A ≤ C ( Q + 1 )
The exit durations are iid copies of a positive random variable σ \sigma σ s > 0 s > 0 s > 0
P ( Q ≥ k ) ≤ e 2 s ( E e − s σ ) k . \mathbb{P}(Q \ge k) \le e^{2s}(\mathbb{E}e^{-s\sigma})^k. P ( Q ≥ k ) ≤ e 2 s ( E e − s σ ) k . Since E e − s σ → 0 \mathbb{E}e^{-s\sigma} \to0 E e − s σ → 0 s → ∞ s \to\infty s → ∞ Q Q Q t ≤ 1 t \le1 t ≤ 1 J A > 0 J_A > 0 J A > 0 ( d ( y , A ) − 12 ) + (d(y,A)-12)_+ ( d ( y , A ) − 12 ) + e λ J A − 1 ≤ λ J A e λ J A e^{\lambda J_A} - 1 \le\lambda J_A e^{\lambda J_A} e λ J A − 1 ≤ λ J A e λ J A
E [ ( e λ J A − 1 ) 1 D 1 { J A > 0 } ] ≤ λ ( E [ J A 3 e 3 λ J A ] ) 1 / 3 ε D 1 / 3 P ( J A > 0 ) 1 / 3 . \mathbb{E}\left[(e^{\lambda J_A}-1)\mathbf{1}_{\mathcal{D}}\mathbf{1}_{\{J_A>0\}}\right]
\le\lambda\left(\mathbb{E}[J_A^3e^{3\lambda J_A}]\right)^{1/3}\varepsilon_{\mathcal{D}}^{1/3}\mathbb{P}(J_A>0)^{1/3}. E [ ( e λ J A − 1 ) 1 D 1 { J A > 0 } ] ≤ λ ( E [ J A 3 e 3 λ J A ] ) 1/3 ε D 1/3 P ( J A > 0 ) 1/3 . The exponential moment and Gaussian travel bound imply Equation (34) , after changing its constant.
Lemma 4.6 (Interacting visit moments). If e C t α ζ ≤ 1 e^{C t^\alpha}\zeta\le1 e C t α ζ ≤ 1 then
E R exp ( λ ∑ i J A ( ω i ) 1 D ( ω i ) ) ≤ exp ( C e C t α ζ D ∣ A ∣ ) . (35) \mathbb{E}_{\mathcal{R}}\exp\left(\lambda\sum_i J_A(\omega_i)\mathbf{1}_{\mathcal{D}}(\omega_i)\right) \le\exp(Ce^{C t^\alpha}\zeta D\lvert A\rvert). \tag*{(35)} E R exp ( λ i ∑ J A ( ω i ) 1 D ( ω i ) ) ≤ exp ( C e C t α ζ D ∣ A ∣) . (35) Proof. Set h ( ω ) = e λ J A ( ω ) 1 D ( ω ) − 1 h(\omega)=e^{\lambda J_A(\omega)1_D(\omega)}-1 h ( ω ) = e λ J A ( ω ) 1 D ( ω ) − 1 ∏ i ( 1 + h ( ω i ) ) \prod_i(1+h(\omega_i)) ∏ i ( 1 + h ( ω i ))
∫ μ ( d Y ) ∏ i [ 1 + e C t α ζ e − d ( Y i , A ) ] ≤ ∫ e C t α ζ f A d μ , f A = ∑ i e − d ( Y i , A ) . \int\mu(\mathrm{d}Y)\prod_i\left[1+e^{C t^\alpha\zeta e^{-d(Y_i,A)}}\right]\leq\int e^{C t^\alpha\zeta f_A}\,\mathrm{d}\mu,\qquad f_A=\sum_i e^{-d(Y_i,A)}. ∫ μ ( d Y ) i ∏ [ 1 + e C t α ζ e − d ( Y i , A ) ] ≤ ∫ e C t α ζ f A d μ , f A = i ∑ e − d ( Y i , A ) . Equation (26) applies with the indicated exponent at most one, and proves Equation (35) .
Cells with a supply of movable paths
Let d i ( X ) = min j ≠ i d ( X i , X j ) d_i(X)=\min_{j\ne i}d(X_i,X_j) d i ( X ) = min j = i d ( X i , X j ) + ∞ +\infty + ∞ N = 1 N=1 N = 1
C v ( X ) = # { i : d ( X i , v ) ≤ 5 , d i ( X ) ≤ s 0 } . C_v(X)=\#\{i:d(X_i,v)\leq5,\ d_i(X)\leq s_0\}. C v ( X ) = # { i : d ( X i , v ) ≤ 5 , d i ( X ) ≤ s 0 } . We call the following conditions a certificate at v v v
n v ( Y ) ≥ ℓ D n_v(Y)\geq\ell D n v ( Y ) ≥ ℓ D C v ( Y ) , C v ( Z ) < θ D C_v(Y),C_v(Z)<\theta D C v ( Y ) , C v ( Z ) < θ D
at most H 1 D H_1D H 1 D v v v θ D \theta D θ D
The certificate is an event of the full path sample. Later it will be used to estimate the probability of suitable exposed data; it will not be added to the conditioning on unexposed trajectories.
Proposition 4.7 (Joint certificate supply). Fix any p 1 > 0 p_1>0 p 1 > 0 H 1 H_1 H 1 t > 0 t>0 t > 0 D 0 D_0 D 0
R { v has no certificate for every v ∈ A } ≤ p 1 ∣ A ∣ (36) \mathcal{R}\{v\text{ has no certificate for every }v\in A\}\leq p_1^{|A|} \tag*{(36)} R { v has no certificate for every v ∈ A } ≤ p 1 ∣ A ∣ (36) for every set A A A D ≥ D 0 D\geq D_0 D ≥ D 0 K K K α \alpha α H 1 H_1 H 1 t t t D 0 D_0 D 0
Proof. We bound simultaneous failures of each individual test first. The low-count estimate in Proposition 3.5 and the close-count estimate in Equation (28) apply at both endpoints by Lemma 4.1. Their per-site parameters tend to zero as D D D
For excessive visiting paths use D \mathcal{D} D λ > 0 \lambda>0 λ > 0 e C t α C λ λ ≤ 1 e^{Ct^\alpha C_\lambda\lambda}\leq1 e C t α C λ λ ≤ 1 t ≤ 1 t\leq1 t ≤ 1 α \alpha α A A A H 1 D H_1D H 1 D ∑ i J A ( ω i ) > H 1 D ∣ A ∣ \sum_i J_A(\omega_i)>H_1D|A| ∑ i J A ( ω i ) > H 1 D ∣ A ∣ (35) and exponential Markov inequality give
R ( this event ) ≤ exp { − [ λ H 1 − C e C t α C λ λ ] D ∣ A ∣ } . \mathcal{R}(\text{this event})\leq\exp\{-[\lambda H_1-Ce^{Ct^\alpha}C_\lambda\lambda]D|A|\}. R ( this event ) ≤ exp { − [ λ H 1 − C e C t α C λ λ ] D ∣ A ∣ } . Take H 1 H_1 H 1
For visitors of diameter greater than one take λ = 1 \lambda=1 λ = 1 D = { diam ω ~ > 1 } \mathcal{D}=\{\operatorname{diam}\tilde{\omega}>1\} D = { diam ω ~ > 1 } C e − c / t Ce^{-c/t} C e − c / t t t t (35) is at most θ / 2 \theta/2 θ /2 A A A θ D \theta D θ D θ D ∣ A ∣ \theta D|A| θ D ∣ A ∣ e − θ D ∣ A ∣ / 2 e^{-\theta D|A|/2} e − θ D ∣ A ∣/2
Finally let D \mathcal{D} D λ \lambda λ D ζ → 0 D\zeta\to0 D ζ → 0 A A A ∣ A ∣ |A| ∣ A ∣ (35) bounds this probability by exp [ − ( λ − o ( 1 ) ) ∣ A ∣ ] \exp[-(\lambda-o(1))|A|] exp [ − ( λ − o ( 1 )) ∣ A ∣ ] λ \lambda λ D D D D D D
There are six failure types. If every site of A A A 6 ∣ A ∣ 6^{|A|} 6 ∣ A ∣ ∣ A ∣ / 6 |A|/6 ∣ A ∣/6 ϵ \epsilon ϵ ( 6 ϵ 1 / 6 ) ∣ A ∣ (6\epsilon^{1/6})^{|A|} ( 6 ϵ 1/6 ) ∣ A ∣ ϵ ≤ ( p 1 / 6 ) 6 \epsilon\le(p_1/6)^6 ϵ ≤ ( p 1 /6 ) 6 (36) . The free estimates remain valid when the already chosen t t t D 0 D_0 D 0
We have obtained an unconditional joint bound for the cells where the full sample is unsuitable. The next section estimates a trial path against a deterministic collection of such trajectories. We will then expose most paths in a way that preserves an exact conditional product representation for the paths still to be changed.
Moving a path through a neighboring cell
We next estimate the cost of replacing one particle’s trajectory while preserving its two outer endpoints. The replacement passes through a point chosen uniformly in its home cell or a face-neighbor cell. Two estimates serve different purposes: averaging over many separated endpoints makes the total collision probability against order D D D
We use the regularity conditions of Definition 4.3. Thus r = α / D r=\alpha/D r = α / D s 0 = b D − 1 / 3 s_0=bD^{-1/3} s 0 = b D − 1/3 h 0 = D − 1 h_0=D^{-1} h 0 = D − 1 γ = 1 / 12 \gamma=1/12 γ = 1/12 [ 0 , 2 t ] [0,2t] [ 0 , 2 t ] m t = ⌈ t / r 2 ⌉ m_t=\lceil t/r^2\rceil m t = ⌈ t / r 2 ⌉ τ = t / m t \tau=t/m_t τ = t / m t c c c k k k r A k ( c ) rA_k(c) r A k ( c )
max k A k ( c ) ≤ D γ , ∑ k = 1 2 m t ( 1 + A k ( c ) ) 3 ≤ 2 C r e g m t . (37) \max_k A_k(c) \le D^\gamma,\qquad\sum_{k=1}^{2m_t}(1+A_k(c))^3 \le2C_{\mathrm{reg}}m_t. \tag*{(37)} k max A k ( c ) ≤ D γ , k = 1 ∑ 2 m t ( 1 + A k ( c ) ) 3 ≤ 2 C reg m t . (37) Its diameter in each outer time window of length h 0 h_0 h 0 s 0 / 8 s_0/8 s 0 /8 0 < t ≤ 1 0<t\le1 0 < t ≤ 1 D D D α ∈ [ α 0 / 2 , 2 α 0 ] \alpha\in[\alpha_0/2,2\alpha_0] α ∈ [ α 0 /2 , 2 α 0 ]
τ ≤ h 0 / 2 , h 0 < t / 4 , r D γ < 1 , C τ / t ≤ r \tau\le h_0/2,\qquad h_0<t/4,\qquad rD^\gamma<1,\qquad C\tau/t\le r τ ≤ h 0 /2 , h 0 < t /4 , r D γ < 1 , C τ / t ≤ r for any fixed geometric constant C C C
Fix a cell B v = v + [ − 1 / 2 , 1 / 2 ) 3 B_v=v+[-1/2,1/2)^3 B v = v + [ − 1/2 , 1/2 ) 3 y , z y,z y , z y ∈ B v y\in B_v y ∈ B v ∣ z − v ∣ ≤ 3 |z-v|\le3 ∣ z − v ∣ ≤ 3 B v B_v B v u u u B B B Q y , z u Q^u_{y,z} Q y , z u Δ \Delta Δ y y y u u u u u u z z z t t t
Q y , z B = ∫ B Q y , z u d u , Q y , z 0 = the Euclidean bridge from y to z of duration 2 t . Q^B_{y,z}=\int_B Q^u_{y,z}\,\mathrm{d}u,\qquad Q^0_{y,z}=\text{the Euclidean bridge from }y\text{ to }z\text{ of duration }2t. Q y , z B = ∫ B Q y , z u d u , Q y , z 0 = the Euclidean bridge from y to z of duration 2 t . Both laws are projected to the torus. Unit cells have volume one, so Q y , z B Q^B_{y,z} Q y , z B C v C_v C v 10 10 10 v v v c c c
H ( c ) = { ω : d ( ω ( s ) , c ( s ) ) ≤ r for some s ∈ [ 0 , 2 t ] } . \mathcal{H}(c)=\{\omega:d(\omega(s),c(s))\le r\text{ for some }s\in[0,2t]\}. H ( c ) = { ω : d ( ω ( s ) , c ( s )) ≤ r for some s ∈ [ 0 , 2 t ]} . The estimates concern C v ∩ H ( c ) C_v\cap\mathcal{H}(c) C v ∩ H ( c )
Separated endpoints and Gaussian densities
Lemma 5.1 (Averaged Gaussian density). Let y 1 , … , y m ∈ R 3 y_1,\ldots,y_m \in\mathbb{R}^3 y 1 , … , y m ∈ R 3 s 0 s_0 s 0 m ≥ ℓ D / 2 m \ge\ell D/2 m ≥ ℓ D /2 s 0 = b D − 1 / 3 s_0 = bD^{-1/3} s 0 = b D − 1/3 u ∈ R 3 u \in\mathbb{R}^3 u ∈ R 3 0 < s ≤ t / 2 0 < s \le t/2 0 < s ≤ t /2 s s s t t t y i y_i y i u u u
C ( 1 + 1 D s 3 / 2 ) . C\left(1+\frac{1}{Ds^{3/2}}\right). C ( 1 + D s 3/2 1 ) . Here C C C b , ℓ b,\ell b , ℓ t , s , D t,s,D t , s , D u u u
Proof. The means are a y i + ( 1 − a ) u ay_i+(1-a)u a y i + ( 1 − a ) u a = 1 − s / t ≥ 1 / 2 a=1-s/t\ge1/2 a = 1 − s / t ≥ 1/2 2 s ( 1 − s / t ) 2s(1-s/t) 2 s ( 1 − s / t ) s 0 / 2 s_0/2 s 0 /2 q i q_i q i s 0 / 2 s_0/2 s 0 /2 s 0 / 4 s_0/4 s 0 /4
# { i : ∣ q i − x ∣ ≤ R } ≤ C ( 1 + R / s 0 ) 3 . \#\{i:|q_i-x|\le R\}\le C(1+R/s_0)^3. # { i : ∣ q i − x ∣ ≤ R } ≤ C ( 1 + R / s 0 ) 3 . Split the Gaussian sum into the ball of radius s \sqrt{s} s x x x j s , ( j + 1 ) s j\sqrt{s},(j+1)\sqrt{s} j s , ( j + 1 ) s j ≥ 1 j\ge1 j ≥ 1
∑ i exp ( − ∣ x − q i ∣ 2 4 s ) ≤ C ( 1 + s 3 / 2 s 0 3 ) . \sum_i \exp\left(-\frac{|x-q_i|^2}{4s}\right)\le C\left(1+\frac{s^{3/2}}{s_0^3}\right). i ∑ exp ( − 4 s ∣ x − q i ∣ 2 ) ≤ C ( 1 + s 0 3 s 3/2 ) . The covariance lies between s s s 2 s 2s 2 s
C m s 3 / 2 + C m s 0 3 ≤ C ( 1 D s 3 / 2 + 1 ) . \frac{C}{ms^{3/2}}+\frac{C}{ms_0^3}\le C\left(\frac{1}{Ds^{3/2}}+1\right). m s 3/2 C + m s 0 3 C ≤ C ( D s 3/2 1 + 1 ) . Collision with a deterministic obstacle
Proposition 5.2 (Averaged collision bound). Fix m ≥ ℓ D / 2 m\ge\ell D/2 m ≥ ℓ D /2 ( y i , z i ) (y_i,z_i) ( y i , z i ) y i ∈ B v y_i\in B_v y i ∈ B v ∣ z i − v ∣ ≤ 3 |z_i-v|\le3 ∣ z i − v ∣ ≤ 3 s 0 s_0 s 0 c c c s 0 s_0 s 0 B B B
1 m ∑ i = 1 m Q y i , z i B ( C v ∩ H ( c ) ) ≤ C t r + C r D 3 γ D h 0 . (38) \frac{1}{m}\sum_{i=1}^m Q_{y_i,z_i}^{B}(C_v\cap H(c))\le Ctr+C\frac{rD^{3\gamma}}{D\sqrt{h_0}}. \tag*{(38)} m 1 i = 1 ∑ m Q y i , z i B ( C v ∩ H ( c )) ≤ C t r + C D h 0 r D 3 γ . (38) The same conclusion holds if c c c m m m C C C b , ℓ , C r e g b,\ell,C_{\mathrm{reg}} b , ℓ , C reg t t t
Proof. We estimate the first leg from an outer endpoint y i y_i y i u u u z i z_i z i u u u C v C_v C v s 0 / 8 s_0/8 s 0 /8 3 s 0 / 4 > r 3s_0/4>r 3 s 0 /4 > r [ 0 , h 0 ] [0,h_0] [ 0 , h 0 ] s ≥ h 0 − τ ≥ h 0 / 2 s\ge h_0-\tau\ge h_0/2 s ≥ h 0 − τ ≥ h 0 /2
We first explain the use of lifts. On a mesh interval, the obstacle has diameter at most r D γ < 1 rD^\gamma<1 r D γ < 1 d ( c ( s ) , v ) < 12 d(c(s),v)<12 d ( c ( s ) , v ) < 12 K > 50 K>50 K > 50 c ~ ( s ) \widetilde{c}(s) c ( s ) 12 12 12 v v v r A k ( c ) rA_k(c) r A k ( c )
The first half of a leg. Suppose s ≤ t / 2 s \le t/2 s ≤ t /2 u u u X s X_s X s s s s τ \tau τ
X s + q = X s + q t − s ( u − X s ) + R q , 0 ≤ q ≤ τ . X_{s+q}=X_s+\frac{q}{t-s}(u-X_s)+R_q,\qquad0\le q\le\tau. X s + q = X s + t − s q ( u − X s ) + R q , 0 ≤ q ≤ τ . Here R R R t − s t-s t − s X s X_s X s u u u s s s R q = W q − q W t − s / ( t − s ) R_q=W_q-qW_{t-s}/(t-s) R q = W q − q W t − s / ( t − s ) W W W Δ \Delta Δ 3 a 2 d a 3a^2\,da 3 a 2 d a
B s = 1 r sup 0 ≤ q ≤ τ ∣ R q ∣ , E B s 3 ≤ C , (39) B_s=\frac{1}{r}\sup_{0\le q\le\tau}|R_q|,\qquad\mathbb{E}B_s^3\le C, \tag*{(39)} B s = r 1 0 ≤ q ≤ τ sup ∣ R q ∣ , E B s 3 ≤ C , (39) because τ ≤ r 2 \tau\le r^2 τ ≤ r 2 τ ≤ t − s \tau\le t-s τ ≤ t − s
On the cutoff event, ∣ X s − v ∣ ≤ 10 |X_s-v|\le10 ∣ X s − v ∣ ≤ 10 2 2 2 v v v (5.5) is thus at most C τ / t ≤ r C\tau/t\le r C τ / t ≤ r
∣ X s − c ~ ( s ) ∣ ≤ r ( 2 + A k ( c ) + B s ) . |X_s-\widetilde c(s)|\le r\left(2+A_k(c)+B_s\right). ∣ X s − c ( s ) ∣ ≤ r ( 2 + A k ( c ) + B s ) . We use the cutoff only to obtain this necessary event. We now discard it and integrate the unrestricted bridge law. By Lemma 5.1, the density of X s X_s X s (5.3) . Since B s B_s B s X s X_s X s (39) , the averaged probability of (5.7) is at most
C r 3 ( 1 + A k ( c ) ) 3 ( 1 + 1 D s 3 / 2 ) . Cr^3(1+A_k(c))^3\left(1+\frac{1}{Ds^{3/2}}\right). C r 3 ( 1 + A k ( c ) ) 3 ( 1 + D s 3/2 1 ) . This estimate holds for each u u u u u u m m m
The second half of a leg. Suppose s > t / 2 s>t/2 s > t /2
X q = ( 1 − q / t ) y i + ( q / t ) U + β q , 0 ≤ q ≤ t , X_q=(1-q/t)y_i+(q/t)U+\beta_q,\qquad0\le q\le t, X q = ( 1 − q / t ) y i + ( q / t ) U + β q , 0 ≤ q ≤ t , where U U U B B B β \beta β U U U β \beta β X s X_s X s ( t / s ) 3 ≤ 8 (t/s)^3\le8 ( t / s ) 3 ≤ 8 C τ / t ≤ r C\tau/t\le r C τ / t ≤ r U ∈ B U\in B U ∈ B
B ~ s = r − 1 sup 0 ≤ q ≤ τ ∣ β s + q − β s ∣ . \widetilde B_s=r^{-1}\sup_{0\le q\le\tau}|\beta_{s+q}-\beta_s|. B s = r − 1 0 ≤ q ≤ τ sup ∣ β s + q − β s ∣. Writing β q = W q − q W t / t \beta_q=W_q-qW_t/t β q = W q − q W t / t E B ~ s 3 ≤ C \mathbb{E}\widetilde B_s^3\le C E B s 3 ≤ C C τ 3 / 2 / r 3 C\tau^{3/2}/r^3 C τ 3/2 / r 3 C τ 3 / ( r 3 t 3 / 2 ) C\tau^3/(r^3t^{3/2}) C τ 3 / ( r 3 t 3/2 ) β \beta β (5.7) , with B s B_s B s B ~ s \widetilde B_s B s U U U U U U β \beta β
C r 3 ( 1 + A k ( c ) ) 3 . Cr^3(1+A_k(c))^3. C r 3 ( 1 + A k ( c ) ) 3 . In particular, no independence between X s X_s X s
Summing the intervals. Apply the two bounds to both legs. The terms without s − 3 / 2 / D s^{-3/2}/D s − 3/2 / D
C r 3 ∑ k = 1 2 m t ( 1 + A k ( c ) ) 3 ≤ C r 3 m t ≤ C t r . Cr^{3}\sum_{k=1}^{2m_t}(1+A_k(c))^3 \le Cr^{3}m_t \le Ctr. C r 3 k = 1 ∑ 2 m t ( 1 + A k ( c ) ) 3 ≤ C r 3 m t ≤ C t r . Here m t = t / τ m_t=t/\tau m t = t / τ τ ≥ r 2 / 2 \tau\ge r^2/2 τ ≥ r 2 /2 ( 1 + A k ( c ) ) 3 ≤ C D 3 γ (1+A_k(c))^3 \le CD^{3\gamma} ( 1 + A k ( c ) ) 3 ≤ C D 3 γ
∑ s = j τ ≥ h 0 / 2 s ≤ t / 2 s − 3 / 2 ≤ C τ h 0 ≤ C r 2 h 0 . \sum_{\substack{s=j\tau\ge h_0/2\\s\le t/2}}s^{-3/2}\le\frac{C}{\tau\sqrt{h_0}}\le\frac{C}{r^2\sqrt{h_0}}. s = j τ ≥ h 0 /2 s ≤ t /2 ∑ s − 3/2 ≤ τ h 0 C ≤ r 2 h 0 C . Their contribution is therefore at most C r 3 D 3 γ / ( D r 2 h 0 ) = C r D 3 γ / ( D h 0 ) Cr^3D^{3\gamma}/(Dr^2\sqrt{h_0})=CrD^{3\gamma}/(D\sqrt{h_0}) C r 3 D 3 γ / ( D r 2 h 0 ) = C r D 3 γ / ( D h 0 ) (38) . Every constant in these sums is independent of t t t t t t D D D (5.2) .
Proposition 5.3 (A single replacement). Let v ∈ B v v\in B_v v ∈ B v ∣ z − v ∣ ≤ 3 |z-v|\le3 ∣ z − v ∣ ≤ 3 c c c d ( y , c ( 0 ) ) > s 0 d(y,c(0))>s_0 d ( y , c ( 0 )) > s 0 d ( z , c ( 2 t ) ) > s 0 d(z,c(2t))>s_0 d ( z , c ( 2 t )) > s 0 Q = Q y , z B Q=Q^B_{y,z} Q = Q y , z B B B B Q = Q y , z 0 Q=Q^0_{y,z} Q = Q y , z 0
Q ( C v ∩ H ( c ) ) ≤ C t r + C r D 3 γ h 0 ≤ C α ( t D − 1 + D − 1 / 4 ) . (40) Q(C_v\cap H(c))\le Ctr+C\frac{rD^{3\gamma}}{\sqrt{h_0}}\le C\alpha(tD^{-1}+D^{-1/4}). \tag*{(40)} Q ( C v ∩ H ( c )) ≤ C t r + C h 0 r D 3 γ ≤ C α ( t D − 1 + D − 1/4 ) . (40) Thus this probability tends to zero with D D D t t t
Proof. For Q y , z B Q^B_{y,z} Q y , z B s ≤ t / 2 s\le t/2 s ≤ t /2 C s − 3 / 2 Cs^{-3/2} C s − 3/2 (5.8) becomes C r 3 ( 1 + A k ( c ) ) 3 s − 3 / 2 Cr^3(1+A_k(c))^3s^{-3/2} C r 3 ( 1 + A k ( c ) ) 3 s − 3/2 (5.9) . The same two sums give the first inequality.
For Q y , z 0 Q^0_{y,z} Q y , z 0 t t t t t t (5.5) holds with 2 t 2t 2 t t t t C τ / t C\tau/t C τ / t s s s C s − 3 / 2 Cs^{-3/2} C s − 3/2 0 < s ≤ t 0<s\le t 0 < s ≤ t [ 0 , h 0 ] [0,h_0] [ 0 , h 0 ] C r D 3 γ / h 0 CrD^{3\gamma}/\sqrt{h_0} C r D 3 γ / h 0 C t r Ctr C t r r = α / D r=\alpha/D r = α / D h 0 = D − 1 h_0=D^{-1} h 0 = D − 1 3 γ = 1 / 4 3\gamma=1/4 3 γ = 1/4
Remark 5.4. The midpoint in Q y , z B Q^B_{y,z} Q y , z B u = c ( t ) u=c(t) u = c ( t ) t t t a > 0 a>0 a > 0 a − 1 a^{-1} a − 1 a a a
Many paths can be moved successfully
The averaged estimate now pays for all nearby obstacles. The following statement separates that deterministic estimate from the random cell certificates that will supply its hypotheses.
Corollary 5.5 (Supply of successful labels). *Consider a collection of deterministic torus paths with endpoints ( y j , z j ) (y_j,z_j) ( y j , z j ) v v v H 1 D H_1D H 1 D 12 12 12 v v v m ≥ ℓ D / 2 m\ge\ell D/2 m ≥ ℓ D /2 y i ∈ B v y_i \in B_v y i ∈ B v ∣ z i − v ∣ ≤ 3 |z_i-v| \le3 ∣ z i − v ∣ ≤ 3 s 0 s_0 s 0
Call a trial for label i i i C v C_v C v i i i t > 0 t>0 t > 0 D D D 9 m / 10 9m/10 9 m /10 B B B
T i , B = { u ∈ B : Q y i , z i u ( successful trial ) ≥ 9 / 10 } (41) T_{i,B}=\left\{u\in B:Q_{y_i,z_i}^{u}(\text{successful trial})\ge9/10\right\} \tag*{(41)} T i , B = { u ∈ B : Q y i , z i u ( successful trial ) ≥ 9/10 } (41) has volume at least 99 / 100 99/100 99/100 v v v α ∈ [ α 0 / 2 , 2 α 0 ] \alpha\in[\alpha_0/2,2\alpha_0] α ∈ [ α 0 /2 , 2 α 0 ]
Proof. The linear means of both bridge legs stay within distance 3 3 3 v v v C e − c / t Ce^{-c/t} C e − c / t ε D ( t ) → 0 \varepsilon_D(t)\to0 ε D ( t ) → 0 12 12 12 r < 1 r<1 r < 1
1 m ∑ i ∫ B Q y i , z i u ( failure ) d u ≤ C e − c / t + ε D ( t ) + C H 1 D ( t r + r D 3 γ D h 0 ) ≤ C e − c / t + ε D ( t ) + C H 1 α ( t + D − 1 / 4 ) . (42) \begin{aligned}
\frac{1}{m}\sum_i\int_B Q_{y_i,z_i}^{u}(\text{failure})\,du
\le Ce^{-c/t}+\varepsilon_D(t)+CH_1D\left(tr+\frac{rD^{3\gamma}}{D\sqrt{h_0}}\right) \\
&\le Ce^{-c/t}+\varepsilon_D(t)+CH_1\alpha\left(t+D^{-1/4}\right). \tag*{(42)}
\end{aligned} m 1 i ∑ ∫ B Q y i , z i u ( failure ) d u ≤ C e − c / t + ε D ( t ) + C H 1 D ( t r + D h 0 r D 3 γ ) ≤ C e − c / t + ε D ( t ) + C H 1 α ( t + D − 1/4 ) . (42) The constant multiplying t t t t t t t t t D D D 1 / 70000 1/70000 1/70000 α \alpha α
If ∣ T i , B ∣ < 99 / 100 |T_{i,B}|<99/100 ∣ T i , B ∣ < 99/100 B B B ( 1 / 100 ) ( 1 / 10 ) = 1 / 1000 (1/100)(1/10)=1/1000 ( 1/100 ) ( 1/10 ) = 1/1000 m / 70 m/70 m /70 m / 10 m/10 m /10
Exposing the environment and retaining movable groups
The bridge estimates provide successful local trials. We now select many disjoint groups of labels and expose everything outside one group. The two outputs are different: most cells retain many useful labels in an unconditional probabilistic sense, while at each fixed exposed datum the unexposed law has an exact constrained-product formula. Keeping these statements separate is essential for the later likelihood comparison.
Eligible labels and their local trial laws
The endpoints Y Y Y Z Z Z i i i v v v
Y i ∈ B v Y_i\in B_v Y i ∈ B v d ( Z i , v ) ≤ 3 d(Z_i,v)\le3 d ( Z i , v ) ≤ 3
d ( Y i , Y j ) > s 0 d(Y_i,Y_j)>s_0 d ( Y i , Y j ) > s 0 d ( Z i , Z j ) > s 0 d(Z_i,Z_j)>s_0 d ( Z i , Z j ) > s 0 j ≠ i j\ne i j = i
its trajectory is regular and remains in the radius-10 ball about v v v
The eligible sets at distinct sites are disjoint, since the starting cells form a partition. When K K K v v v
For each eligible label i i i v v v target cell is B v B_v B v Q i u Q_i^u Q i u t t t Y i Y_i Y i u u u u u u Z i Z_i Z i v v v F F F i i i
C i ( F ) = { the trial is regular and stays within distance 10 of v ; it avoids every path in F at distance r } . (43) C_i(F)=\{\text{the trial is regular and stays within distance 10 of }v;\ \text{it avoids every path in }F\text{ at distance }r\}. \tag*{(43)} C i ( F ) = { the trial is regular and stays within distance 10 of v ; it avoids every path in F at distance r } . (43) Avoidance throughout means strict distance greater than r r r
Lemma 6.1 (Useful labels in a certified cell). At a cell with a certificate there are at least ℓ D / 2 \ell D/2 ℓ D /2 t t t D D D B w B_w B w
∣ { u ∈ B w : Q i u ( C i ( F i a l l ) ) ≥ 0.9 } ∣ ≥ 0.99. (44) \left|\left\{u\in B_w:Q_i^u(C_i(F_i^{\mathrm{all}}))\ge0.9\right\}\right|\ge0.99. \tag*{(44)} { u ∈ B w : Q i u ( C i ( F i all )) ≥ 0.9 } ≥ 0.99. (44) Here F i a l l F_i^{\mathrm{all}} F i all ∣ ⋅ ∣ |\cdot| ∣ ⋅ ∣
Proof. Start with the at least ℓ D \ell D ℓ D Y i ∈ B v Y_i\in B_v Y i ∈ B v θ D \theta D θ D v v v v v v s 0 s_0 s 0 θ D \theta D θ D v v v ( ℓ − 3 θ ) D ≥ ℓ D / 2 (\ell-3\theta)D\ge\ell D/2 ( ℓ − 3 θ ) D ≥ ℓ D /2
We now apply Corollary 5.5 to the entire eligible list at this site, not merely the subset just used to prove its lower cardinality bound. Every eligible label has its initial endpoint in B v B_v B v v v v s 0 s_0 s 0 H 1 D H_1D H 1 D (44) for at least nine tenths of the entire eligible list, with choices uniform in the certificate.
Choose an integer M ≥ 1 M\ge1 M ≥ 1
G D = ⌊ ℓ D 4 M ⌋ . G_D=\left\lfloor\frac{\ell D}{4M}\right\rfloor. G D = ⌊ 4 M ℓ D ⌋ . At a site with at least M G D MG_D M G D M G D MG_D M G D G D G_D G D M M M
Fix one group indexed by g ∈ { 1 , … , G D } g\in\{1,\ldots,G_D\} g ∈ { 1 , … , G D } I \mathcal{I} I exposed datum B \mathcal{B} B
These last trajectories form the frozen family F \mathcal{F} F B \mathcal{B} B B \mathcal{B} B
For i ∈ I i \in\mathcal{I} i ∈ I v v v B i B_i B i 2 t 2t 2 t
P i ( d ω i ) = 1 C i ( F ) B i ( d ω i ) a i , a i = B i ( C i ( F ) ) . (45) P_i(\mathrm{d}\omega_i) = \frac{\mathbf{1}_{C_i(F)} B_i(\mathrm{d}\omega_i)}{a_i}, \qquad a_i = B_i(C_i(F)). \tag*{(45)} P i ( d ω i ) = a i 1 C i ( F ) B i ( d ω i ) , a i = B i ( C i ( F )) . (45) The event C i ( F ) C_i(F) C i ( F ) (43) .
Proposition 6.2 (Exact conditional law). For almost every exposed datum the normalizers in Equation (45) are positive, and the conditional law of the unexposed trajectories is
λ I ( d ω I ) = 1 Z I 1 { all pairs in I avoid each other } ∏ i ∈ I P i ( d ω i ) , Z I > 0. (46) \lambda_{\mathcal{I}}(\mathrm{d}\omega_{\mathcal{I}}) = \frac{1}{Z_{\mathcal{I}}}\mathbf{1}_{\{\text{all pairs in }\mathcal{I}\text{ avoid each other}\}} \prod_{i\in\mathcal{I}} P_i(\mathrm{d}\omega_i), \qquad Z_{\mathcal{I}} > 0. \tag*{(46)} λ I ( d ω I ) = Z I 1 1 { all pairs in I avoid each other } i ∈ I ∏ P i ( d ω i ) , Z I > 0. (46) All reference trajectories obey their own regularity and confinement cutoffs. Each label has at most C M CM C M C C C
Proof. Condition first on the endpoints. By Lemma 4.1, the free reference is a product of torus bridges and the remaining interaction is the hard-avoidance indicator. At fixed endpoints, eligibility of an individual path consists of its own regularity and confinement conditions: its starting cell, endpoint displacement and endpoint separation are already fixed. The probabilities of all allocations depend only on the recorded eligibility indicators, and are consequently constant on the conditional fiber.
Now expose all paths outside I \mathcal{I} I K K K I \mathcal{I} I (46) by disintegration on the continuous-path spaces. Equivalently, integrate the asserted formula against bounded functions of the exposed and unexposed variables and use Fubini; all factors just identified reproduce Equation (30) and the allocation probabilities.
The integral of the conditional density is positive for almost every sampled datum. Its positivity implies positivity of each a i a_i a i Z I Z_{\mathcal{I}} Z I r < 1 r < 1 r < 1 M M M (46) is a conditioned product, not a claim that its trajectories are independent.
Accessibility and the probability of bad cells
For i ∈ I i \in\mathcal{I} i ∈ I B w B_w B w
T i ( w ) = { u ∈ B w : Q i u ( C i ( F ) ) ≥ 0.9 } . T_i(w) = \{u \in B_w : Q_i^u(C_i(F)) \ge0.9\}. T i ( w ) = { u ∈ B w : Q i u ( C i ( F )) ≥ 0.9 } . A label is accessible if ∣ T i ( w ) ∣ ≥ 0.99 |T_i(w)| \ge0.99 ∣ T i ( w ) ∣ ≥ 0.99 T i ( w ) T_i(w) T i ( w ) B \mathcal{B} B A v \mathcal{A}_v A v v v v good if ∣ A v ∣ ≥ M / 2 |A_v| \ge M/2 ∣ A v ∣ ≥ M /2 bad otherwise. The sets in (6.6) are measurable: bridge kernels are measurable in their endpoints, and regularity, confinement and strict avoidance are Borel events of continuous paths.
Proposition 6.3 (Unconditional supply of good cells). For any prescribed p > 0 p > 0 p > 0 M M M M M M D D D K K K
P B { A ⊂ { b a d s i t e s } } ≤ p ∣ A ∣ (47) \mathbb{P}_{\mathcal{B}}\{A \subset\{\mathit{bad\ sites}\}\} \le p^{|A|} \tag*{(47)} P B { A ⊂ { bad sites }} ≤ p ∣ A ∣ (47) for every set A A A α \alpha α M M M D D D
Proof. Choose p 1 < p / 2 p_1 < p/2 p 1 < p /2 ℓ D / 2 ≥ M G D \ell D/2 \ge M G_D ℓ D /2 ≥ M G D (44) . For a fixed group, its M M M k k k k k k 10 − k 10^{-k} 1 0 − k
P { fewer than M / 2 stronger labels in the group ∣ full sample } ≤ q M : = 2 M 10 − M / 2 . (48) \mathbb{P}\{\text{fewer than } M/2 \text{ stronger labels in the group} \mid\text{full sample}\} \le q_M := 2^M 10^{-M/2}. \tag*{(48)} P { fewer than M /2 stronger labels in the group ∣ full sample } ≤ q M := 2 M 1 0 − M /2 . (48) This tends to zero exponentially in M M M
A label with the stronger property is accessible after allocation: its frozen obstacle family is a subset of all other sampled paths, so removing obstacles can only increase its success probabilities. Moreover the auxiliary sampling tests at distinct certified cells are independent conditional on the full sample. For each subset A 0 ⊂ A A_0 \subset A A 0 ⊂ A A 0 A_0 A 0 p 1 ∣ A 0 ∣ q M ∣ A ∖ A 0 ∣ p_1^{|A_0|}q_M^{|A\setminus A_0|} p 1 ∣ A 0 ∣ q M ∣ A ∖ A 0 ∣ (36) . Summing over A 0 A_0 A 0 (47) by ( p 1 + q M ) ∣ A ∣ (p_1+q_M)^{|A|} ( p 1 + q M ) ∣ A ∣ M M M q M < p / 2 q_M < p/2 q M < p /2 □ \square □
One-path comparisons at fixed exposed data
The remaining estimates apply at a fixed datum, even when its unexposed trajectories no longer satisfy any full-sample certificate. Let i i i B w B_w B w U ⊂ T i ( w ) U \subset T_i(w) U ⊂ T i ( w ) 0.98 0.98 0.98 Π i U \Pi_i^U Π i U u u u U U U Q i u Q_i^u Q i u C i ( F ) C_i(F) C i ( F )
Proposition 6.4 (Density and deterministic compatibility). There is a constant G ≥ 2 G \ge2 G ≥ 2 t t t D , K , M D,K,M D , K , M
Π i U ≪ P i , 0 ≤ d Π i U d P i ≤ G , ∫ d Π i U d P i d P i = 1. (49) \Pi_i^U \ll P_i,\qquad0 \le\frac{\mathrm{d}\Pi_i^U}{\mathrm{d}P_i} \le G,\qquad\int\frac{\mathrm{d}\Pi_i^U}{\mathrm{d}P_i}\,\mathrm{d}P_i = 1. \tag*{(49)} Π i U ≪ P i , 0 ≤ d P i d Π i U ≤ G , ∫ d P i d Π i U d P i = 1. (49) There is also η D → 0 \eta_D \to0 η D → 0 I \mathcal{I} I P i P_i P i Π i U \Pi_i^U Π i U η D \eta_D η D
η D = C t ( tr + r D 3 γ h 0 − 1 / 2 ) ≤ C t ′ D − 1 / 4 . (50) \eta_D = C_t\left(\operatorname{tr} + rD^{3\gamma}h_0^{-1/2}\right) \le C'_tD^{-1/4}. \tag*{(50)} η D = C t ( tr + r D 3 γ h 0 − 1/2 ) ≤ C t ′ D − 1/4 . (50) Proof. The midpoint density g i g_i g i B i B_i B i ( Y i + Z i ) / 2 (Y_i+Z_i)/2 ( Y i + Z i ) /2 t I tI t I c t > 0 c_t>0 c t > 0 v v v v v v B i B_i B i Q i u Q_i^u Q i u
a i = ∫ g i ( u ) Q i u ( C i ( F ) ) d u ≥ 0.99 ⋅ 0.9 c t = : a t > 0. a_i=\int g_i(u)Q_i^u(C_i(F))\,\mathrm{d}u\ge0.99\cdot0.9\,c_t=:a_t>0. a i = ∫ g i ( u ) Q i u ( C i ( F )) d u ≥ 0.99 ⋅ 0.9 c t =: a t > 0. At a successful trajectory with midpoint u u u P i P_i P i
a i 1 U ( u ) ∣ U ∣ g i ( u ) Q i u ( C i ( F ) ) ≤ 1 0.98 ⋅ 0.9 c t , \frac{a_i\mathbf{1}_U(u)}{|U|g_i(u)Q_i^u(C_i(F))}\le\frac{1}{0.98\cdot0.9\,c_t}, ∣ U ∣ g i ( u ) Q i u ( C i ( F )) a i 1 U ( u ) ≤ 0.98 ⋅ 0.9 c t 1 , since a i ≤ 1 a_i\le1 a i ≤ 1 G G G (49) ; its final identity holds because both measures are probabilities.
For collision with the fixed opponent, first use the unconditioned bridge B i B_i B i P i P_i P i C ( t r + r D 3 γ h 0 − 1 / 2 ) C(tr+rD^3\gamma h_0^{-1/2}) C ( t r + r D 3 γ h 0 − 1/2 ) a i ≥ a t a_i\ge a_t a i ≥ a t P i P_i P i U U U ∣ U ∣ ≥ 0.98 |U|\ge0.98 ∣ U ∣ ≥ 0.98 u u u 0.9 0.9 0.9 i i i
Finally γ = 1 / 12 \gamma=1/12 γ = 1/12 h 0 = D − 1 h_0=D^{-1} h 0 = D − 1 r = α / D r=\alpha/D r = α / D r D 3 γ h 0 − 1 / 2 = α D − 1 / 4 rD^3\gamma h_0^{-1/2}=\alpha D^{-1/4} r D 3 γ h 0 − 1/2 = α D − 1/4 t r = O t ( D − 1 ) tr=O_t(D^{-1}) t r = O t ( D − 1 ) α \alpha α (50) , with constants depending on the already fixed t t t M M M
Each changed path has only C M CM C M C M η D CM\eta_D C M η D M M M G G G D D D
Paths through a sparse set of obstacles
The comparison of particle laws will move labels along paths of usable cells. Two such paths may each be long, but their mutual encounters must have a bounded exponential moment. We combine a purely geometric skeleton law from the companion article with the joint bad-cell bound. The detours and their probability estimates are proved here; the bad cells need not be independent or translation invariant.
Write L K = ( Z / K Z ) 3 \mathcal{L}_K=(\mathbb{Z}/K\mathbb{Z})^3 L K = ( Z / K Z ) 3 d ∞ d_\infty d ∞ ∗ * ∗ D \mathcal{D} D B ⊂ L K \mathcal{B}\subset\mathcal{L}_K B ⊂ L K
P ( A ⊂ B ) ≤ p ∣ A ∣ ( A ⊂ L K ) . (51) \mathbb{P}(A\subset\mathcal{B})\le p^{|A|}\qquad(A\subset\mathcal{L}_K). \tag*{(51)} P ( A ⊂ B ) ≤ p ∣ A ∣ ( A ⊂ L K ) . (51) An ordered pair ( v , w ) (v,w) ( v , w ) n n n ⌈ K / 10 ⌉ ≤ n ≤ ⌊ K / 8 ⌋ \lceil K/10\rceil\le n\le\lfloor K/8\rfloor ⌈ K /10 ⌉ ≤ n ≤ ⌊ K /8 ⌋ d 2 , d 3 d_2,d_3 d 2 , d 3 ∣ d i ∣ ≤ n |d_i| \le n ∣ d i ∣ ≤ n ( n , d 2 , d 3 ) (n,d_2,d_3) ( n , d 2 , d 3 ) v 0 = 1 / 2000 v_0=1/2000 v 0 = 1/2000 π , π ′ \pi,\pi' π , π ′
J R ( π , π ′ ) = ∑ z ∈ π ∑ z ′ ∈ π ′ 1 { d ∞ ( z , z ′ ) ≤ R } . (52) J_R(\pi,\pi')=\sum_{z\in\pi}\sum_{z'\in\pi'}\mathbf{1}_{\{d_\infty(z,z')\le R\}}. \tag*{(52)} J R ( π , π ′ ) = z ∈ π ∑ z ′ ∈ π ′ ∑ 1 { d ∞ ( z , z ′ ) ≤ R } . (52) Here a simple path visits each vertex at most once.
Proposition 7.1 (Good paths with bounded encounters). Fix an integer R ≥ 1 R \ge1 R ≥ 1 p 0 , κ > 0 p_0,\kappa> 0 p 0 , κ > 0 C p a t h < ∞ C_{\mathrm{path}} < \infty C path < ∞ K 0 K_0 K 0 R R R K ≥ K 0 K \ge K_0 K ≥ K 0 (51) holds with p ≤ p 0 p \le p_0 p ≤ p 0 ( v , w ) (v,w) ( v , w ) D D D E v , w E_{v,w} E v , w 1 / 2 1/2 1/2 v v v w w w D D D
E [ 1 E v , w E [ e κ J R ( π , π ′ ) ∣ D ] ] ≤ C p a t h . (53) \mathbb{E}\left[\mathbf{1}_{E_{v,w}}\mathbb{E}\left[e^{\kappa J_R(\pi,\pi')} \mid D\right]\right]\le C_{\mathrm{path}}. \tag*{(53)} E [ 1 E v , w E [ e κ J R ( π , π ′ ) ∣ D ] ] ≤ C path . (53) The constants are uniform in the datum, its law, and the endpoint pair. For every sufficiently large K K K v 0 K 6 v_0K^6 v 0 K 6
The event is determined before drawing the paths. The expectation in (53) averages over one common environment and two fresh path samples; it is not a bound for every fixed environment. We will use the proposition with R = 30 R=30 R = 30
The geometric input is the explicit law in [29 ] , Section 8.1, pp. 22–24; Lemma 8.1 and Equations (24)–(25), p. 23]. Its use of unpredictability and intersection moments follows Benjamini, Pemantle and Peres [5 ] ; finite-endpoint synchronization averages were developed by Abbe, Massoulié, Montanari, Sly and Srivastava [1 ] . Garban and Spencer [21 ] adapt this path-averaging method to symmetry breaking in classical disordered spin systems. We restate the law and both estimates independently of any gas parameters.
Fix an admissible pair, choose a lift of v v v w w w v + ( n , d 2 , d 3 ) v+(n,d_2,d_3) v + ( n , d 2 , d 3 ) j ∈ { 2 , 3 } j\in\{2,3\} j ∈ { 2 , 3 } U ℓ , k ( j ) U_{\ell,k}^{(j)} U ℓ , k ( j ) [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] ℓ , k ≥ 0 \ell,k\ge0 ℓ , k ≥ 0
v i ( j ) = c v e l ∑ ℓ ≥ 0 2 − ℓ / 4 U ℓ , ⌊ ( i − 1 ) / 2 ℓ ⌋ ( j ) , Z t ( j ) = ∑ i = 1 t v i ( j ) , Z 0 ( j ) = 0. (54) v_i^{(j)}=c_{\mathrm{vel}}\sum_{\ell\ge0}2^{-\ell/4}U_{\ell,\lfloor(i-1)/2^\ell\rfloor}^{(j)},\qquad Z_t^{(j)}=\sum_{i=1}^{t}v_i^{(j)},\qquad Z_0^{(j)}=0. \tag*{(54)} v i ( j ) = c vel ℓ ≥ 0 ∑ 2 − ℓ /4 U ℓ , ⌊( i − 1 ) / 2 ℓ ⌋ ( j ) , Z t ( j ) = i = 1 ∑ t v i ( j ) , Z 0 ( j ) = 0. (54) As in the companion, fix c v e l = ( 1 − 2 − 1 / 4 ) / 2 c_{\mathrm{vel}}=(1-2^{-1/4})/2 c vel = ( 1 − 2 − 1/4 ) /2 c v e l ∑ ℓ ≥ 0 2 − ℓ / 4 = 1 / 2 c_{\mathrm{vel}}\sum_{\ell\ge0}2^{-\ell/4}=1/2 c vel ∑ ℓ ≥ 0 2 − ℓ /4 = 1/2
u t = v + ( t , ⌊ t d 2 / n + Z min ( t , n − t ) ( 2 ) ⌋ , ⌊ t d 3 / n + Z min ( t , n − t ) ( 3 ) ⌋ ) , 0 ≤ t ≤ n . u_t=v+\left(t,\left\lfloor td_2/n+Z_{\min(t,n-t)}^{(2)}\right\rfloor,\left\lfloor td_3/n+Z_{\min(t,n-t)}^{(3)}\right\rfloor\right),\qquad0\le t\le n. u t = v + ( t , ⌊ t d 2 / n + Z m i n ( t , n − t ) ( 2 ) ⌋ , ⌊ t d 3 / n + Z m i n ( t , n − t ) ( 3 ) ⌋ ) , 0 ≤ t ≤ n . The endpoints are v , w v,w v , w Γ \Gamma Γ
For two independent copies Γ , Γ ′ \Gamma, \Gamma' Γ , Γ ′ R s ≥ 1 R_s \ge1 R s ≥ 1
J o c c ( R s ) = ∑ z ∈ Γ 1 { d ∞ ( z , Γ ′ ) ≤ R s } + ∑ z ′ ∈ Γ ′ 1 { d ∞ ( z ′ , Γ ) ≤ R s } . (55) J_{\mathrm{occ}}^{(R_s)} = \sum_{z \in\Gamma} \mathbf{1}_{\{d_\infty(z,\Gamma') \le R_s\}} + \sum_{z' \in\Gamma'} \mathbf{1}_{\{d_\infty(z',\Gamma) \le R_s\}}. \tag*{(55)} J occ ( R s ) = z ∈ Γ ∑ 1 { d ∞ ( z , Γ ′ ) ≤ R s } + z ′ ∈ Γ ′ ∑ 1 { d ∞ ( z ′ , Γ ) ≤ R s } . (55) Both sums count occurrences, including any repetitions. For a fixed B > 0 B > 0 B > 0
S B ( Γ , Γ ′ ) = ∑ z ∈ Γ e − d ∞ ( z , Γ ′ ) / B , S_B(\Gamma,\Gamma') = \sum_{z \in\Gamma} e^{-d_\infty(z,\Gamma')/B}, S B ( Γ , Γ ′ ) = z ∈ Γ ∑ e − d ∞ ( z , Γ ′ ) / B , again counting occurrences. The companion estimates are
E e α s J o c c ( R s ) ≤ C h i t , α s = c h i t ( 1 + R s ) 2 , (56) \mathbb{E}e^{\alpha_s J_{\mathrm{occ}}^{(R_s)}} \le C_{\mathrm{hit}}, \qquad\alpha_s = \frac{c_{\mathrm{hit}}}{(1+R_s)^2}, \tag*{(56)} E e α s J occ ( R s ) ≤ C hit , α s = ( 1 + R s ) 2 c hit , (56) E e ξ ( B ) S B ( Γ , Γ ′ ) ≤ C ( B ) , ξ ( B ) > 0. (57) \mathbb{E}e^{\xi(B)S_B(\Gamma,\Gamma')} \le C(B), \qquad\xi(B)>0. \tag*{(57)} E e ξ ( B ) S B ( Γ , Γ ′ ) ≤ C ( B ) , ξ ( B ) > 0. (57) Here c h i t > 0 c_{\mathrm{hit}} > 0 c hit > 0 C h i t < ∞ C_{\mathrm{hit}} < \infty C hit < ∞ B B B K K K D \mathcal{D} D
For later use we convert the occurrence count to our pair count. Every first-coordinate level of a lifted skeleton contains at most five occurrences: level zero has only the initial vertex, and all vertices created in the t t t v 1 + t v_1+t v 1 + t n ≤ K / 8 n \le K/8 n ≤ K /8 Γ \Gamma Γ 5 ( 2 R + 1 ) 5(2R+1) 5 ( 2 R + 1 ) R R R R s = R R_s=R R s = R
J R ( Γ , Γ ′ ) ≤ 5 ( 2 R + 1 ) J o c c ( R ) . (58) J_R(\Gamma,\Gamma') \le5(2R+1)J_{\mathrm{occ}}^{(R)}. \tag*{(58)} J R ( Γ , Γ ′ ) ≤ 5 ( 2 R + 1 ) J occ ( R ) . (58) The left side uses the double occurrence sum from Equation (52) . This conversion uses the actual refinement law, as well as its moment bounds.
Finite components and their exterior boundaries
We record the deterministic facts used to move a lattice path around bad vertices. For a finite *-connected nonempty set C ⊂ Z 3 C \subset\mathbb{Z}^3 C ⊂ Z 3 fill ( C ) \operatorname{fill}(C) fill ( C ) C C C Z 3 ∖ C \mathbb{Z}^3 \setminus C Z 3 ∖ C C C C
Lemma 7.2 (Exterior shell). *The set fill ( C ) \operatorname{fill}(C) fill ( C ) C C C C C C ∂ ∗ e x t C \partial^{\mathrm{ext}}_* C ∂ ∗ ext C 26 ∣ C ∣ 26|C| 26∣ C ∣ C C C -component of a larger bad set, every vertex of this shell is good.
Proof. A vertex outside the coordinate box has a coordinate ray to infinity that misses C C C [34 ] , arXiv version 2, Theorem 4: for a finite *-connected subset of Z 3 \mathbb{Z}^3 Z 3 C C C C C C C C C
For an integer a ≥ 1 a \ge1 a ≥ 1 a a a a a a
Lemma 7.3 (Small periodic components). Suppose K > 2 a K > 2a K > 2 a a a a L K \mathcal{L}_K L K m m m a m < K am < K am < K Z 3 \mathbb{Z}^3 Z 3 a ( m − 1 ) a(m-1) a ( m − 1 )
Proof. Choose a spanning tree, lift its root, and lift each tree edge by its unique displacement in [ − a , a ] 3 [-a,a]^3 [ − a , a ] 3 a ( m − 1 ) a(m-1) a ( m − 1 ) K Z 3 K\mathbb{Z}^3 K Z 3 a m < K am<K am < K K Z 3 K\mathbb{Z}^3 K Z 3
We will also use a basic counting bound. For each fixed range a a a A a A_a A a m m m A a m A_a^m A a m Z 3 \mathbb{Z}^3 Z 3 2 ( m − 1 ) 2(m-1) 2 ( m − 1 ) A a A_a A a m = 1 m=1 m = 1 (51) , this gives
P ( a range- a bad component has at least m vertices ) ≤ K 3 ( A a p ) m . (59) \mathbb{P}(\text{a range-}a\text{ bad component has at least }m\text{ vertices}) \le K^3(A_a p)^m. \tag*{(59)} P ( a range- a bad component has at least m vertices ) ≤ K 3 ( A a p ) m . (59) A component of size at least m m m m m m
Fix a 1 = R + 2 a_1=R+2 a 1 = R + 2 a 1 a_1 a 1
m K = ⌊ K 10 a 1 ⌋ . m_K=\left\lfloor\frac{K}{10a_1}\right\rfloor. m K = ⌊ 10 a 1 K ⌋ . For large K K K E v , w \mathcal{E}_{v,w} E v , w
If the fill of a lifted *-component of size m m m m m m C m 3 Cm^3 C m 3 K K K (51) thus yields
P ( E v , w ) ≤ K 3 ( A a 1 p ) m K + C ∑ m ≥ 1 m 3 ( A 1 p ) m . (60) \mathbb{P}(\mathcal{E}_{v,w}) \le K^3(A_{a_1}p)^{m_K}+C\sum_{m\ge1}m^3(A_1p)^m. \tag*{(60)} P ( E v , w ) ≤ K 3 ( A a 1 p ) m K + C m ≥ 1 ∑ m 3 ( A 1 p ) m . (60) The constant includes both endpoints. First make p 0 p_0 p 0 1 / 4 1/4 1/4 K 0 K_0 K 0 1 / 4 1/4 1/4 p ≤ p 0 p\le p_0 p ≤ p 0 K ≥ K 0 K\ge K_0 K ≥ K 0 P ( E v , w ) ≥ 1 / 2 \mathbb{P}(\mathcal{E}_{v,w})\ge1/2 P ( E v , w ) ≥ 1/2 p 0 p_0 p 0
Nearest-neighbor detours
Draw the skeleton of Section 7.1 independently of the datum. Its folding in Equation (7.5) fixes the final endpoint without conditioning the random process on a future value. We now adapt this nearest-neighbor walk to the bad set while keeping the endpoint rejection event fixed.
On E v , w \mathcal{E}_{v,w} E v , w C C C C C C ∂ ∗ C \partial^* C ∂ ∗ C C C C C C C fill ( C ) \operatorname{fill}(C) fill ( C )
Every component treated in this procedure was met by the original nominal refinement: any remaining bad vertex still belongs to that original walk. Project the final walk to the torus and erase loops in chronological order. The result is a simple good nearest-neighbor path from v v v w w w ∗ * ∗ Γ \Gamma Γ
We now bound the encounters introduced by this construction. Use two independent nominal samples Γ , Γ ′ \Gamma,\Gamma' Γ , Γ ′ π , π ′ \pi,\pi' π , π ′ H H H a 1 a_1 a 1 C R = 52 ( 2 R + 1 ) 3 C_R=52(2R+1)^3 C R = 52 ( 2 R + 1 ) 3 5 ( 2 R + 1 ) 5(2R+1) 5 ( 2 R + 1 ) (58) . On E v , w \mathcal{E}_{v,w} E v , w
J R ( π , π ′ ) ≤ J R ( Γ , Γ ′ ) + C R ∑ H ∣ H ∣ 1 { d ∞ ( H , Γ ) ≤ a 1 , d ∞ ( H , Γ ′ ) ≤ a 1 } . (61) J_R(\pi,\pi') \le J_R(\Gamma,\Gamma')
+ C_R \sum_H |H| \mathbf{1}_{\{d_\infty(H,\Gamma)\le a_1,\ d_\infty(H,\Gamma')\le a_1\}}. \tag*{(61)} J R ( π , π ′ ) ≤ J R ( Γ , Γ ′ ) + C R H ∑ ∣ H ∣ 1 { d ∞ ( H , Γ ) ≤ a 1 , d ∞ ( H , Γ ′ ) ≤ a 1 } . (61) For nominal walks, J R J_R J R
To prove (61) , first count encounters where both vertices are nominal. These are bounded by the first term. If a nonnominal vertex belongs to the shell of C C C C C C R + 1 ≤ a 1 R+1\le a_1 R + 1 ≤ a 1 C C C C , C ′ C,C' C , C ′ d ∞ ( C , C ′ ) ≤ R + 2 ≤ a 1 d_\infty(C,C')\le R+2\le a_1 d ∞ ( C , C ′ ) ≤ R + 2 ≤ a 1 a 1 a_1 a 1 H H H H H H 26 ∣ H ∣ 26|H| 26∣ H ∣ ∗ * ∗ H H H ( 2 R + 1 ) 3 (2R+1)^3 ( 2 R + 1 ) 3
Averaging the environment and applying both moments
Fix both complete nominal paths. Their independence from D \mathcal{D} D (61) as a product over the actual range-a 1 a_1 a 1 1 + x ≤ e x 1+x\le e^x 1 + x ≤ e x
E D [ 1 E v , w exp ( κ C R ∑ H ∣ H ∣ 1 { d ∞ ( H , Γ ) , d ∞ ( H , Γ ′ ) ≤ a 1 } ) ] ≤ exp ( ∑ F ( p e κ C R ) ∣ F ∣ 1 { d ∞ ( F , Γ ) , d ∞ ( F , Γ ′ ) ≤ a 1 } ) . (62) \mathbb{E}_{\mathcal{D}}\left[
\mathbf{1}_{\mathcal{E}_{v,w}}
\exp\left(
\kappa C_R \sum_H |H|
\mathbf{1}_{\{d_\infty(H,\Gamma),d_\infty(H,\Gamma')\le a_1\}}
\right)
\right]
\le
\exp\left(
\sum_F \left(p e^{\kappa C_R}\right)^{|F|}
\mathbf{1}_{\{d_\infty(F,\Gamma),d_\infty(F,\Gamma')\le a_1\}}
\right). \tag*{(62)} E D [ 1 E v , w exp ( κ C R H ∑ ∣ H ∣ 1 { d ∞ ( H , Γ ) , d ∞ ( H , Γ ′ ) ≤ a 1 } ) ] ≤ exp ( F ∑ ( p e κ C R ) ∣ F ∣ 1 { d ∞ ( F , Γ ) , d ∞ ( F , Γ ′ ) ≤ a 1 } ) . (62) where F F F a 1 a_1 a 1 e κ C R ∣ F ∣ − 1 e^{\kappa C_R|F|}-1 e κ C R ∣ F ∣ − 1 e κ C R ∣ F ∣ e^{\kappa C_R|F|} e κ C R ∣ F ∣
We bound the exponent by the soft-distance statistic in (57) . If a contributing set F F F m m m z ∈ Γ z \in\Gamma z ∈ Γ a 1 a_1 a 1 F F F ( 2 a 1 + 1 ) 3 (2a_1+1)^3 ( 2 a 1 + 1 ) 3 z z z A a 1 m A_{a_1}^m A a 1 m F F F m − 1 m-1 m − 1 a 1 a_1 a 1 F F F a 1 a_1 a 1 Γ ′ \Gamma' Γ ′
d ∞ ( z , Γ ′ ) ≤ 2 a 1 + a 1 ( m − 1 ) = a 1 ( m + 1 ) . d_\infty(z,\Gamma') \le2a_1+a_1(m-1)=a_1(m+1). d ∞ ( z , Γ ′ ) ≤ 2 a 1 + a 1 ( m − 1 ) = a 1 ( m + 1 ) . This argument applies even to wrapping candidate sets; only the actual components used for detours required the earlier lift and rejection test. Absorb the root choices into a constant A R ≥ 1 A_R \ge1 A R ≥ 1 (62) is at most
∑ z ∈ Γ ∑ m ≥ 1 q m 1 { d ∞ ( z , Γ ′ ) ≤ a 1 ( m + 1 ) } , q = A R p e κ C R . (63) \sum_{z\in\Gamma}\sum_{m\ge1}q^m\mathbf{1}_{\{d_\infty(z,\Gamma')\le a_1(m+1)\}}, \qquad q=A_Rpe^{\kappa C_R}. \tag*{(63)} z ∈ Γ ∑ m ≥ 1 ∑ q m 1 { d ∞ ( z , Γ ′ ) ≤ a 1 ( m + 1 )} , q = A R p e κ C R . (63) All constants here depend only on the fixed R R R
Require q ≤ e − 2 q\le e^{-2} q ≤ e − 2 d ≤ a 1 ( m + 1 ) d\le a_1(m+1) d ≤ a 1 ( m + 1 ) q m q^m q m
q m = q m / 2 q m / 2 ≤ q m / 2 e − m ≤ e q m / 2 e − d / a 1 . q^m=q^{m/2}q^{m/2}\le q^{m/2}e^{-m}\le eq^{m/2}e^{-d/a_1}. q m = q m /2 q m /2 ≤ q m /2 e − m ≤ e q m /2 e − d / a 1 . Summing the first factor over m ≥ 1 m\ge1 m ≥ 1
∑ m ≥ 1 q m 1 { d ≤ a 1 ( m + 1 ) } ≤ η ( p , κ ) e − d / a 1 , η ( p , κ ) = e q 1 − q . (64) \sum_{m\ge1}q^m\mathbf{1}_{\{d\le a_1(m+1)\}}\le\eta(p,\kappa)e^{-d/a_1},\qquad\eta(p,\kappa)=\frac{e\sqrt q}{1-\sqrt q}. \tag*{(64)} m ≥ 1 ∑ q m 1 { d ≤ a 1 ( m + 1 )} ≤ η ( p , κ ) e − d / a 1 , η ( p , κ ) = 1 − q e q . (64) In particular η ( p , κ ) → 0 \eta(p,\kappa)\to0 η ( p , κ ) → 0 p → 0 p\to0 p → 0 κ \kappa κ B = a 1 B=a_1 B = a 1 (61) , (58) , and (62) –(64) give
E D [ 1 E v , w e κ J R ( π , π ′ ) ∣ Γ , Γ ′ ] ≤ exp ( κ C R J o c c ( R ) + η ( p , κ ) S B ( Γ , Γ ′ ) ) . (65) \mathbb{E}_{D}\left[\mathbf{1}_{\mathcal{E}_{v,w}}e^{\kappa J_R(\pi,\pi')}\mid\Gamma,\Gamma'\right]\le\exp\left(\kappa C_RJ_{\mathrm{occ}}^{(R)}+\eta(p,\kappa)S_B(\Gamma,\Gamma')\right). \tag*{(65)} E D [ 1 E v , w e κ J R ( π , π ′ ) ∣ Γ , Γ ′ ] ≤ exp ( κ C R J occ ( R ) + η ( p , κ ) S B ( Γ , Γ ′ ) ) . (65) On the left, the adapted paths are used only on E v , w \mathcal{E}_{v,w} E v , w
First choose κ > 0 \kappa>0 κ > 0 2 κ C R ≤ α R 2\kappa C_R\le\alpha_R 2 κ C R ≤ α R α R = c h i t / ( 1 + R ) 2 \alpha_R=c_{\mathrm{hit}}/(1+R)^2 α R = c hit / ( 1 + R ) 2 p 0 > 0 p_0>0 p 0 > 0 q ≤ e − 2 q\le e^{-2} q ≤ e − 2 2 η ( p , κ ) ≤ ξ ( B ) 2\eta(p,\kappa)\le\xi(B) 2 η ( p , κ ) ≤ ξ ( B ) p ≤ p 0 p\le p_0 p ≤ p 0 K 0 K_0 K 0
E exp ( κ C R J o c c ( R ) + η S B ) ≤ ( E e 2 κ C R J o c c ( R ) ) 1 / 2 ( E e 2 η S B ) 1 / 2 ≤ c h i t C ( B ) = : C p a t h < ∞ . \mathbb{E}\exp\left(\kappa C_RJ_{\mathrm{occ}}^{(R)}+\eta S_B\right)
\le\left(\mathbb{E}e^{2\kappa C_RJ_{\mathrm{occ}}^{(R)}}\right)^{1/2}
\left(\mathbb{E}e^{2\eta S_B}\right)^{1/2}
\le\sqrt{c_{\mathrm{hit}}C(B)}=:C_{\mathrm{path}}<\infty. E exp ( κ C R J occ ( R ) + η S B ) ≤ ( E e 2 κ C R J occ ( R ) ) 1/2 ( E e 2 η S B ) 1/2 ≤ c hit C ( B ) =: C path < ∞. Together with (65) , this proves the annealed bound (53) . Both path draws use one common datum and independent skeleton seeds. The event retains its original mass; there is no renormalization by conditioning on acceptance and no assertion of a bound for each fixed environment.
Finally, for K ≥ 80 K\ge80 K ≥ 80 [ ⌈ K / 10 ⌉ , ⌊ K / 8 ⌋ ] [\lceil K/10\rceil,\lfloor K/8\rfloor] [⌈ K /10 ⌉ , ⌊ K /8 ⌋] K / 40 − 1 ≥ K / 80 K/40-1\ge K/80 K /40 − 1 ≥ K /80 n n n ( 2 n + 1 ) 2 ≥ K 2 / 25 (2n+1)^2\ge K^2/25 ( 2 n + 1 ) 2 ≥ K 2 /25 K K K 2 n < K 2n<K 2 n < K K 3 K^3 K 3 K 6 / 2000 = υ 0 K 6 K^6/2000=\upsilon_0K^6 K 6 /2000 = υ 0 K 6
We now prove that the nonnegative ground function Φ \Phi Φ
All constants in this section are independent of D D D K K K
Affinity and a common finite measure
For finite nonnegative measures P , Q P,Q P , Q
H ( P , Q ) = ∫ p q d ξ , p = d P d ξ , q = d Q d ξ , ξ = P + Q . H(P,Q)=\int\sqrt{pq}\,\mathrm{d}\xi,\qquad p=\frac{\mathrm{d}P}{\mathrm{d}\xi},\qquad q=\frac{\mathrm{d}Q}{\mathrm{d}\xi},\qquad\xi=P+Q. H ( P , Q ) = ∫ pq d ξ , p = d ξ d P , q = d ξ d Q , ξ = P + Q . Using another dominating measure gives the same value. We will use three elementary properties. If P ≥ P ′ P\ge P' P ≥ P ′ Q ≥ Q ′ Q\ge Q' Q ≥ Q ′ H ( P , Q ) ≥ H ( P ′ , Q ′ ) H(P,Q)\ge H(P',Q') H ( P , Q ) ≥ H ( P ′ , Q ′ )
H ( ∑ j P j , ∑ j Q j ) ≥ ∑ j H ( P j , Q j ) . (66) H\left(\sum_j P_j,\sum_j Q_j\right)\ge\sum_j H(P_j,Q_j). \tag*{(66)} H ( j ∑ P j , j ∑ Q j ) ≥ j ∑ H ( P j , Q j ) . (66) Indeed, pointwise Cauchy–Schwarz gives ∑ j p j q j ≤ ( ∑ j p j ) ( ∑ j q j ) \sum_j\sqrt{p_jq_j}\le\sqrt{(\sum_jp_j)(\sum_jq_j)} ∑ j p j q j ≤ ( ∑ j p j ) ( ∑ j q j ) E ( p q ∣ output ) ≤ E ( p ∣ output ) E ( q ∣ output ) \mathbb{E}(\sqrt{pq}\mid\text{output})\le\sqrt{\mathbb{E}(p\mid\text{output})\mathbb{E}(q\mid\text{output})} E ( pq ∣ output ) ≤ E ( p ∣ output ) E ( q ∣ output )
Lemma 8.1 (A common finite measure). Let P , Q P,Q P , Q ν \nu ν m > 0 m>0 m > 0
∫ ( d ν d P ) 2 d P ≤ C P , ∫ ( d ν d Q ) 2 d Q ≤ C Q , \int\left(\frac{\mathrm{d}\nu}{\mathrm{d}P}\right)^2\mathrm{d}P\le C_P,\qquad
\int\left(\frac{\mathrm{d}\nu}{\mathrm{d}Q}\right)^2\mathrm{d}Q\le C_Q, ∫ ( d P d ν ) 2 d P ≤ C P , ∫ ( d Q d ν ) 2 d Q ≤ C Q , then H ( P , Q ) ≥ m 2 / C P C Q H(P,Q)\ge m^2/\sqrt{C_PC_Q} H ( P , Q ) ≥ m 2 / C P C Q
Proof. With densities p , q , s p,q,s p , q , s 4 , 4 , 2 4,4,2 4 , 4 , 2
m = ∫ ( s 2 p ) 1 / 4 ( s 2 q ) 1 / 4 ( p q ) 1 / 4 ≤ C P 1 / 4 C Q 1 / 4 H ( P , Q ) 1 / 2 . m=\int\left(\frac{s^2}{p}\right)^{1/4}\left(\frac{s^2}{q}\right)^{1/4}(pq)^{1/4}\le C_P^{1/4}C_Q^{1/4}H(P,Q)^{1/2}. m = ∫ ( p s 2 ) 1/4 ( q s 2 ) 1/4 ( pq ) 1/4 ≤ C P 1/4 C Q 1/4 H ( P , Q ) 1/2 . The ratios can be set to zero where their denominators vanish, since s s s
Recall V = K 3 V = K^3 V = K 3 D = N / V D = N/V D = N / V B v B_v B v Y = ( x 2 , … , x N ) Y = (x_2,\ldots,x_N) Y = ( x 2 , … , x N ) v , w v,w v , w ( x , y , Y ) ∈ B v × B w × Λ K N − 1 (x,y,Y) \in B_v \times B_w \times\Lambda_K^{N-1} ( x , y , Y ) ∈ B v × B w × Λ K N − 1
d P v w = V Φ ( x , Y ) 2 d x d y d Y , d Q v w = V Φ ( y , Y ) 2 d x d y d Y . (67) dP_{vw} = V\Phi(x,Y)^2\,dx\,dy\,dY,\qquad dQ_{vw} = V\Phi(y,Y)^2\,dx\,dy\,dY. \tag*{(67)} d P v w = V Φ ( x , Y ) 2 d x d y d Y , d Q v w = V Φ ( y , Y ) 2 d x d y d Y . (67) The occupation formula gives the exact identity
B ( Φ ) = 1 V 2 ∑ v , w H ( P v w , Q v w ) . (68) B(\Phi) = \frac{1}{V^2}\sum_{v,w} H(P_{vw},Q_{vw}). \tag*{(68)} B ( Φ ) = V 2 1 v , w ∑ H ( P v w , Q v w ) . (68) Fix a group index 1 ≤ g ≤ G D : = ⌊ ℓ D / ( 4 M ) ⌋ 1 \leq g \leq G_D := \lfloor\ell D/(4M)\rfloor 1 ≤ g ≤ G D := ⌊ ℓ D / ( 4 M )⌋ B \mathcal{B} B I \mathcal{I} I λ I \lambda_{\mathcal{I}} λ I P ^ g , Q ^ g \widehat{P}_g,\widehat{Q}_g P g , Q g B \mathcal{B} B ( x , y , Y ) (x,y,Y) ( x , y , Y ) P ^ g \widehat{P}_g P g i ∈ I i \in\mathcal{I} i ∈ I x = ω i ( t ) x=\omega_i(t) x = ω i ( t ) x ∈ B v x \in B_v x ∈ B v N − 1 N-1 N − 1 Y Y Y y y y B w B_w B w Q ^ g \widehat{Q}_g Q g y ∈ B w y \in B_w y ∈ B w x x x B v B_v B v
Lemma 8.2 (Tag normalization). For each ordered pair of cells,
H ( P v w , Q v w ) ≥ 1 D ∑ g = 1 G D H ( P ^ g , Q ^ g ) . (69) H(P_{vw},Q_{vw}) \geq\frac{1}{D}\sum_{g=1}^{G_D} H(\widehat{P}_g,\widehat{Q}_g). \tag*{(69)} H ( P v w , Q v w ) ≥ D 1 g = 1 ∑ G D H ( P g , Q g ) . (69) Proof. Put all groups on the original stationary sample, with all allocation randomizations and a fresh uniform permutation of the full label set. After deleting a tag, the restricted permutation uniformly orders its bath. Each group’s conditional construction is a disintegration of this common sample with respect to its own exposed data. Thus the differing data spaces are projected away before the following comparison; no conditional symmetry of the unexposed-path law is asserted. For every sampled path configuration, group label sets are disjoint. Their sum of tag contributions is bounded by the sum over all N N N Φ 2 \Phi^2 Φ 2 Φ ( x , Y ) 2 \Phi(x,Y)^2 Φ ( x , Y ) 2 N N N V / N = 1 / D V/N=1/D V / N = 1/ D
1 D ∑ g proj P ^ g ≤ P v w , 1 D ∑ g proj Q ^ g ≤ Q v w . \frac{1}{D}\sum_g \operatorname{proj}\widehat{P}_g \leq P_{vw},\qquad
\frac{1}{D}\sum_g \operatorname{proj}\widehat{Q}_g \leq Q_{vw}. D 1 g ∑ proj P g ≤ P v w , D 1 g ∑ proj Q g ≤ Q v w . Use monotonicity, (66) , and the increase of affinity under projection. Its homogeneity supplies the factor 1 / D 1/D 1/ D
The remaining task is to find, for each retained group and each of many ordered cell pairs, a common measure of mass at least 1 / 2 1/2 1/2 P ^ g \widehat{P}_g P g Q ^ g \widehat{Q}_g Q g C 2 C_2 C 2 1 / ( 4 C 2 ) 1/(4C_2) 1/ ( 4 C 2 ) 1 / D 1/D 1/ D (69) can be paid because the number of disjoint groups is proportional to D D D
Two midpoint assignments with the same observed bath
Take R = 30 R=30 R = 30 v , w v,w v , w ( n , d 2 , d 3 ) (n,d_2,d_3) ( n , d 2 , d 3 ) ⌈ K / 10 ⌉ ≤ n ≤ ⌊ K / 8 ⌋ \lceil K/10\rceil\le n \le\lfloor K/8\rfloor ⌈ K /10 ⌉ ≤ n ≤ ⌊ K /8 ⌋ ∣ d i ∣ ≤ n |d_i|\le n ∣ d i ∣ ≤ n E v w \mathcal{E}_{vw} E v w 1 / 2 1/2 1/2
z 0 = v , z 1 , … , z d = w z_0=v,z_1,\ldots,z_d=w z 0 = v , z 1 , … , z d = w through good sites. For each k < d k<d k < d i k i_k i k z k z_k z k B B B
We construct two path configurations that share their observed ( x , y , Y ) (x,y,Y) ( x , y , Y ) λ I \lambda_I λ I z k z_k z k i k i_k i k k < d k<d k < d i k − 1 i_{k-1} i k − 1 k > 0 k>0 k > 0 T i k ( z k ) T_{i_k}(z_k) T i k ( z k ) T i k − 1 ( z k ) T_{i_{k-1}}(z_k) T i k − 1 ( z k ) A k A_k A k U k U_k U k A k A_k A k U k U_k U k d + 1 d+1 d + 1 Figure 1 shows the two midpoint assignments.
Use U d U_d U d U 0 U_0 U 0 λ I \lambda_I λ I
We verify that this conditional construction is defined. A vertex factor carries at most two paths, each with a marginal of the form Π i A k \Pi_i^{A_k} Π i A k M M M C c o l M C_{\mathrm{col}}M C col M C c o l C_{\mathrm{col}} C col
χ D = C c o l M η D ⟶ 0 ( D ⟶ ∞ , M fixed ) . (70) \chi_D=C_{\mathrm{col}}M\eta_D\longrightarrow0 \qquad(D\longrightarrow\infty,\ M\ \text{fixed}). \tag*{(70)} χ D = C col M η D ⟶ 0 ( D ⟶ ∞ , M fixed ) . (70) For any fixed regular and confined values of the other paths with their recorded endpoints, the factor passes every incident check with probability at least 1 − χ D 1-\chi_D 1 − χ D A k A_k A k d + 1 d+1 d + 1 ( 1 − χ D ) d + 1 > 0 (1-\chi_D)^{d+1}>0 ( 1 − χ D ) d + 1 > 0 D D D
Tag i 0 i_0 i 0 i d − 1 i_{d-1} i d − 1 x = U 0 , y = U d x=U_0,y=U_d x = U 0 , y = U d U 1 , … , U d − 1 U_1,\ldots,U_{d-1} U 1 , … , U d − 1 ν ^ \hat{\nu} ν ^ ( B , x , y , Y ) (\mathcal{B},x,y,Y) ( B , x , y , Y ) E v w \mathcal{E}_{vw} E v w
ν ^ ( a l l ) = P ( E v w ) ≥ 1 2 . (71) \hat{\nu}(\mathrm{all})=\mathbb{P}(\mathcal{E}_{vw})\ge\frac{1}{2}. \tag*{(71)} ν ^ ( all ) = P ( E v w ) ≥ 2 1 . (71) Cancellation of conditional normalizers
The midpoint construction has two path copies in each internal vertex factor. To estimate its likelihood, designate one copy as primary and regard the other as auxiliary. At an internal vertex z k z_k z k 1 ≤ k < d 1 \le k < d 1 ≤ k < d i k i_k i k i k − 1 i_{k-1} i k − 1
The next lemma concerns finitely many coordinates in standard Borel spaces, each with a probability reference P i P_i P i check will mean an indicator imposing a constraint on one or two coordinates. The joint reference law λ \lambda λ ⨂ i P i \bigotimes_i P_i ⨂ i P i
A change indexed by a set S S S S S S S S S Π S , i \Pi_{S,i} Π S , i i ∈ S i \in S i ∈ S primary coordinate in the space of P i P_i P i g S , i P i g_{S,i}P_i g S , i P i
0 ≤ g S , i ≤ G , ∫ g S , i d P i = 1. (72) 0 \le g_{S,i} \le G,\qquad\int g_{S,i}\,\mathrm{d}P_i = 1. \tag*{(72)} 0 ≤ g S , i ≤ G , ∫ g S , i d P i = 1. (72) The factors are fixed before the outside coordinates are sampled. The product is conditioned on specified checks, including every reference check involving the primary coordinates in S S S λ \lambda λ L S L_S L S
For each primary and auxiliary coordinate, fix a measurable allowed set on which all its relevant reference and proposal marginals are concentrated. These sets, like the proposal factors, are fixed before sampling outside coordinates. A configuration is individually permitted when each coordinate lies in its allowed set; this imposes no pairwise checks. The compatibility hypothesis below is uniform on these sets, including configurations that fail some pairwise checks.
We make explicit the deletion convention for this construction. When some factors are removed, remove their coordinates and all checks incident to them. Do not replace a removed auxiliary coordinate by an original reference coordinate. Coordinates fixed outside the sets under discussion remain as obstacles in all the checks in which they occur. Coordinates here are path instances in a specified copy. If an auxiliary with label j j j j j j j j j
Lemma 8.3 (Cancellation away from an overlap). Consider two changes S , T S,T S , T G ≥ 1 G \ge1 G ≥ 1 U = S ∪ T U = S \cup T U = S ∪ T
(i) There is χ ∈ [ 0 , 1 ) \chi\in[0,1) χ ∈ [ 0 , 1 ) U U U P i P_i P i Π S , i \Pi_{S,i} Π S , i Π T , i \Pi_{T,i} Π T , i 1 − χ 1-\chi 1 − χ
(ii) There are decompositions S = C S ∪ F S S = C_S \cup F_S S = C S ∪ F S T = C T ∪ F T T = C_T \cup F_T T = C T ∪ F T F S ∩ T = F T ∩ S = ∅ F_S \cap T = F_T \cap S = \varnothing F S ∩ T = F T ∩ S = ∅ F S F_S F S T T T T T T S , T S,T S , T F S , F T F_S,F_T F S , F T
Then
∫ L S L T d λ ≤ G ∣ S ∩ T ∣ ( 1 − χ ) − 2 ( ∣ C S ∣ + ∣ C T ∣ ) . (73) \int L_S L_T \,d\lambda\le G^{|S\cap T|}(1-\chi)^{-2(|C_S|+|C_T|)}. \tag*{(73)} ∫ L S L T d λ ≤ G ∣ S ∩ T ∣ ( 1 − χ ) − 2 ( ∣ C S ∣ + ∣ C T ∣ ) . (73) It is enough that the assumptions hold for almost every fixed admissible configuration outside U U U
Proof. Put a χ = 1 − χ a_\chi= 1-\chi a χ = 1 − χ c = ∣ C S ∣ + ∣ C T ∣ c = |C_S|+|C_T| c = ∣ C S ∣ + ∣ C T ∣ U U U
For any selected set, its reference normalizer is the integral of its reference-check indicator against its product references. In particular, write Z U Z_U Z U U U U S S S Z S Z_S Z S S S S Z S Z_S Z S T ∖ S T\setminus S T ∖ S Y S Y_S Y S J S J_S J S ⨂ i ∈ S Π S , i \bigotimes_{i\in S}\Pi_{S,i} ⨂ i ∈ S Π S , i
Disintegrate each factor in the form
Π S , i ( d x i d ζ i ) = P i ( d x i ) K S , i ( x i , d ζ i ) , K S , i ( x i , all ) = g S , i ( x i ) . \Pi_{S,i}(dx_i\,d\zeta_i)=P_i(dx_i)K_{S,i}(x_i,d\zeta_i),\qquad K_{S,i}(x_i,\text{all})=g_{S,i}(x_i). Π S , i ( d x i d ζ i ) = P i ( d x i ) K S , i ( x i , d ζ i ) , K S , i ( x i , all ) = g S , i ( x i ) . Here ζ i \zeta_i ζ i
H S ( x S ; x S c ) = ∫ J S ∏ i ∈ S K S , i ( x i , d ζ i ) . H_S(x_S;x_{S^c})=\int J_S\prod_{i\in S}K_{S,i}(x_i,d\zeta_i). H S ( x S ; x S c ) = ∫ J S i ∈ S ∏ K S , i ( x i , d ζ i ) . On the reference-admissible configurations, the exact likelihood is
L S = Z S Y S H S . L_S=\frac{Z_S}{Y_S}H_S. L S = Y S Z S H S . Indeed, the conditional reference density is its check indicator divided by Z S Z_S Z S H S / Y S H_S/Y_S H S / Y S ⨂ i ∈ S P i \bigotimes_{i\in S}P_i ⨂ i ∈ S P i
Now retain only the factors indexed by F S F_S F S U U U Z F S , Y F S , H F S Z_{F_S},Y_{F_S},H_{F_S} Z F S , Y F S , H F S F T F_T F T T T T U U U H F S , H F T H_{F_S},H_{F_T} H F S , H F T
Z U ≥ a χ c Z F S Z F T . (74) Z_U \ge a_\chi^c Z_{F_S} Z_{F_T}. \tag*{(74)} Z U ≥ a χ c Z F S Z F T . (74) Z S ≤ Z F S , Y S ≥ a χ ∣ C S ∣ Y F S , (75) Z_S \le Z_{F_S}, \qquad Y_S \ge a_\chi^{|C_S|}Y_{F_S}, \tag*{(75)} Z S ≤ Z F S , Y S ≥ a χ ∣ C S ∣ Y F S , (75) Z T ≤ Z F T , Y T ≥ a χ ∣ C T ∣ Y F T . (76) Z_T \le Z_{F_T}, \qquad Y_T \ge a_\chi^{|C_T|}Y_{F_T}. \tag*{(76)} Z T ≤ Z F T , Y T ≥ a χ ∣ C T ∣ Y F T . (76) For the first inequality, the reference integral over the two far sets factorizes; add every remaining distinct coordinate with its uniform compatibility bound. There are at most c c c Z S Z_S Z S C S C_S C S T ∖ S T \setminus S T ∖ S F S F_S F S C S C_S C S T ∖ S T \setminus S T ∖ S a χ a_\chi a χ T T T Z S Z_S Z S Y S Y_S Y S
The insertion bounds also prove positivity of the divisors. In particular, integrating coordinates one at a time gives a positive product lower bound for every reduced normalizer. The full reference conditional is defined on the outside configurations under consideration.
Deleting proposal checks involving a near factor gives
H S ≤ H F S ∏ i ∈ C S g S , i , H T ≤ H F T ∏ i ∈ C T g T , i . H_S \le H_{F_S}\prod_{i\in C_S}g_{S,i}, \qquad H_T \le H_{F_T}\prod_{i\in C_T}g_{T,i}. H S ≤ H F S i ∈ C S ∏ g S , i , H T ≤ H F T i ∈ C T ∏ g T , i . After dropping the reference admissibility indicator in an upper bound, the two far primary integrals equal Y F S Y_{F_S} Y F S Y F T Y_{F_T} Y F T G G G (72) . Therefore
∫ H S H T 1 a d m i s s i b l e ∏ i ∈ U P i ( d x i ) ≤ Y F S Y F T G ∣ S ∩ T ∣ . (77) \int H_S H_T \mathbf{1}_{\mathrm{admissible}}\prod_{i\in U}P_i(\mathrm{d}x_i) \le Y_{F_S}Y_{F_T}G^{|S\cap T|}. \tag*{(77)} ∫ H S H T 1 admissible i ∈ U ∏ P i ( d x i ) ≤ Y F S Y F T G ∣ S ∩ T ∣ . (77) The conditional reference law on U U U Z U Z_U Z U (8.9) –(77) . The two pointwise ratio bounds contribute a χ c Z F S Z F T / ( Y F S Y F T ) a_\chi^c Z_{F_S}Z_{F_T}/(Y_{F_S}Y_{F_T}) a χ c Z F S Z F T / ( Y F S Y F T ) Z U Z_U Z U a χ − c / ( Z F S Z F T ) a_\chi^{-c}/(Z_{F_S}Z_{F_T}) a χ − c / ( Z F S Z F T ) G ∣ S ∩ T ∣ a χ − 2 c G^{|S\cap T|}a_\chi^{-2c} G ∣ S ∩ T ∣ a χ − 2 c (73) .
The likelihood cost is confined to encounters
Fix the data, the route and the selected labels. Regard either copy as the primary one. The reference for that copy is λ I ⊗ P ∗ \lambda_I \otimes P_* λ I ⊗ P ∗ P ∗ P_* P ∗
S = { i 0 , … , i d − 1 } ∪ { ∗ } . S=\{i_0,\ldots,i_{d-1}\}\cup\{*\}. S = { i 0 , … , i d − 1 } ∪ { ∗ } . The objects in Lemma 8.3 now have the following meaning:
Table 1.
Index a vertex factor by its primary label, using ∗ * ∗ z k z_k z k
By Proposition 6.4, the primary marginal of each path factor has density at most G G G Π i \Pi_i Π i 1 A k / ∣ A k ∣ 1_{A_k}/|A_k| 1 A k /∣ A k ∣ P ∗ P_* P ∗ 1 / 0.98 1/0.98 1/0.98 G G G
The union bound proving (70) verifies hypothesis (i) for each proposal factor. The same argument with P i P_i P i Π i A k \Pi_i^{A_k} Π i A k
Take two independent route and label choices at the same data and endpoint pair, and denote their selected index sets by S , T S,T S , T S S S R R R T T T C S , F S , C T , F T C_S,F_S,C_T,F_T C S , F S , C T , F T
All paths carried by a factor lie within distance eleven of its center. The original path of an opposing real label also lies within distance eleven of that opposing factor’s center. Since R = 30 R=30 R = 30
For clarity, a reduced far proposal retains exactly the paths carried by its far vertex factors and the fixed paths outside S ∪ T S\cup T S ∪ T
Write L S , L T L_S,L_T L S , L T
J R = # { ( j , k ) : d ∞ ( z j , z k ′ ) ≤ R } , J_R=\#\{(j,k):d_\infty(z_j,z'_k)\le R\}, J R = # {( j , k ) : d ∞ ( z j , z k ′ ) ≤ R } , then ∣ C S ∣ + ∣ C T ∣ ≤ 2 J R |C_S|+|C_T|\le2J_R ∣ C S ∣ + ∣ C T ∣ ≤ 2 J R
∫ L S L T d ( λ T ⊗ P ∗ ) ≤ G ∣ S ∩ T ∣ ( 1 − χ D ) − 4 J R . (78) \int L_SL_T\,\mathrm{d}(\lambda_T\otimes P_*)\le G^{|S\cap T|}(1-\chi_D)^{-4J_R}. \tag*{(78)} ∫ L S L T d ( λ T ⊗ P ∗ ) ≤ G ∣ S ∩ T ∣ ( 1 − χ D ) − 4 J R . (78) In particular, no factor is paid for a long portion of one route that stays away from the other.
Averaging labels and forgetting the paths
Condition on the data and the two lattice paths before choosing their labels. Each common departure cell has at least M / 2 M/2 M /2 2 / M 2/M 2/ M G G G
E l a b e l s G ∣ S ∩ T ∣ ≤ G exp ( 2 ( G − 1 ) M J R ) . (79) \mathbb{E}_{\mathrm{labels}}G^{|S\cap T|}\le G\exp\left(\frac{2(G-1)}{M}J_R\right). \tag*{(79)} E labels G ∣ S ∩ T ∣ ≤ G exp ( M 2 ( G − 1 ) J R ) . (79) Choose the fixed integer M M M
2 ( G − 1 ) M ≤ κ 2 . (80) \frac{2(G-1)}{M}\le\frac{\kappa}{2}. \tag*{(80)} M 2 ( G − 1 ) ≤ 2 κ . (80) Then increase D 0 D_0 D 0 D ≥ D 0 D \ge D_0 D ≥ D 0 α \alpha α
− 4 log ( 1 − χ D ) ≤ κ 2 , G D D ≥ ℓ 8 M . (81) -4\log(1-\chi_D) \le\frac{\kappa}{2}, \qquad\frac{G_D}{D} \ge\frac{\ell}{8M}. \tag*{(81)} − 4 log ( 1 − χ D ) ≤ 2 κ , D G D ≥ 8 M ℓ . (81) The first requirement is possible by (8.5); the second follows once ℓ D / ( 4 M ) ≥ 2 \ell D/(4M) \ge2 ℓ D / ( 4 M ) ≥ 2 G G G κ \kappa κ C p a t h C_{\mathrm{path}} C path
The following use of an averaged likelihood and a two-route second moment also occurs in the point-reassignment argument of Peres and Sly [31 ] (Proposition 2.1). Here Lemma 8.3 supplies the additional conditional-normalizer estimate required by the hard-core path law.
Let L ‾ \overline{L} L E v w \mathcal{E}_{vw} E v w
E B ∫ L ‾ 2 d ( λ I ⊗ P ∗ ) ≤ G E [ 1 E v w e κ J R ] ≤ C 2 , C 2 : = G C p a t h . (82) \mathbb{E}_B\int\overline{L}^{\,2}\,\mathrm{d}(\lambda_I \otimes P_*) \le G\mathbb{E}\left[\mathbf{1}_{\mathcal{E}_{vw}}e^{\kappa J_R}\right] \le C_2,\qquad C_2 := GC_{\mathrm{path}}. \tag*{(82)} E B ∫ L 2 d ( λ I ⊗ P ∗ ) ≤ G E [ 1 E v w e κ J R ] ≤ C 2 , C 2 := G C path . (82) Here the two route randomizations share the same data; the expectation over the data is taken only once. The event is not conditioned to have probability one. Its indicator is part of both likelihoods, and its square is itself. The calculation applies separately with either copy primary, with the same constant C 2 C_2 C 2
We next check carefully that tagging and observing the bath do not increase this bound. At fixed data, split L ‾ = ∑ i ∈ I f i \overline{L}=\sum_{i\in I}f_i L = ∑ i ∈ I f i f i f_i f i i i i f i f_i f i ω i ( t ) ∈ B v \omega_i(t)\in B_v ω i ( t ) ∈ B v ω i ( t ) ∈ B w \omega_i(t)\in B_w ω i ( t ) ∈ B w
∑ i f i 2 ≤ ( ∑ i f i ) 2 , \sum_i f_i^2 \le\left(\sum_i f_i\right)^2, i ∑ f i 2 ≤ ( i ∑ f i ) 2 , the measure that retains the tag index has squared likelihood integral at most C 2 C_2 C 2 i i i B , x , y , Y B,x,y,Y B , x , y , Y f f f P P P f f f ∫ f 2 d P \int f^2\,\mathrm{d}P ∫ f 2 d P
For either primary copy, the resulting measure is the common observed ν ^ \widehat{\nu} ν P ^ g \widehat{P}_g P g Q ^ g \widehat{Q}_g Q g
∫ ( d ν ^ d P ^ g ) 2 d P ^ g ≤ C 2 , ∫ ( d ν ^ d Q ^ g ) 2 d Q ^ g ≤ C 2 . (83) \int\left(\frac{\mathrm{d}\widehat{\nu}}{\mathrm{d}\widehat{P}_g}\right)^2\mathrm{d}\widehat{P}_g \le C_2,\qquad
\int\left(\frac{\mathrm{d}\widehat{\nu}}{\mathrm{d}\widehat{Q}_g}\right)^2\mathrm{d}\widehat{Q}_g \le C_2. \tag*{(83)} ∫ ( d P g d ν ) 2 d P g ≤ C 2 , ∫ ( d Q g d ν ) 2 d Q g ≤ C 2 . (83) Together with (8.6), Lemma 8.1 now gives
H ( P ^ g , Q ^ g ) ≥ 1 4 C 2 . (84) H(\widehat{P}_g,\widehat{Q}_g) \ge\frac{1}{4C_2}. \tag*{(84)} H ( P g , Q g ) ≥ 4 C 2 1 . (84) This lower bound is uniform in the group index and in every admissible ordered pair of cells.
Proposition 8.4 (Uniform modulus occupation). After the fixed choices above, there are constants D 0 < ∞ D_0<\infty D 0 < ∞
c ∗ : = v 0 ℓ 32 M G C p a t h > 0 , v 0 = 1 2000 (85) c_* := \frac{v_0\ell}{32MGC_{\mathrm{path}}} > 0,\qquad v_0=\frac{1}{2000} \tag*{(85)} c ∗ := 32 M G C path v 0 ℓ > 0 , v 0 = 2000 1 (85) such that, whenever D ≥ D 0 D \ge D_0 D ≥ D 0 α = D r ∈ [ α 0 / 2 , 2 α 0 ] \alpha= Dr \in[\alpha_0/2,2\alpha_0] α = D r ∈ [ α 0 /2 , 2 α 0 ] K K K B ( Φ ) ≥ c ∗ B(\Phi) \ge c_* B ( Φ ) ≥ c ∗ c ∗ c_* c ∗ D D D D D D [ D 0 , ∞ ) [D_0,\infty) [ D 0 , ∞ )
Proof. For large K K K v 0 V 2 v_0V^2 v 0 V 2 (81) and (84) give
H ( P v w , Q v w ) ≥ G D 4 D C 2 ≥ ℓ 32 M C 2 . \mathrm{H}(P_{vw},Q_{vw}) \ge\frac{G_D}{4DC_2} \ge\frac{\ell}{32MC_2}. H ( P v w , Q v w ) ≥ 4 D C 2 G D ≥ 32 M C 2 ℓ . Sum these nonnegative contributions in (68) to obtain B ( Φ ) ≥ v 0 ℓ / ( 32 M C 2 ) B(\Phi) \ge v_0\ell/(32MC_2) B ( Φ ) ≥ v 0 ℓ / ( 32 M C 2 ) (85) . All ingredients are uniform in the ground vector. The threshold and local-uniformity assertions follow from the preceding path and group estimates and Proposition 7.1. No parameter entering c ∗ c_* c ∗ D 0 D_0 D 0
The modulus occupation has now been bounded below by a fixed number. To treat the original complex ground vector, it remains to show that a small part of each one-coordinate slice contains every possible phase obstruction. That estimate may require a larger lower bound on D D D
A common component in each one-particle slice
We now construct the retained sets required by Lemma 2.3. With the bath fixed, we cover its forbidden balls by microscopic cubes and fill the bounded pockets enclosed by each bad cluster. The retained points will lie in one component of the open allowed slice. To bound the removed volume, we separate clusters whose coordinate spans are at most one from larger clusters. The former cost at most C r 2 N Cr^2N C r 2 N 1 / r 1/r 1/ r
We retain the scaled torus Λ K \Lambda_K Λ K V = K 3 V=K^3 V = K 3 B v B_v B v v ∈ ( Z / K Z ) 3 v \in(\mathbb{Z}/K\mathbb{Z})^3 v ∈ ( Z / K Z ) 3
Y = ( x 2 , … , x N ) , m ( Y ) = ∫ Λ K Φ ( x , Y ) 2 d x , ν ( d Y ) = m ( Y ) d Y . Y=(x_2,\ldots,x_N), \qquad m(Y)=\int_{\Lambda_K}\Phi(x,Y)^2\,\mathrm{d}x,\qquad\nu(\mathrm{d}Y)=m(Y)\,\mathrm{d}Y. Y = ( x 2 , … , x N ) , m ( Y ) = ∫ Λ K Φ ( x , Y ) 2 d x , ν ( d Y ) = m ( Y ) d Y . Thus ν \nu ν μ ( d X ) = Φ ( X ) 2 d X \mu(\mathrm{d}X)=\Phi(X)^2\,\mathrm{d}X μ ( d X ) = Φ ( X ) 2 d X d ( x i , x j ) > r d(x_i,x_j)>r d ( x i , x j ) > r 2 ≤ i < j ≤ N 2\le i<j\le N 2 ≤ i < j ≤ N m ( Y ) = 0 m(Y)=0 m ( Y ) = 0
S Y = Λ K ∖ ⋃ j = 2 N B ‾ ( x j , r ) . (86) S_Y=\Lambda_K\setminus\bigcup_{j=2}^{N}\overline{B}(x_j,r). \tag*{(86)} S Y = Λ K ∖ j = 2 ⋃ N B ( x j , r ) . (86) We first work deterministically with admissible baths. The probability estimate at the end uses the joint close-neighbor bound from (28) .
Microcubes and filled components
Put
q r = ⌈ 1 / r ⌉ , h = q r − 1 , J = K q r . q_r=\lceil1/r\rceil,\qquad h=q_r^{-1},\qquad J=Kq_r. q r = ⌈ 1/ r ⌉ , h = q r − 1 , J = K q r . For r ≤ 1 r \le1 r ≤ 1 r / 2 ≤ h ≤ r r/2 \le h \le r r /2 ≤ h ≤ r q r 3 q_r^3 q r 3
Q z = h ( z + [ 0 , 1 ] 3 ) − ( 1 / 2 , 1 / 2 , 1 / 2 ) , z ∈ Z 3 . Q_z = h\left(z + [0,1]^3\right) - (1/2,1/2,1/2), \qquad z \in\mathbb{Z}^3. Q z = h ( z + [ 0 , 1 ] 3 ) − ( 1/2 , 1/2 , 1/2 ) , z ∈ Z 3 . Their index period is J J J bad if its closure intersects a closed ball in (86) ; otherwise it is good . Marking is periodic. Distinct index sites are ∗ \ast ∗ ∗ \ast ∗ C C C H ∞ ( C ) H_{\infty}(C) H ∞ ( C ) Z 3 ∖ C \mathbb{Z}^3 \setminus C Z 3 ∖ C
Fill ( C ) = Z 3 ∖ H ∞ ( C ) . \operatorname{Fill}(C) = \mathbb{Z}^3 \setminus H_{\infty}(C). Fill ( C ) = Z 3 ∖ H ∞ ( C ) . Thus the fill adds all finite nearest-neighbor holes to C C C A ( Y ) ⊂ Λ K A(Y) \subset\Lambda_K A ( Y ) ⊂ Λ K A ( Y ) = Λ K A(Y) = \Lambda_K A ( Y ) = Λ K
G ( Y ) = Λ K ∖ A ( Y ) . G(Y) = \Lambda_K \setminus A(Y). G ( Y ) = Λ K ∖ A ( Y ) . The fills remove bounded pockets behind the obstacles. This is the same discrete filling operation used in Lemma 7.2; here its scale is comparable to the exclusion distance. Phase transfer requires the points of G ( Y ) G(Y) G ( Y ) S Y S_Y S Y Figure 2 distinguishes bad cubes, their fill, and such a route.
Figure 2. A planar schematic of the filling operation. The argument uses three-dimensional microcubes. Each closed good cube avoids every closed forbidden ball, so a finite route through good cubes has positive clearance. The picture concerns a single-coordinate slice with the bath fixed.
Lemma 9.1 (Connectivity with strict clearance). For every bath Y Y Y G ( Y ) G(Y) G ( Y ) S Y S_Y S Y
Proof. There is nothing to prove when G ( Y ) G(Y) G ( Y )
Start with any finite nearest-neighbor path between these sites. If it visits a bad component C C C u u u w w w u u u C C C u u u w w w ∗ \ast ∗ C C C ∗ \ast ∗ C C C C C C
Join successive cube centers through their common faces, and join the original points to their cube centers. A point on a grid face can be assigned any adjacent cube: membership outside A ( Y ) \mathcal{A}(Y) A ( Y )
Small fills and distinct close-particle witnesses
Call a finite microcomponent C C C small if h diam ∞ C ≤ 1 h\operatorname{diam}_{\infty} C \le1 h diam ∞ C ≤ 1 large . For a small fill, each occupied column has cross-sectional area h 2 h^2 h 2 μ \mu μ
Lemma 9.2 (Volume of small fills). For K > 10 K > 10 K > 10 r ≤ 1 r \le1 r ≤ 1 C r 2 N Cr^2N C r 2 N C C C
Proof. The fill of a finite component lies in its coordinate bounding box by Lemma 7.2. Moreover, every column parallel to the first coordinate axis that meets the fill also meets the component. Otherwise that entire column would avoid the component and connect each of its vertices to infinity, precluding a finite hole there.
For a small component C C C ∣ C ∣ |C| ∣ C ∣ 1 + 2 h 1+2h 1 + 2 h h 2 h^2 h 2
( 1 + 2 h ) h 2 ∣ C ∣ ≤ 3 h 2 ∣ C ∣ . (1+2h)h^2|C| \le3h^2|C|. ( 1 + 2 h ) h 2 ∣ C ∣ ≤ 3 h 2 ∣ C ∣. A closed radius-r r r C 0 = 343 C_0=343 C 0 = 343 r / h ≤ 2 r/h \le2 r / h ≤ 2
Choose one representative of each small component modulo the torus period. Its projection is injective because each physical coordinate span is at most one and K > 10 K>10 K > 10 C 0 ( N − 1 ) C_0(N-1) C 0 ( N − 1 ) (9.2) , and allowing overlaps of their fills in this upper bound, gives 3 C 0 h 2 ( N − 1 ) ≤ C r 2 N 3C_0h^2(N-1) \le Cr^2N 3 C 0 h 2 ( N − 1 ) ≤ C r 2 N
Lemma 9.3 (Distinct witnesses in a large component). There are absolute constants r 1 , c g > 0 r_1,c_g>0 r 1 , c g > 0 r ≤ r 1 r \le r_1 r ≤ r 1 K > 10 K>10 K > 10 B v B_v B v c g / r c_g/r c g / r v v v 6 r 6r 6 r c g = 1 / ( 8 C 0 ) c_g=1/(8C_0) c g = 1/ ( 8 C 0 ) C 0 = 343 C_0=343 C 0 = 343
Proof. Fix a lifted microcube index z z z B v B_v B v 1 / 2 1/2 1/2 z z z z z z 1 / 4 1/4 1/4 1 / ( 4 h ) 1/(4h) 1/ ( 4 h ) 1 / 4 + h 1/4+h 1/4 + h z z z
Choose one bath ball witnessing badness at each path vertex. A fixed ball witnesses at most C 0 C_0 C 0 v v v r r r K K K 1 / ( 4 C 0 h ) ≥ 1 / ( 4 C 0 r ) 1/(4C_0h) \ge1/(4C_0r) 1/ ( 4 C 0 h ) ≥ 1/ ( 4 C 0 r ) r 1 r_1 r 1 c g c_g c g
Repeated labels do not destroy the close-partner conclusion. Form a graph whose vertices are the distinct witness labels, joining different labels whenever they occur consecutively in the sequence. This graph is connected and has at least two vertices, so none is isolated. Two adjacent closed microcubes meet. The centers of balls intersecting those cubes are therefore at distance at most
2 r + 2 3 h ≤ 6 r . 2r+2\sqrt{3}h \le6r. 2 r + 2 3 h ≤ 6 r . Every distinct label thus has a different close partner, including labels that recur in separated portions of the witness sequence.
We now fix the geometric restriction on the close-count threshold:
0 < θ < c g 2 α 0 . (87) 0<\theta<\frac{c_g}{2\alpha_0}. \tag*{(87)} 0 < θ < 2 α 0 c g . (87) This restriction is imposed before the choice of b b b r = α / D r=\alpha/D r = α / D α ≤ 2 α 0 \alpha\le2\alpha_0 α ≤ 2 α 0 D D D
6 r < s 0 = b D − 1 / 3 , r ≤ r 1 . 6r<s_0=bD^{-1/3},\qquad r\le r_1. 6 r < s 0 = b D − 1/3 , r ≤ r 1 . For a full configuration X = ( x , Y ) X=(x,Y) X = ( x , Y ) (27) :
Q ( X ) = { v : # { i : d ( x i , v ) ≤ 5 , min j ≠ i d ( x i , x j ) ≤ s 0 } ≥ θ D } . (88) \mathcal{Q}(X)=\left\{v:\#\left\{i:d(x_i,v)\le5,\ \min_{j\ne i}d(x_i,x_j)\le s_0\right\}\ge\theta D\right\}. \tag*{(88)} Q ( X ) = { v : # { i : d ( x i , v ) ≤ 5 , j = i min d ( x i , x j ) ≤ s 0 } ≥ θ D } . (88) By Equation (28) , for every deterministic set S S S
μ { S ⊂ Q ( X ) } ≤ p D ∣ S ∣ , p D ⟶ 0. (89) \mu\{S\subset\mathcal{Q}(X)\}\le p_D^{|S|},\qquad p_D\longrightarrow0. \tag*{(89)} μ { S ⊂ Q ( X )} ≤ p D ∣ S ∣ , p D ⟶ 0. (89) Lemma 9.3 and Equations (87) –(9.4) show that every parent cell of a large microcomponent belongs to Q ( X ) \mathcal{Q}(X) Q ( X ) c g / r > θ D c_g/r>\theta D c g / r > θ D x x x x x x ν \nu ν ( N − 1 ) (N-1) ( N − 1 )
Periodic animals and the omitted volume
Proposition 9.4 (Small omitted volume). Fix the constants as above and take D D D C , C a n C,C_{\mathrm{an}} C , C an K ≥ 100 K\ge100 K ≥ 100 C a n p D ≤ 1 / 2 C_{\mathrm{an}}p_D\le1/2 C an p D ≤ 1/2
δ g e o m : = ∫ ∣ A ( Y ) ∣ V ν ( d Y ) ≤ C r 2 D + C p D + K 3 ( C a n p D ) ⌊ K / 10 ⌋ . (90) \delta_{\mathrm{geom}}:=\int\frac{|\mathcal{A}(Y)|}{V}\,\nu(\mathrm{d}Y)\le Cr^2D+Cp_D+K^3(C_{\mathrm{an}}p_D)^{\lfloor K/10\rfloor}. \tag*{(90)} δ geom := ∫ V ∣ A ( Y ) ∣ ν ( d Y ) ≤ C r 2 D + C p D + K 3 ( C an p D ) ⌊ K /10 ⌋ . (90) The retained points belong to one component of the open slice S Y S_Y S Y α ∈ [ α 0 / 2 , 2 α 0 ] \alpha\in[\alpha_0/2,2\alpha_0] α ∈ [ α 0 /2 , 2 α 0 ]
Proof. We estimate the bath-dependent volume by integrating under the full configuration law:
δ g e o m = ∫ Λ K N ∣ A ( Y ) ∣ V μ ( d x d Y ) . (91) \delta_{\mathrm{geom}}=\int_{\Lambda_K^N}\frac{|\mathcal{A}(Y)|}{V}\,\mu(\mathrm{d}x\,\mathrm{d}Y). \tag*{(91)} δ geom = ∫ Λ K N V ∣ A ( Y ) ∣ μ ( d x d Y ) . (91) This is the marginal identity ∫ Φ ( x , Y ) 2 d x = m ( Y ) \int\Phi(x,Y)^2\,\mathrm{d}x=m(Y) ∫ Φ ( x , Y ) 2 d x = m ( Y ) x x x
Set m K = ⌊ K / 10 ⌋ m_K=\lfloor K/10\rfloor m K = ⌊ K /10 ⌋ E K E_K E K (88) has a ∗ \ast ∗ m K m_K m K m m m C a n m C_{\mathrm{an}}^m C an m 2 ( m − 1 ) 2(m-1) 2 ( m − 1 ) 26 2 ( m − 1 ) 26^{2(m-1)} 2 6 2 ( m − 1 ) m K m_K m K m K m_K m K m K m_K m K (89) and a union bound over roots give
μ ( E K ) ≤ K 3 ( C a n p D ) m K . (92) \mu(E_K)\le K^3(C_{\mathrm{an}}p_D)^{m_K}. \tag*{(92)} μ ( E K ) ≤ K 3 ( C an p D ) m K . (92) On E K c E_K^c E K c K / 10 K/10 K /10 m m m 2 m − 1 < K 2m-1<K 2 m − 1 < K K K K m − 1 m-1 m − 1
The parent-cell map sends adjacent microcubes to equal or ∗ \ast ∗ E K c E_K^c E K c q 3 q^3 q 3
Consequently, if a prescribed unit cell is affected by a large fill, there is a coarse bad animal of some size m ≥ 1 m\ge1 m ≥ 1 C m Cm C m C m 3 Cm^3 C m 3 (89) give
μ { E K c and the cell meets a large fill } ≤ ∑ m ≥ 1 C m 3 ( C a n p D ) m ≤ C ′ p D . (93) \mu\{E_K^c\text{ and the cell meets a large fill}\}\le\sum_{m\ge1}Cm^3(C_{\mathrm{an}}p_D)^m\le C'p_D. \tag*{(93)} μ { E K c and the cell meets a large fill } ≤ m ≥ 1 ∑ C m 3 ( C an p D ) m ≤ C ′ p D . (93) The last inequality follows by summing the convergent power series on C a n p D ≤ 1 / 2 C_{\mathrm{an}}p_D\le1/2 C an p D ≤ 1/2
Sum Equation (93) over unit cells and divide by V V V E K E_K E K C r 2 N / V = C r 2 D Cr^2N/V=Cr^2D C r 2 N / V = C r 2 D (92) proves Equation (90) , by Equation (91) . Lemma 9.1 gives the connectivity assertion.
All sets used here are measurable. Bad-cube membership is a finite family of distance tests in one period. A finite periodic component has a finite lift precisely when its spanning-tree lift closes along every additional edge. In that case its fill is determined by connectivity in a finite bounding box, with the exterior attached. These finite procedures determine the projected discarded cubes and their volume from the bath positions. □ \square □
The right side of Equation (90) has the needed order of limits. Since r 2 D = α 2 / D r^2D=\alpha^2/D r 2 D = α 2 / D D D D K K K D D D C a n p D ≤ 1 / 2 C_{\mathrm{an}}p_D\le1/2 C an p D ≤ 1/2
Choice of constants and the thermodynamic limit
We now combine the uniform modulus bound with the phase estimate. The order of the choices is essential: the modulus fraction is fixed before we make the discarded volume small enough to preserve it.
A bound in the scaled variables
Proposition 10.1. There are absolute constants α 0 \alpha_0 α 0 D ∗ D_* D ∗ c ∗ > 0 c_*>0 c ∗ > 0 D = N / K 3 ≥ D ∗ D=N/K^3\ge D_* D = N / K 3 ≥ D ∗ r = α / D r=\alpha/D r = α / D α ∈ [ α 0 / 2 , 2 α 0 ] \alpha\in[\alpha_0/2,2\alpha_0] α ∈ [ α 0 /2 , 2 α 0 ]
B ( Ψ ) ≥ c ∗ / 2 B(\Psi)\ge c_*/2 B ( Ψ ) ≥ c ∗ /2 for all sufficiently large integer K K K D D D [ D ∗ , ∞ ) [D_*,\infty) [ D ∗ , ∞ )
Proof. We first list the choices that yield Proposition 8.4. Fix the local cutoff profiles and the exponent γ = 1 / 12 \gamma=1/12 γ = 1/12 α 0 \alpha_0 α 0 ℓ \ell ℓ θ \theta θ
3 θ < ℓ / 2 , θ < c g / ( 2 α 0 ) . 3\theta<\ell/2,\qquad\theta<c_g/(2\alpha_0). 3 θ < ℓ /2 , θ < c g / ( 2 α 0 ) . The first inequality leaves the required eligible labels; the second is the witness condition in (87) . Next fix the separation constant b b b s 0 = b D − 1 / 3 s_0=bD^{-1/3} s 0 = b D − 1/3
At the fixed encounter radius R = 30 R=30 R = 30 p p p κ > 0 \kappa>0 κ > 0 C p a t h C_{\mathrm{path}} C path H 1 H_1 H 1 t > 0 t>0 t > 0 D D D G ≥ 2 G\ge2 G ≥ 2 M M M
2 ( G − 1 ) M ≤ κ 2 . \frac{2(G-1)}{M}\le\frac{\kappa}{2}. M 2 ( G − 1 ) ≤ 2 κ . Finally choose D 0 D_0 D 0 ℓ D ≥ 8 M \ell D\ge8M ℓ D ≥ 8 M
− 4 log ( 1 − χ D ) ≤ κ 2 , χ D = C c o l M η D . -4\log(1-\chi_D)\le\frac{\kappa}{2},\qquad\chi_D=C_{\mathrm{col}}M\eta_D. − 4 log ( 1 − χ D ) ≤ 2 κ , χ D = C col M η D . Here η D → 0 \eta_D\to0 η D → 0 D ≥ D 0 D\ge D_0 D ≥ D 0
B ( Φ ) ≥ c ∗ : = v 0 ℓ 32 M G C p a t h > 0 , v 0 = 1 2000 . (94) B(\Phi)\ge c_*:=\frac{v_0\ell}{32MGC_{\mathrm{path}}}>0,\qquad v_0=\frac{1}{2000}. \tag*{(94)} B ( Φ ) ≥ c ∗ := 32 M G C path v 0 ℓ > 0 , v 0 = 2000 1 . (94) None of v 0 v_0 v 0 ℓ \ell ℓ M M M G G G C p a t h C_{\mathrm{path}} C path D D D
By Proposition 9.4, the retained points lie in a common component of each allowed slice, and their discarded-volume expectation satisfies
δ g e o m ≤ C α 2 D + C p D + C K 3 ( C a n p D ) ⌊ K / 10 ⌋ , p D ⟶ 0. (95) \delta_{\mathrm{geom}}\le C\frac{\alpha^2}{D}+Cp_D+CK^3(C_{\mathrm{an}}p_D)^{\lfloor K/10\rfloor},\qquad p_D\longrightarrow0. \tag*{(95)} δ geom ≤ C D α 2 + C p D + C K 3 ( C an p D ) ⌊ K /10 ⌋ , p D ⟶ 0. (95) The constants in this estimate use only choices already fixed. Increase D 0 D_0 D 0 D ∗ D_* D ∗ D ≥ D ∗ D \ge D_* D ≥ D ∗ α \alpha α
C α 2 D + C p D ≤ c ∗ 2 256 , C a n p D ≤ 1 2 . C\frac{\alpha^2}{D}+C_{\mathrm{pD}}\le\frac{c_*^2}{256},\qquad C_{\mathrm{anpD}}\le\frac{1}{2}. C D α 2 + C pD ≤ 256 c ∗ 2 , C anpD ≤ 2 1 . For sufficiently large K K K (95) is at most c ∗ 2 / 256 c_*^2/256 c ∗ 2 /256 δ g e o m ≤ c ∗ 2 / 64 \delta_{\mathrm{geom}} \le c_*^2/64 δ geom ≤ c ∗ 2 /64 B ( Ψ ) ≥ c ∗ / 2 B(\Psi) \ge c_*/2 B ( Ψ ) ≥ c ∗ /2 D D D α \alpha α D ≥ D ∗ D \ge D_* D ≥ D ∗ C K 3 2 − ⌊ K / 10 ⌋ CK^3 2^{-\lfloor K/10\rfloor} C K 3 2 − ⌊ K /10 ⌋ D D D
Return to fixed exclusion distance and density
Proof of Theorem 1.1. Let α 0 \alpha_0 α 0 D ∗ D_* D ∗ c ∗ c_* c ∗
ε 0 = α 0 3 4 D ∗ 2 , c 0 = c ∗ / 2. \varepsilon_0=\frac{\alpha_0^3}{4D_*^2},\qquad c_0=c_*/2. ε 0 = 4 D ∗ 2 α 0 3 , c 0 = c ∗ /2. Decrease ε 0 \varepsilon_0 ε 0 a , ρ > 0 a,\rho>0 a , ρ > 0 ρ a 3 < ε 0 \rho a^3<\varepsilon_0 ρ a 3 < ε 0 N k , L k , Ψ k N_k,L_k,\Psi_k N k , L k , Ψ k α 0 \alpha_0 α 0 (6) will then turn the small physical gas parameter into a large scaled density. For all sufficiently large k k k
q k = a N k α 0 L k , K k = ⌊ q k ⌋ , r k = a K k L k , D k = N k K k 3 . q_k=\sqrt{\frac{aN_k}{\alpha_0L_k}},\qquad K_k=\lfloor q_k\rfloor,\qquad r_k=\frac{aK_k}{L_k},\qquad D_k=\frac{N_k}{K_k^3}. q k = α 0 L k a N k , K k = ⌊ q k ⌋ , r k = L k a K k , D k = K k 3 N k . Since q k / L k → a ρ / α 0 > 0 q_k/L_k\to\sqrt{a\rho/\alpha_0}>0 q k / L k → a ρ / α 0 > 0 K k → ∞ K_k\to\infty K k → ∞ K k / q k → 1 K_k/q_k\to1 K k / q k → 1
α k : = D k r k = a N k L k K k 2 = α 0 ( q k K k ) 2 ⟶ α 0 , \alpha_k:=D_kr_k=\frac{aN_k}{L_kK_k^2}=\alpha_0\left(\frac{q_k}{K_k}\right)^2\longrightarrow\alpha_0, α k := D k r k = L k K k 2 a N k = α 0 ( K k q k ) 2 ⟶ α 0 , D k ⟶ D ∞ : = α 0 3 / 2 ρ a 3 > 2 D ∗ . D_k\longrightarrow D_\infty:=\frac{\alpha_0^{3/2}}{\sqrt{\rho a^3}}>2D_*. D k ⟶ D ∞ := ρ a 3 α 0 3/2 > 2 D ∗ . Thus α k \alpha_k α k D k D_k D k ( D ∗ , ∞ ) (D_*,\infty) ( D ∗ , ∞ ) k k k s k = L k / K k s_k=L_k/K_k s k = L k / K k Ψ k \Psi_k Ψ k Λ K k \Lambda_{K_k} Λ K k r k r_k r k B B B B ( Ψ k ) ≥ c 0 B(\Psi_k)\ge c_0 B ( Ψ k ) ≥ c 0 k k k (1.2) for every such sequence.
Proof of Corollary 1.2. The bosonic ground eigenspace is finite dimensional by compactness of the form embedding. Every density operator supported on it is a convex combination of rank-one projections onto normalized ground vectors. For all sufficiently large members of the sequence, the proof of Theorem 1.1 gives the same lower bound c 0 c_0 c 0 Π k / Tr Π k \Pi_k/\operatorname{Tr}\Pi_k Π k / Tr Π k
