For a homogeneous gas, Bose–Einstein condensation means that one orbital contains a positive fraction of all particles in the equilibrium state. We consider the spatially constant orbital and keep both the physical interaction and a positive density fixed as the volume grows. The bound below uses the same positive fraction throughout a sufficiently dilute interval of densities.
Let v : R 3 → [ 0 , ∞ ) v : \mathbb{R}^{3} \to[0,\infty) v : R 3 → [ 0 , ∞ ) v v v Λ L = ( R / L Z ) 3 \Lambda_L = (\mathbb{R}/L\mathbb{Z})^{3} Λ L = ( R / L Z ) 3 V = L 3 V = L^{3} V = L 3 v L ( x ) = ∑ k ∈ Z 3 v ( x + L k ) v_L(x) = \sum_{k\in\mathbb{Z}^{3}} v(x + Lk) v L ( x ) = ∑ k ∈ Z 3 v ( x + L k )
H N , L = − ∑ i = 1 N Δ i + ∑ 1 ≤ i < j ≤ N v L ( x i − x j ) on L s y m 2 ( Λ L N ) . H_{N,L} = -\sum_{i=1}^{N} \Delta_i + \sum_{1\leq i<j\leq N} v_L(x_i-x_j)\quad\text{on }L^2_{\mathrm{sym}}(\Lambda_L^N). H N , L = − i = 1 ∑ N Δ i + 1 ≤ i < j ≤ N ∑ v L ( x i − x j ) on L sym 2 ( Λ L N ) . The units are ℏ 2 / ( 2 m p a r t i c l e ) = k B = 1 \hbar^2/(2m_{\mathrm{particle}}) = k_B = 1 ℏ 2 / ( 2 m particle ) = k B = 1 T > 0 T > 0 T > 0 Γ N , L , T = e − H N , L / T / Tr e − H N , L / T \Gamma_{N,L,T} = e^{-H_{N,L}/T}/\operatorname{Tr}e^{-H_{N,L}/T} Γ N , L , T = e − H N , L / T / Tr e − H N , L / T T = 0 T = 0 T = 0 γ N , L , T ( 1 ) \gamma^{(1)}_{N,L,T} γ N , L , T ( 1 ) N N N u 0 = V − 1 / 2 u_0 = V^{-1/2} u 0 = V − 1/2
Theorem 1.1. For every potential v v v ρ ∗ ( v ) > 0 \rho_*(v) > 0 ρ ∗ ( v ) > 0 c ∗ ( v ) > 0 c_*(v) > 0 c ∗ ( v ) > 0 0 < ρ < ρ ∗ ( v ) 0 < \rho< \rho_*(v) 0 < ρ < ρ ∗ ( v ) L 0 ( ρ , v ) < ∞ L_0(\rho,v) < \infty L 0 ( ρ , v ) < ∞ L ≥ L 0 ( ρ , v ) L \geq L_0(\rho,v) L ≥ L 0 ( ρ , v ) N ≥ 1 N \geq1 N ≥ 1 ρ / 2 ≤ N / L 3 ≤ 2 ρ \rho/2 \leq N/L^3 \leq2\rho ρ /2 ≤ N / L 3 ≤ 2 ρ 0 ≤ T ≤ ρ 2 0 \leq T \leq\rho^2 0 ≤ T ≤ ρ 2
⟨ u 0 , γ N , L , T ( 1 ) u 0 ⟩ N ≥ c ∗ ( v ) . \frac{\langle u_0,\gamma^{(1)}_{N,L,T}u_0\rangle}{N} \geq c_*(v). N ⟨ u 0 , γ N , L , T ( 1 ) u 0 ⟩ ≥ c ∗ ( v ) . Consequently, for each such fixed density ρ \rho ρ 0 ≤ T ≤ ρ 2 0 \leq T \leq\rho^2 0 ≤ T ≤ ρ 2 L → ∞ L \to\infty L → ∞ N / L 3 → ρ N/L^3 \to\rho N / L 3 → ρ c ∗ ( v ) c_*(v) c ∗ ( v )
The volume threshold may depend on the density; the condensate lower bound does not. The Hamiltonian is the exact interacting Hamiltonian, and its potential is not rescaled with the number of particles. The conclusion is macroscopic occupation of the constant orbital with a positive lower bound throughout the admitted dilute interval. The zero-temperature state is unambiguous: positivity of the finite-volume heat kernel gives a unique bosonic ground state, as shown at the end of the proof.
Context and antecedents
The ideal-gas theory begins with Bose’s counting of light quanta [8 ] and Einstein’s extension to a monatomic ideal gas, including the condensation prediction in his second paper [15 , 16 ] . Bogoliubov’s treatment of weakly interacting bosons introduced an approximate quasiparticle description based on macroscopic occupation of one mode [7 ] .
For interacting particles, Penrose and Onsager formulated condensation through a macroscopic eigenvalue of the one-particle density matrix and applied this viewpoint to liquid helium [31 ] . Yang developed the broader framework of off-diagonal long-range order, covering Bose condensation and fermion pairing [36 ] . These criteria identify the order to be proved; establishing it for a specified interacting Hamiltonian remains a separate mathematical problem.
Rigorous lattice antecedents include Dyson, Lieb, and Simon’s low-temperature long-range order for the spin-1 2 \frac{1}{2} 2 1 d ≥ 3 d \ge3 d ≥ 3 [14 ] , and Kennedy, Lieb, and Shastry’s ground-state long-range order for all spins on hypercubic lattices in dimensions d ≥ 2 d \ge2 d ≥ 2 [26 ] . For spin-1 2 \frac{1}{2} 2 1
The distinction between the thermodynamic and Gross–Pitaevskii limits is essential. Lieb and Seiringer proved complete condensation in the Gross–Pitaevskii limit, with the interaction scaled so that N a Na N a a a a [29 ] . Deuchert and Seiringer subsequently treated positive-temperature homogeneous gases with scattering length a N = a v L / N a_N = a_vL/N a N = a v L / N a v a_v a v [11 ] . These results allow the interaction scale to vary with particle number. The problem considered here instead fixes the potential and a positive density before taking the volume to infinity.
The limits between Gross–Pitaevskii scaling and a fixed-density thermodynamic limit have several forms. Brennecke, Brooks, Caraci, and Oldenburg proved complete ground-state condensation in the Gross–Pitaevskii regime and beyond [9 ] . Fournais treated low-energy states on growing boxes in a joint dilute and increasing-volume limit [19 ] , Theorem 1.2. Chong, Liang, and Nam gave simpler proofs of low-energy condensation using kinetic localization and Poincaré-type inequalities [10 ] . For the unit-box interaction N 2 − 2 κ v ( N 1 − κ x ) N^{2-2\kappa}v(N^{1-\kappa}x) N 2 − 2 κ v ( N 1 − κ x ) N 3 κ − 2 N^{3\kappa-2} N 3 κ − 2 κ < 2 / 3 \kappa< 2/3 κ < 2/3 [25 ] , Corollary 6.
The thermodynamic energy problem has a separate history. Dyson’s hard-sphere bounds and the leading dilute asymptotics of Lieb and Yngvason established the first-order energy scale [13 , 30 ] . The next correction was predicted by Lee, Huang, and Yang’s pseudopotential calculation [27 ] . Fournais and Solovej proved the corresponding lower bound for nonnegative radial finite-range potentials, including hard cores [20 ] , Theorems 1.1 and 1.3; recent upper bounds attain the same coefficient for hard spheres and for general nonnegative even finite-range potentials [3 , 4 ] . These energy results, and low-temperature free-energy estimates for strong potentials [18 ] , do not by themselves determine occupation of a single orbital: the one-particle kinetic gap tends to zero as the volume grows.
A distinct thermodynamic-limit result appears in Sütő’s two-step preprint [33 ] . Its Theorem 1.2 assumes nonnegativity both of the potential and of its Fourier transform, together with integrability, a Fourier-moment condition, and spatial decay, and states condensation above a temperature-dependent density threshold. Fourier positivity is an additional restriction: for example, a nonnegative spherical square barrier has a Fourier transform that changes sign. That result therefore does not cover all bounded nonnegative radial finite-range potentials, and its fixed-temperature threshold does not alone provide the joint low-density and low-temperature uniformity sought here.
Galanda and Pinamonti construct algebraic equilibrium states with a condensate background under quantitative convergence conditions for a spatially localized interaction [21 ] , Theorem 6.8. Their subsequent analysis treats renormalized correlation functions in Gross–Pitaevskii scaling, with a zero-temperature limit in its final convergence theorem [22 ] , Theorem 7.3. These constructions use a different state formulation and coupled interaction, regulator, and temperature limits from the fixed-potential canonical thermodynamic limit considered here.
The trajectory formulation descends from Feynman’s treatment of the Bose gas [17 ] ; for the canonical Feynman–Kac representation and its symmetrization and one-particle normalizations, see [35 ] . The random-path comparison has a different ancestry. Benjamini, Pemantle, and Peres constructed paths with exponentially decaying intersection tails [5 ] . Häggström and Mossel developed a hierarchical random-drift construction with low predictability; its independent block variables are a close antecedent of the dyadic velocities used below [24 ] . Abbe, Massoulié, Montanari, Sly, and Srivastava used finite-endpoint path averages and overlap estimates for group synchronization [1 ] ; Garban and Spencer used this approach for classical disordered-spin symmetry breaking along the Nishimori line [23 ] . Here the finite cell paths, their thickened intersection estimates, and the continuum bridge comparison require separate proofs.
How the comparison works
Let q 0 = ⟨ u 0 , γ ( 1 ) u 0 ⟩ / N q_0 = \langle u_0, \gamma^{(1)}u_0\rangle/N q 0 = ⟨ u 0 , γ ( 1 ) u 0 ⟩ / N μ \mu μ q 0 q_0 q 0 ν \nu ν a > 0 a > 0 a > 0 f = d ν / d μ f = \mathrm{d}\nu/\mathrm{d}\mu f = d ν / d μ
a 2 ≤ q 0 ∫ f 2 d μ . a^2 \le q_0 \int f^2 \,\mathrm{d}\mu. a 2 ≤ q 0 ∫ f 2 d μ . Thus the main problem is a comparison of positive path measures.
Partition space into cells of side s s s ρ − 1 / 2 \rho^{-1/2} ρ − 1/2 δ = s 2 \delta= s^2 δ = s 2 m = N s 3 / V m = Ns^3/V m = N s 3 / V ρ − 1 / 2 \rho^{-1/2} ρ − 1/2 1 / T 1/T 1/ T
The local analysis must control simultaneous failures in specified families of cells. Energy and entropy estimates first give occupation tails and separation of most endpoints at the shorter scale r = ζ ρ − 1 / 3 r = \zeta\rho^{-1/3} r = ζ ρ − 1/3 ζ > 0 \zeta> 0 ζ > 0
Between two distant cells, choose a random lattice path, called a skeleton, whose encounters with an independent copy have bounded exponential moments. Here an encounter records a vertex within a fixed cell distance of the other path; a second estimate weights all vertices by exponentially decaying distance. These two geometric bounds are established independently of the gas in Section 8.1. Detour around unusable components. Select one eligible particle from each cell on the resulting simple path, and cyclically reassign their starting points. One end is opened and the selected bridges are sampled from their exactly normalized interacting law. This defines ν \nu ν
The second moment involves two independently chosen paths in related configurations. Estimating both changes against the original closed configuration would incur a cost proportional to their lengths. Instead, a reverse insertion supplies a baseline in which contributions far from encounters cancel. Two effects remain small as ρ \rho ρ r − 1 / 2 r^{-1/2} r − 1/2 1 / m 1/m 1/ m
Changing trajectories also changes which cells are usable and which particles are eligible, so the two routes cannot be treated as independent of the gas. A stricter cell test in the reverse configuration controls these changes: away from encounters the selection weights agree, while rare components give a controlled list of possible detours. Keeping those selection weights throughout the comparison yields the required second-moment bound after averaging over the environment and the two paths.
Sections 3–7 prove the local analytic bounds. Section 8 constructs the paths and detours. Section 9 defines the change of measure, and Section 10 proves its second-moment estimate with all normalizations and selection constraints retained. Section 11 fixes the parameters in order and passes the uniform positive-temperature bound to zero temperature.
Scales and conventions
We use the Hamiltonian of Section 1 also on labelled coordinates, and suppress L L L v v v R v R_v R v ∥ v ∥ 1 , ∥ v ∥ ∞ \lVert v\rVert_1,\lVert v\rVert_\infty ∥ v ∥ 1 , ∥ v ∥ ∞ R 3 \mathbb{R}^3 R 3 v v v v L v_L v L
The orthogonal symmetrizer is included exactly once in a trace kernel. If k t k_t k t N N N
k t s y m ( X , Y ) = 1 N ! ∑ π ∈ S N k t ( X , π Y ) , Z N = ∫ k β s y m ( X , X ) d X . k_t^{\mathrm{sym}}(X,Y)=\frac{1}{N!}\sum_{\pi\in\mathfrak{S}_N} k_t(X,\pi Y),\qquad Z_N=\int k_\beta^{\mathrm{sym}}(X,X)\,\mathrm{d}X. k t sym ( X , Y ) = N ! 1 π ∈ S N ∑ k t ( X , π Y ) , Z N = ∫ k β sym ( X , X ) d X . Thus integration over labelled coordinates carries no further factorial. We work with ρ / 2 ≤ N / V ≤ 2 ρ \rho/2\leq N/V\leq2\rho ρ /2 ≤ N / V ≤ 2 ρ 0 < T ≤ ρ 2 0<T\leq\rho^2 0 < T ≤ ρ 2 β = T − 1 \beta=T^{-1} β = T − 1 ρ \rho ρ C , c > 0 C,c>0 C , c > 0 v v v ρ , L , N , T \rho,L,N,T ρ , L , N , T
We use a periodic grid G = ( Z / n Z ) 3 \mathcal{G}=(\mathbb{Z}/n\mathbb{Z})^3 G = ( Z / n Z ) 3
n = ⌊ L K ρ − 1 / 2 ⌋ , s = L / n ∈ [ K ρ − 1 / 2 , 2 K ρ − 1 / 2 ] , M = n 3 , m = N / M . n=\left\lfloor\frac{L}{K\rho^{-1/2}}\right\rfloor,\qquad s=L/n\in[K\rho^{-1/2},2K\rho^{-1/2}],\qquad M=n^3,\qquad m=N/M. n = ⌊ K ρ − 1/2 L ⌋ , s = L / n ∈ [ K ρ − 1/2 , 2 K ρ − 1/2 ] , M = n 3 , m = N / M . Here K ≥ 1 K\geq1 K ≥ 1 ρ s 3 / 2 ≤ m ≤ 2 ρ s 3 \rho s^3/2\leq m\leq2\rho s^3 ρ s 3 /2 ≤ m ≤ 2 ρ s 3
A union U U U u ≥ 1 u\geq1 u ≥ 1 ∣ U ∣ = u s 3 |U|=us^3 ∣ U ∣ = u s 3 h s hs h s h ≥ 0 h\geq0 h ≥ 0 C ( h + 2 ) 3 u C(h+2)^3u C ( h + 2 ) 3 u d ( ⋅ , ⋅ ) d(\cdot,\cdot) d ( ⋅ , ⋅ ) p p p n p ( X ) = ∑ i p ( x i ) n_p(X)=\sum_i p(x_i) n p ( X ) = ∑ i p ( x i ) n U n_U n U U U U
Energy and free-energy estimates
This section gives the particle-number costs needed to localize the gas. Throughout, 0 < T ≤ ρ 2 0<T\leq\rho^2 0 < T ≤ ρ 2 β = T − 1 \beta=T^{-1} β = T − 1 ρ / 2 ≤ N / V ≤ 2 ρ \rho/2\leq N/V\leq2\rho ρ /2 ≤ N / V ≤ 2 ρ ρ \rho ρ − W -W − W H j , L ( − W ) H_{j,L}^{(-W)} H j , L ( − W ) Z j ( − W ) Z_j^{(-W)} Z j ( − W ) F j ( − W ) = − T log Z j ( − W ) F_j^{(-W)}=-T\log Z_j^{(-W)} F j ( − W ) = − T log Z j ( − W ) Z 0 = 1 Z_0=1 Z 0 = 1 F 0 = 0 F_0=0 F 0 = 0 σ \sigma σ F ( − W ) ( σ ) = Tr H ( − W ) σ − T S ( σ ) \mathcal{F}^{(-W)}(\sigma)=\operatorname{Tr} H^{(-W)}\sigma-T S(\sigma) F ( − W ) ( σ ) = Tr H ( − W ) σ − T S ( σ ) S ( σ ) = − Tr ( σ log σ ) S(\sigma)=-\operatorname{Tr}(\sigma\log\sigma) S ( σ ) = − Tr ( σ log σ ) 0 log 0 = 0 0\log0=0 0 log 0 = 0 v v v ρ \rho ρ L L L N N N T T T
Lemma 3.1 (A Neumann cell bound). There are b 0 , c v > 0 b_0,c_v>0 b 0 , c v > 0 Q Q Q b ≥ b 0 b\ge b_0 b ≥ b 0 y ∈ Q y\in Q y ∈ Q f ∈ H 1 ( Q ) f\in H^1(Q) f ∈ H 1 ( Q )
∫ Q ( ∣ ∇ f ∣ 2 + v ( x − y ) ∣ f ∣ 2 ) d x ≥ c v b − 3 ∫ Q ∣ f ∣ 2 d x . (1) \int_Q\left(\lvert\nabla f\rvert^2+v(x-y)\lvert f\rvert^2\right)\,dx\ge c_vb^{-3}\int_Q\lvert f\rvert^2\,dx. \tag*{(1)} ∫ Q ( ∣ ∇ f ∣ 2 + v ( x − y ) ∣ f ∣ 2 ) d x ≥ c v b − 3 ∫ Q ∣ f ∣ 2 d x . (1) The same conclusion holds for fixed positive coefficients in front of the two terms, with changed constants. Consequently, for disjoint cubes Q Q Q b b b
c v b − 3 ∑ Q n Q 1 { n Q ≥ 2 } . (2) c_vb^{-3}\sum_Q n_Q\mathbf{1}_{\{n_Q\ge2\}}. \tag*{(2)} c v b − 3 Q ∑ n Q 1 { n Q ≥ 2 } . (2) Both statements apply to torus cubes smaller than the torus, and to Dirichlet states extended by zero.
Proof. Choose ν > 0 \nu>0 ν > 0 A = { x : v ( x ) ≥ ν } A=\{x:v(x)\ge\nu\} A = { x : v ( x ) ≥ ν } b 0 b_0 b 0 b 0 / 2 > R v b_0/2>R_v b 0 /2 > R v b 0 ≥ 1 b_0\ge1 b 0 ≥ 1 y ∈ Q y\in Q y ∈ Q b / 2 b/2 b /2 y + A y+A y + A Q Q Q ∣ A ∣ / 8 = : a v > 0 \lvert A\rvert/8=:a_v>0 ∣ A ∣ /8 =: a v > 0 A y A_y A y f 0 = ∣ Q ∣ − 1 ∫ Q f f_0=\lvert Q\rvert^{-1}\int_Qf f 0 = ∣ Q ∣ − 1 ∫ Q f ∥ f − f 0 ∥ 6 ≤ C ∥ ∇ f ∥ 2 \lVert f-f_0\rVert_6\le C\lVert\nabla f\rVert_2 ∥ f − f 0 ∥ 6 ≤ C ∥ ∇ f ∥ 2
a v ∣ f 0 ∣ 2 ≤ 2 ∫ A y ∣ f ∣ 2 + 2 ∫ A y ∣ f − f 0 ∣ 2 ≤ 2 ν − 1 ∫ Q v ( x − y ) ∣ f ∣ 2 + C a v 2 / 3 ∫ Q ∣ ∇ f ∣ 2 . a_v\lvert f_0\rvert^2\le2\int_{A_y}\lvert f\rvert^2+2\int_{A_y}\lvert f-f_0\rvert^2\le2\nu^{-1}\int_Qv(x-y)\lvert f\rvert^2+Ca_v^{2/3}\int_Q\lvert\nabla f\rvert^2. a v ∣ f 0 ∣ 2 ≤ 2 ∫ A y ∣ f ∣ 2 + 2 ∫ A y ∣ f − f 0 ∣ 2 ≤ 2 ν − 1 ∫ Q v ( x − y ) ∣ f ∣ 2 + C a v 2/3 ∫ Q ∣ ∇ f ∣ 2 . Moreover ∥ f − f 0 ∥ 2 2 ≤ C b 2 ∥ ∇ f ∥ 2 2 \lVert f-f_0\rVert_2^2\le Cb^2\lVert\nabla f\rVert_2^2 ∥ f − f 0 ∥ 2 2 ≤ C b 2 ∥ ∇ f ∥ 2 2 b 2 ≤ b 3 b^2\le b^3 b 2 ≤ b 3 b ≥ 1 b\ge1 b ≥ 1
For (2), cut configuration space according to the cube containing each coordinate and discard cross-cube interactions. This is Neumann bracketing: the restrictions of an H 1 H^1 H 1 j ≥ 2 j\ge2 j ≥ 2 ⌊ j / 2 ⌋ ≥ j / 3 \lfloor j/2\rfloor\ge j/3 ⌊ j /2 ⌋ ≥ j /3 v v v □ \square □
Lemma 3.2 (An entropy bound). If Ω \Omega Ω α > 0 \alpha>0 α > 0 j j j
α Tr ( ∑ i = 1 j − Δ i ) σ − T S ( σ ) ≥ − T j − C α T ∣ Ω ∣ ( V − 1 + T 3 / 2 ) . (3) \alpha\operatorname{Tr}\left(\sum_{i=1}^{j}-\Delta_i\right)\sigma-T S(\sigma)\ge-Tj-C_\alpha T\lvert\Omega\rvert\left(V^{-1}+T^{3/2}\right). \tag*{(3)} α Tr ( i = 1 ∑ j − Δ i ) σ − T S ( σ ) ≥ − T j − C α T ∣ Ω ∣ ( V − 1 + T 3/2 ) . (3) In particular a state of finite kinetic energy has finite entropy.
Proof. The periodic heat kernel satisfies K t ( 0 ) ≤ C ( V − 1 + t − 3 / 2 ) K_t(0)\le C(V^{-1}+t^{-3/2}) K t ( 0 ) ≤ C ( V − 1 + t − 3/2 ) ∂ Ω \partial\Omega ∂ Ω Tr e t Δ Ω ≤ C α ∣ Ω ∣ ( V − 1 + t − 3 / 2 ) \operatorname{Tr}e^{t\Delta_\Omega}\le C_\alpha\lvert\Omega\rvert(V^{-1}+t^{-3/2}) Tr e t Δ Ω ≤ C α ∣ Ω ∣ ( V − 1 + t − 3/2 ) e − 1 e^{-1} e − 1
∑ l ≥ 1 e − l l Tr e l β α Δ Ω ≤ C α ∣ Ω ∣ ( V − 1 + T 3 / 2 ) . \sum_{l\ge1}\frac{e^{-l}}{l}\operatorname{Tr}e^{l\beta\alpha\Delta_\Omega}\le C_\alpha\lvert\Omega\rvert\left(V^{-1}+T^{3/2}\right). l ≥ 1 ∑ l e − l Tr e l β α Δ Ω ≤ C α ∣ Ω ∣ ( V − 1 + T 3/2 ) . Its j j j e − j e^{-j} e − j
Proposition 3.3 (Particle-number free-energy bounds). Uniformly in the temperature range under consideration,
F N ≥ c v ρ N , (4) F_N \ge c_v\rho N, \tag*{(4)} F N ≥ c v ρN , (4) F N + k ≥ F N + c v ρ k ( k ≥ 0 ) . (5) F_{N+k} \ge F_N+c_v\rho k \qquad(k \ge0). \tag*{(5)} F N + k ≥ F N + c v ρ k ( k ≥ 0 ) . (5) For every bounded W ≥ 0 W \ge0 W ≥ 0
F j ( − W ) − F j − 1 ( − W ) ≤ C v ρ ( 1 ≤ j ≤ 4 ρ V ) . (6) F_j^{(-W)}-F_{j-1}^{(-W)} \le C_v\rho\qquad(1 \le j \le4\rho V). \tag*{(6)} F j ( − W ) − F j − 1 ( − W ) ≤ C v ρ ( 1 ≤ j ≤ 4 ρ V ) . (6) The constant in (6) is independent of W W W
Proof. Partition the torus into equal cubes of side in [ b , 2 b ] [b,2b] [ b , 2 b ] b = ( 4 / ρ ) 1 / 3 b=(4/\rho)^{1/3} b = ( 4/ ρ ) 1/3 V / b 3 = ρ V / 4 ≤ N / 2 V/b^3=\rho V/4\le N/2 V / b 3 = ρ V /4 ≤ N /2 L L L ∑ Q n Q 1 { n Q ≥ 2 } ≥ N − # { Q } ≥ N / 2 \sum_Q n_Q\mathbf{1}_{\{n_Q\ge2\}}\ge N-\#\{Q\}\ge N/2 ∑ Q n Q 1 { n Q ≥ 2 } ≥ N − # { Q } ≥ N /2 c v ρ N c_v\rho N c v ρN T N + C T ( 1 + V T 3 / 2 ) TN+CT(1+VT^{3/2}) T N + C T ( 1 + V T 3/2 ) ρ N \rho N ρN ρ ↓ 0 \rho\downarrow0 ρ ↓ 0 T / ρ ≤ ρ T/\rho\le\rho T / ρ ≤ ρ T / ( ρ N ) ≤ ρ / N T/(\rho N)\le\rho/N T / ( ρN ) ≤ ρ / N V T 5 / 2 / ( ρ N ) ≤ 2 ρ 3 VT^{5/2}/(\rho N)\le2\rho^3 V T 5/2 / ( ρN ) ≤ 2 ρ 3 c v c_v c v
We next show that F j / j F_j/j F j / j j ≥ 1 j\ge1 j ≥ 1 j j j S i S_i S i i i i S 0 = 0 S_0=0 S 0 = 0 [28 ] applied to two overlapping i i i i − 1 i-1 i − 1 2 S i ≥ S i − 1 + S i + 1 2S_i\ge S_{i-1}+S_{i+1} 2 S i ≥ S i − 1 + S i + 1 i = 1 i=1 i = 1 S i S_i S i S j / j ≤ S j − 1 / ( j − 1 ) S_j/j\le S_{j-1}/(j-1) S j / j ≤ S j − 1 / ( j − 1 ) j ≥ 2 j\ge2 j ≥ 2 ( j − 1 ) (j-1) ( j − 1 ) ( j − 1 ) / 2 (j-1)/2 ( j − 1 ) /2 ( j − 2 ) / 2 (j-2)/2 ( j − 2 ) /2 j = 2 j=2 j = 2 F j / j ≥ F j − 1 / ( j − 1 ) F_j/j\ge F_{j-1}/(j-1) F j / j ≥ F j − 1 / ( j − 1 ) F N + k ≥ ( N + k ) F N / N F_{N+k}\ge(N+k)F_N/N F N + k ≥ ( N + k ) F N / N
The lower increments now quantify the free-energy gain from adding particles. To obtain the complementary upper cost, uniformly even in an attractive external field, we insert one particle directly in the path representation.
Retain in the symmetrized Feynman–Kac formula only those permutations in which the extra label is fixed. There are ( j − 1 ) ! (j-1)! ( j − 1 )! 1 / j ! 1/j! 1/ j ! V K β ( 0 ) VK_\beta(0) V K β ( 0 ) β ( j − 1 ) ∥ v ∥ 1 / V \beta(j-1)\|v\|_1/V β ( j − 1 ) ∥ v ∥ 1 / V W ≥ 0 W\ge0 W ≥ 0
Z j ( − W ) ≥ V K β ( 0 ) j e − β ( j − 1 ) ∥ v ∥ 1 / V Z j − 1 ( − W ) . Z_j^{(-W)} \ge\frac{VK_\beta(0)}{j}e^{-\beta(j-1)\|v\|_1/V}Z_{j-1}^{(-W)}. Z j ( − W ) ≥ j V K β ( 0 ) e − β ( j − 1 ) ∥ v ∥ 1 / V Z j − 1 ( − W ) . Since K β ( 0 ) ≥ ( 4 π β ) − 3 / 2 K_\beta(0)\ge(4\pi\beta)^{-3/2} K β ( 0 ) ≥ ( 4 π β ) − 3/2 T log + ( C ρ T − 3 / 2 ) T\log_+(C\rho T^{-3/2}) T log + ( C ρ T − 3/2 ) ρ \rho ρ t ↦ t log ( C ρ t − 3 / 2 ) t\mapsto t\log(C\rho t^{-3/2}) t ↦ t log ( C ρ t − 3/2 ) 0 < t ≤ ρ 2 0<t\le\rho^2 0 < t ≤ ρ 2 log ( C ρ t − 3 / 2 ) − 3 / 2 ≥ log ( C ρ − 2 ) − 3 / 2 ≥ 0 \log(C\rho t^{-3/2})-3/2\ge\log(C\rho^{-2})-3/2\ge0 log ( C ρ t − 3/2 ) − 3/2 ≥ log ( C ρ − 2 ) − 3/2 ≥ 0 C ρ 2 ( 1 + ∣ log ρ ∣ ) ≤ C ρ C\rho^2(1+|\log\rho|)\le C\rho C ρ 2 ( 1 + ∣ log ρ ∣ ) ≤ C ρ 4 ∥ v ∥ 1 ρ 4\|v\|_1\rho 4∥ v ∥ 1 ρ T ↓ 0 T\downarrow0 T ↓ 0
Localization, occupation, and isolation
We now obtain simultaneous probability bounds for specified collections of cells. The important scale is s ≍ K ρ − 1 / 2 s \asymp K\rho^{-1/2} s ≍ K ρ − 1/2 m = N s 3 / V ≍ ρ s 3 m = Ns^3/V \asymp\rho s^3 m = N s 3 / V ≍ ρ s 3 m m m
The estimates use the same variational comparison. Spatial localization gives a lower bound for the free-energy cost of a tilted state, while diagonal tilting gives an upper bound for that cost in terms of a gradient error and the tilt’s normalization. Comparing the two bounds controls the derivative of a logarithmic exponential moment. Positive density tilts control excess occupation, negative tilts control shortages, and a tilt of the close-particle count will give isolation.
Lemma 4.1 (Two-color localization and diagonal tilting). Let p 2 + q 2 = 1 p^2+q^2=1 p 2 + q 2 = 1 p p p H 0 1 ( Ω ) H_0^1(\Omega) H 0 1 ( Ω ) Ω \Omega Ω J J J σ \sigma σ w j w_j w j p p p σ i n , j , σ o u t , j \sigma_{\mathrm{in},j},\sigma_{\mathrm{out},j} σ in , j , σ out , j
F ( − W ) ( σ ) + Tr n ∣ ∇ p ∣ 2 + ∣ ∇ q ∣ 2 σ ≥ ∑ j = 0 J w j [ F ( − W ) ( σ i n , j ) + F ( − W ) ( σ o u t , j ) ] − T H ( w ) . (7) \mathcal{F}^{(-W)}(\sigma)+\operatorname{Tr} n|\nabla p|^2+|\nabla q|^2\sigma
\ge\sum_{j=0}^{J}w_j\left[\mathcal{F}^{(-W)}(\sigma_{\mathrm{in},j})+\mathcal{F}^{(-W)}(\sigma_{\mathrm{out},j})\right]-T H(w). \tag*{(7)} F ( − W ) ( σ ) + Tr n ∣∇ p ∣ 2 + ∣∇ q ∣ 2 σ ≥ j = 0 ∑ J w j [ F ( − W ) ( σ in , j ) + F ( − W ) ( σ out , j ) ] − T H ( w ) . (7) Here H ( w ) = − ∑ j w j log w j H(w)=-\sum_j w_j\log w_j H ( w ) = − ∑ j w j log w j 0 log 0 = 0 0\log0=0 0 log 0 = 0 H ( w ) ≤ C + ∑ j j w j H(w)\le C+\sum_j j w_j H ( w ) ≤ C + ∑ j j w j ∑ j j w j = Tr n p 2 σ \sum_j j w_j=\operatorname{Tr}np^2\sigma ∑ j j w j = Tr n p 2 σ
Let Γ = Z N , L − 1 e − β H N , L \Gamma=Z_{N,L}^{-1}e^{-\beta H_{N,L}} Γ = Z N , L − 1 e − β H N , L P P P G G G g = e G g=e^G g = e G M G = E P e 2 G M_G=\mathbb{E}_P e^{2G} M G = E P e 2 G σ = g Γ g / M G \sigma=g\Gamma g/M_G σ = g Γ g / M G E g \mathbb{E}_g E g
F ( σ ) − F N ≤ E g ∑ i ∣ ∇ i G ∣ 2 + T ( 2 E g G − log M G ) . (8) \mathcal{F}(\sigma)-F_N\le\mathbb{E}_g\sum_i|\nabla_iG|^2+T\left(2\mathbb{E}_gG-\log M_G\right). \tag*{(8)} F ( σ ) − F N ≤ E g i ∑ ∣ ∇ i G ∣ 2 + T ( 2 E g G − log M G ) . (8) Proof. The geometric localization and entropy split behind (7) also appear in [12, Section 3, Lemma 3.1]. We give the decomposition into fixed color-number sectors explicitly. The one-particle map f ↦ ( p f , q f ) f\mapsto(pf,qf) f ↦ ( p f , q f ) j j j Sym j L 2 ( Ω ) ⊗ Sym J − j L 2 ( Λ L ) \operatorname{Sym}^j L^2(\Omega)\otimes\operatorname{Sym}^{J-j}L^2(\Lambda_L) Sym j L 2 ( Ω ) ⊗ Sym J − j L 2 ( Λ L ) p ∇ p + q ∇ q = 0 p\nabla p+q\nabla q=0 p ∇ p + q ∇ q = 0 H ( w ) + ∑ j w j [ S ( σ i n , j ) + S ( σ o u t , j ) ] H(w)+\sum_jw_j[S(\sigma_{\mathrm{in},j})+S(\sigma_{\mathrm{out},j})] H ( w ) + ∑ j w j [ S ( σ in , j ) + S ( σ out , j )] w w w ( 1 − e − 1 ) e − j (1-e^{-1})e^{-j} ( 1 − e − 1 ) e − j H ( w ) ≤ − log ( 1 − e − 1 ) + E j H(w)\le-\log(1-e^{-1})+\mathbb{E}j H ( w ) ≤ − log ( 1 − e − 1 ) + E j p ( x i ) 2 p(x_i)^2 p ( x i ) 2 q ( x i ) 2 q(x_i)^2 q ( x i ) 2
For (8), define τ = Γ 1 / 2 g 2 Γ 1 / 2 / M G \tau=\Gamma^{1/2}g^2\Gamma^{1/2}/M_G τ = Γ 1/2 g 2 Γ 1/2 / M G τ \tau τ σ \sigma σ B ∗ B B^*B B ∗ B B B ∗ BB^* B B ∗ B = g Γ 1 / 2 / M G B=g\Gamma^{1/2}/\sqrt{M_G} B = g Γ 1/2 / M G H ψ = E ψ H\psi=E\psi H ψ = E ψ g 2 ψ g^2\psi g 2 ψ
⟨ g ψ , H g ψ ⟩ = E ⟨ ψ , g 2 ψ ⟩ + ⟨ ψ , ∑ i ∣ ∇ i g ∣ 2 ψ ⟩ . \langle g\psi,Hg\psi\rangle=E\langle\psi,g^2\psi\rangle+\langle\psi,\sum_i|\nabla_i g|^2\psi\rangle. ⟨ g ψ , H g ψ ⟩ = E ⟨ ψ , g 2 ψ ⟩ + ⟨ ψ , i ∑ ∣ ∇ i g ∣ 2 ψ ⟩ . After summing in an eigenbasis, Tr H σ = Tr H τ + E g ∑ i ∣ ∇ i G ∣ 2 \operatorname{Tr} H\sigma= \operatorname{Tr} H\tau+ \mathbb{E}_g\sum_i \lvert\nabla_i G\rvert^2 Tr H σ = Tr H τ + E g ∑ i ∣ ∇ i G ∣ 2 g g g N , L N,L N , L
To compare relative entropies, approximate g 2 g^2 g 2 E E E p E = Tr Γ 1 E p_E=\operatorname{Tr}\Gamma1_E p E = Tr Γ 1 E p E > 0 p_E>0 p E > 0 τ E = Γ 1 / 2 1 E Γ 1 / 2 / p E \tau_E=\Gamma^{1/2}1_E\Gamma^{1/2}/p_E τ E = Γ 1/2 1 E Γ 1/2 / p E τ E \tau_E τ E p E p_E p E p E g E 2 / M G p_Eg_E^2/M_G p E g E 2 / M G [28 ] , Corollary (3.2) and Section IV) therefore bounds D ( τ ∥ Γ ) D(\tau\Vert\Gamma) D ( τ ∥Γ ) g g g D ( τ ∥ Γ ) ≤ 2 E g G − log M G D(\tau\Vert\Gamma)\le2\mathbb{E}_gG-\log M_G D ( τ ∥Γ ) ≤ 2 E g G − log M G F ( τ ) − F N = T D ( τ ∥ Γ ) \mathcal{F}(\tau)-F_N=TD(\tau\Vert\Gamma) F ( τ ) − F N = T D ( τ ∥Γ )
For a nonempty union U U U ϕ U ( x ) = e − d ( x , U ) / s \phi_U(x)=e^{-d(x,U)/s} ϕ U ( x ) = e − d ( x , U ) / s h s hs h s U U U C ( h + 2 ) 3 C(h+2)^3 C ( h + 2 ) 3 U U U
Lemma 4.2 (Localized density bounds). Let D 0 = H 0 ρ D_0=H_0\rho D 0 = H 0 ρ H 0 H_0 H 0 J ≤ N J\le N J ≤ N 0 ≤ W ≤ ρ 0\le W\le\rho 0 ≤ W ≤ ρ J J J
F ( − W ) ( σ ) − F J ( − W ) ≥ ( c v D 0 − C v ρ − C s s − 2 ) E σ n ϕ U − C v D 0 2 ∣ U ∣ . (9) \mathcal{F}^{(-W)}(\sigma)-F_J^{(-W)}\ge(c_vD_0-C_v\rho-C_ss^{-2})\mathbb{E}_\sigma n\phi_U-C_vD_0^2\lvert U\rvert. \tag*{(9)} F ( − W ) ( σ ) − F J ( − W ) ≥ ( c v D 0 − C v ρ − C s s − 2 ) E σ n ϕ U − C v D 0 2 ∣ U ∣ . (9) In the corresponding equilibrium state,
E n U ≤ C v ρ ∣ U ∣ . (10) \mathbb{E}n_U\le C_v\rho\lvert U\rvert. \tag*{(10)} E n U ≤ C v ρ ∣ U ∣ . (10) For the unmodified N N N
log E P e 2 λ n ϕ U ≤ C v λ ρ ∣ U ∣ ( 0 ≤ λ ≤ 1 ) . (11) \log\mathbb{E}_P e^{2\lambda n\phi_U}\le C_v\lambda\rho\lvert U\rvert\qquad(0\le\lambda\le1). \tag*{(11)} log E P e 2 λn ϕ U ≤ C v λ ρ ∣ U ∣ ( 0 ≤ λ ≤ 1 ) . (11) In particular E P n Q 2 ≤ C m 2 \mathbb{E}_P n_Q^2\le Cm^2 E P n Q 2 ≤ C m 2 Q Q Q
Proof. For h ≥ 0 h\ge0 h ≥ 0 p h = 1 p_h=1 p h = 1 h s hs h s U U U h s hs h s ( h + 1 ) s (h+1)s ( h + 1 ) s q h = ( 1 − p h 2 ) 1 / 2 q_h=(1-p_h^2)^{1/2} q h = ( 1 − p h 2 ) 1/2 ∣ ∇ p h ∣ 2 + ∣ ∇ q h ∣ 2 ≤ C s − 2 \lvert\nabla p_h\rvert^2+\lvert\nabla q_h\rvert^2\le Cs^{-2} ∣ ∇ p h ∣ 2 + ∣ ∇ q h ∣ 2 ≤ C s − 2 Ω h \Omega_h Ω h ( h + 2 ) s (h+2)s ( h + 2 ) s C ( h + 4 ) 3 ∣ U ∣ C(h+4)^3\lvert U\rvert C ( h + 4 ) 3 ∣ U ∣
Apply (7). Inside, partition the torus into cubes of side in [ D 0 − 1 / 3 , 2 D 0 − 1 / 3 ] [D_0^{-1/3},2D_0^{-1/3}] [ D 0 − 1/3 , 2 D 0 − 1/3 ] b 0 b_0 b 0 s s s Ω h \Omega_h Ω h C D 0 ( h + 4 ) 3 ∣ U ∣ CD_0(h+4)^3\lvert U\rvert C D 0 ( h + 4 ) 3 ∣ U ∣ c v D 0 j − C D 0 2 ( h + 4 ) 3 ∣ U ∣ c_vD_0j-CD_0^2(h+4)^3\lvert U\rvert c v D 0 j − C D 0 2 ( h + 4 ) 3 ∣ U ∣ ρ j \rho j ρ j F J − j ( − W ) ≥ F J ( − W ) − C v ρ j F_{J-j}^{(-W)}\ge F_J^{(-W)}-C_v\rho j F J − j ( − W ) ≥ F J ( − W ) − C v ρ j T ( C + j ) T(C+j) T ( C + j ) T ≤ ρ 2 T\le\rho^2 T ≤ ρ 2 j j j c v D 0 c_vD_0 c v D 0 C v ρ C_v\rho C v ρ C T [ 1 + ∣ Ω h ∣ ( V − 1 + T 3 / 2 ) ] CT[1+\lvert\Omega_h\rvert(V^{-1}+T^{3/2})] C T [ 1 + ∣ Ω h ∣ ( V − 1 + T 3/2 )] C v D 0 2 ( h + 4 ) 3 ∣ U ∣ C_vD_0^2(h+4)^3\lvert U\rvert C v D 0 2 ( h + 4 ) 3 ∣ U ∣ ∣ U ∣ ≥ s 3 \lvert U\rvert\ge s^3 ∣ U ∣ ≥ s 3 V ≥ s 3 V\ge s^3 V ≥ s 3
Average in h h h e − h d h e^{-h}\,dh e − h d h x x x ∫ 0 ∞ p h ( x ) 2 e − h d h ≥ ϕ U ( x ) \int_0^\infty p_h(x)^2e^{-h}\,dh\ge\phi_U(x) ∫ 0 ∞ p h ( x ) 2 e − h d h ≥ ϕ U ( x ) C s s − 2 ϕ U ( x ) C_ss^{-2}\phi_U(x) C s s − 2 ϕ U ( x ) h ∈ [ max ( 0 , d ( x , U ) / s − 1 ) , d ( x , U ) / s ] h\in[\max(0,d(x,U)/s-1),d(x,U)/s] h ∈ [ max ( 0 , d ( x , U ) / s − 1 ) , d ( x , U ) / s ] ( h + 4 ) 3 (h+4)^3 ( h + 4 ) 3 H 0 H_0 H 0 c v D 0 − C v ρ c_vD_0-C_v\rho c v D 0 − C v ρ n U ≤ n ϕ U n_U\le n_{\phi_U} n U ≤ n ϕ U
For (11), take G = λ n ϕ U G=\lambda n\phi_U G = λn ϕ U ∑ i ∣ ∇ i G ∣ 2 ≤ λ 2 s − 2 n ϕ U \sum_i\lvert\nabla_iG\rvert^2\le\lambda^2s^{-2}n_{\phi_U} ∑ i ∣ ∇ i G ∣ 2 ≤ λ 2 s − 2 n ϕ U G ≥ 0 G\ge0 G ≥ 0 log M G ≥ 0 \log M_G\ge0 log M G ≥ 0 2 T λ E n ϕ U 2T\lambda\mathbb{E}n_{\phi_U} 2 T λ E n ϕ U H 0 H_0 H 0 E g n ϕ U ≤ C v ρ ∣ U ∣ \mathbb{E}g n_{\phi_U} \le C_v\rho|U| E g n ϕ U ≤ C v ρ ∣ U ∣ 0 ≤ λ ≤ 1 0 \le\lambda\le1 0 ≤ λ ≤ 1 ∂ λ log M G = 2 E g n ϕ U \partial_\lambda\log M_G = 2\mathbb{E}g n_{\phi_U} ∂ λ log M G = 2 E g n ϕ U P ( n Q ≥ t ) ≤ min ( 1 , e C m − 2 t ) P(n_Q \ge t) \le\min(1,e^{C m-2t}) P ( n Q ≥ t ) ≤ min ( 1 , e C m − 2 t ) 2 t P ( n Q > t ) 2tP(n_Q > t) 2 tP ( n Q > t ) E n Q 2 ≤ C ( m + 1 ) 2 \mathbb{E}n_Q^2 \le C(m+1)^2 E n Q 2 ≤ C ( m + 1 ) 2 C m 2 Cm^2 C m 2 m → ∞ m \to\infty m → ∞
The mean bound alone does not guarantee that a cell supplies a particle to a spatial route. We next control simultaneous shortages and excesses in any specified family of cells. The lower tail uses the free-energy gain from placing extra particles in an underoccupied region; the upper tail follows from the exponential density estimate.
Lemma 4.3 (Simultaneous lower and upper occupation tails). There are fixed 0 < b − < 1 / 4 0 < b_- < 1/4 0 < b − < 1/4 b + > 4 b_+ > 4 b + > 4 U U U u ≥ 1 u \ge1 u ≥ 1 { n Q < 2 b − m \{n_Q < 2b_-m { n Q < 2 b − m Q ⊂ U } Q \subset U\} Q ⊂ U } { n Q > b + m / 2 \{n_Q > b_+m/2 { n Q > b + m /2 Q ⊂ U } Q \subset U\} Q ⊂ U }
C e − c m u . (12) Ce^{-cmu}. \tag*{(12)} C e − c m u . (12) The constants and the required large fixed K K K v v v
Proof. In each cell of U U U p = 1 p=1 p = 1 s / 4 s/4 s /4 s / 3 s/3 s /3 C s − 2 Cs^{-2} C s − 2 0 ≤ ϕ ≤ 1 0 \le\phi\le1 0 ≤ ϕ ≤ 1 ∣ ∇ ϕ ∣ 2 ≤ C ϕ / s 2 |\nabla\phi|^2 \le C_\phi/s^2 ∣∇ ϕ ∣ 2 ≤ C ϕ / s 2 p p p s / 8 s/8 s /8 φ \varphi φ ∫ ∣ ∇ φ ∣ 2 ≤ C s − 2 \int|\nabla\varphi|^2 \le Cs^{-2} ∫ ∣∇ φ ∣ 2 ≤ C s − 2 ∥ φ ∥ ∞ 2 ≤ C / ∣ U ∣ \|\varphi\|_\infty^2 \le C/|U| ∥ φ ∥ ∞ 2 ≤ C /∣ U ∣ R v R_v R v supp q \operatorname{supp}q supp q
The conditional outside state is supported in ( supp q ) N − j (\operatorname{supp}q)^{N-j} ( supp q ) N − j j j j k k k φ \varphi φ
ψ ⟼ ( N − j + k k ) 1 / 2 P s y m ( ψ ⊗ φ ⊗ k ) . \psi\longmapsto\binom{N-j+k}{k}^{1/2}P_{\mathrm{sym}}(\psi\otimes\varphi^{\otimes k}). ψ ⟼ ( k N − j + k ) 1/2 P sym ( ψ ⊗ φ ⊗ k ) . Distinct assignments of the new labels have disjoint supports, so this is an isometry. It preserves entropy and has additive kinetic energy; the separated supports also eliminate all old–new interactions. A new–new pair has expectation at most ∥ v ∥ 1 ∥ φ ∥ ∞ 2 \|v\|_1\|\varphi\|_\infty^2 ∥ v ∥ 1 ∥ φ ∥ ∞ 2
F ( σ o u t , j ) ≥ F N − j + k − C k / s 2 − C v k 2 / ∣ U ∣ . \mathcal{F}(\sigma_{\mathrm{out},j}) \ge F_{N-j+k}-Ck/s^2-C_vk^2/|U|. F ( σ out , j ) ≥ F N − j + k − C k / s 2 − C v k 2 /∣ U ∣. Choose a sufficiently small fixed η > 0 \eta>0 η > 0 k = ⌊ η ρ ∣ U ∣ ⌋ k=\lfloor\eta\rho|U|\rfloor k = ⌊ η ρ ∣ U ∣ ⌋ ρ ∣ U ∣ ≥ ρ s 3 → ∞ \rho|U|\ge\rho s^3\to\infty ρ ∣ U ∣ ≥ ρ s 3 → ∞ k ≥ η ρ ∣ U ∣ / 2 k\ge\eta\rho|U|/2 k ≥ η ρ ∣ U ∣/2 N + k ≤ 4 ρ V N+k\le4\rho V N + k ≤ 4 ρ V j j j F N − j + k ≥ F N + c v ρ k − C v ρ j F_{N-j+k}\ge F_N+c_v\rho k-C_v\rho j F N − j + k ≥ F N + c v ρ k − C v ρ j η \eta η C v k 2 / ∣ U ∣ C_vk^2/|U| C v k 2 /∣ U ∣ K K K C k / s 2 Ck/s^2 C k / s 2 c v ρ k c_v\rho k c v ρ k ρ 2 ∣ U ∣ \rho^2|U| ρ 2 ∣ U ∣ p 2 ≤ ϕ p^2\le\phi p 2 ≤ ϕ
F ( σ ) − F N ≥ c v ′ ρ 2 ∣ U ∣ − C v ρ E σ n ϕ . \mathcal{F}(\sigma)-F_N \ge c'_v\rho^2|U|-C_v\rho\mathbb{E}_\sigma n_\phi. F ( σ ) − F N ≥ c v ′ ρ 2 ∣ U ∣ − C v ρ E σ n ϕ . Here the localization term has been absorbed in the latter error, since s − 2 ≤ ρ / K 2 s^{-2}\le\rho/K^2 s − 2 ≤ ρ / K 2
Now set G = − λ n ϕ G=-\lambda n_\phi G = − λ n ϕ 0 ≤ λ ≤ 1 0\le\lambda\le1 0 ≤ λ ≤ 1 C λ s − 2 E g n ϕ C\lambda s^{-2}\mathbb{E}_g n_\phi C λ s − 2 E g n ϕ − T log M G -T\log M_G − T log M G G ≤ 0 G\le0 G ≤ 0 C v ρ E g n ϕ − T log M G ≥ c v ′ ρ 2 ∣ U ∣ C_v\rho\mathbb{E}_g n_\phi-T\log M_G\ge c'_v\rho^2|U| C v ρ E g n ϕ − T log M G ≥ c v ′ ρ 2 ∣ U ∣ f ( λ ) = − log M G f(\lambda)=-\log M_G f ( λ ) = − log M G f ′ ( λ ) = 2 E g n ϕ f'(\lambda)=2\mathbb{E}_g n_\phi f ′ ( λ ) = 2 E g n ϕ f ( λ ) ≤ a ρ ∣ U ∣ f(\lambda) \le a\rho|U| f ( λ ) ≤ a ρ ∣ U ∣ T f ′ ( λ ) ≤ a ρ 3 ∣ U ∣ Tf'(\lambda) \le a\rho^3|U| T f ′ ( λ ) ≤ a ρ 3 ∣ U ∣ f ′ ( λ ) ≥ c v ′ ′ ρ ∣ U ∣ f'(\lambda) \ge c''_v\rho|U| f ′ ( λ ) ≥ c v ′′ ρ ∣ U ∣ 0 < a < c v ′ ′ 0<a<c''_v 0 < a < c v ′′ f ( 0 ) = 0 f(0)=0 f ( 0 ) = 0 f ( 1 ) ≥ a ρ ∣ U ∣ f(1)\ge a\rho|U| f ( 1 ) ≥ a ρ ∣ U ∣ E e − 2 n ϕ ≤ e − a ρ ∣ U ∣ \mathbb{E}e^{-2n_\phi}\le e^{-a\rho|U|} E e − 2 n ϕ ≤ e − a ρ ∣ U ∣
On the lower-count event, n ϕ ≤ 2 b − m u n_\phi\le2b_-m_u n ϕ ≤ 2 b − m u e 4 b − m u − a ρ ∣ U ∣ e^{4b_-m_u-a\rho|U|} e 4 b − m u − a ρ ∣ U ∣ m u / 2 ≤ ρ ∣ U ∣ ≤ 2 m u m_u/2\le\rho|U|\le2m_u m u /2 ≤ ρ ∣ U ∣ ≤ 2 m u b − b_- b − e − c m u e^{-cm_u} e − c m u n ϕ U ≥ b + m u / 2 n_{\phi_U}\ge b_+m_u/2 n ϕ U ≥ b + m u /2 λ = 1 \lambda=1 λ = 1 e c v ρ ∣ U ∣ − b + m u e^{c_v\rho|U|-b_+m_u} e c v ρ ∣ U ∣ − b + m u b + b_+ b +
Occupation control supplies many particles per cell. The bridge comparison will additionally require well-separated endpoints. The next estimate shows that a prescribed region rarely contains many particles with a neighbor at the shorter scale r r r
Lemma 4.4 (Isolation at a shorter scale). Fix 0 < ζ ≤ 1 0<\zeta\le1 0 < ζ ≤ 1 r = ζ ρ − 1 / 3 r=\zeta\rho^{-1/3} r = ζ ρ − 1/3 λ 0 = r ρ \lambda_0=r\sqrt{\rho} λ 0 = r ρ r r r S S S k ′ ≥ 0 k'\ge0 k ′ ≥ 0
P ( # { nonisolated particles in S } ≥ k ′ ) ≤ exp { C v λ 0 ζ 3 ρ ∣ S ∣ − λ 0 k ′ } . (13) \mathbb{P}\bigl(\#\{\text{nonisolated particles in }S\}\ge k'\bigr)
\le\exp\{C_v\lambda_0\zeta^3\rho|S|-\lambda_0k'\}. \tag*{(13)} P ( # { nonisolated particles in S } ≥ k ′ ) ≤ exp { C v λ 0 ζ 3 ρ ∣ S ∣ − λ 0 k ′ } . (13) The constant C v C_v C v ζ \zeta ζ ζ \zeta ζ
Proof. Partition the torus into cubes of side in [ r , 2 r ] [r,2r] [ r , 2 r ] S S S 0 ≤ χ ≤ 1 0\le\chi\le1 0 ≤ χ ≤ 1 ∣ ∇ χ ∣ ≤ C / r |\nabla\chi|\le C/r ∣∇ χ ∣ ≤ C / r Q = ∑ x ( n χ − 1 ) + Q=\sum_x(n_\chi-1)_+ Q = ∑ x ( n χ − 1 ) + a a a n χ ≥ max ( a , 2 ) n_\chi\ge\max(a,2) n χ ≥ max ( a , 2 ) a ≤ 2 ( n χ − 1 ) + a\le2(n_\chi-1)_+ a ≤ 2 ( n χ − 1 ) + # { nonisolated in S } ≤ 2 Q \#\{\text{nonisolated in }S\}\le2Q # { nonisolated in S } ≤ 2 Q ∑ i ∣ ∇ i Q ∣ 2 ≤ C r − 2 n U 0 \sum_i|\nabla_iQ|^2\le Cr^{-2}n_{U_0} ∑ i ∣ ∇ i Q ∣ 2 ≤ C r − 2 n U 0 U 0 U_0 U 0 ∣ U 0 ∣ ≤ C ∣ S ∣ |U_0|\le C|S| ∣ U 0 ∣ ≤ C ∣ S ∣
Repeat the proof of (9) for U 0 U_0 U 0 p h = 1 p_h=1 p h = 1 U 0 U_0 U 0 ( n χ − 1 ) + ≤ n supp χ 1 { n supp χ ≥ 2 } (n_\chi-1)_+\le n_{\operatorname{supp}\chi}\mathbf{1}_{\{n_{\operatorname{supp}\chi}\ge2\}} ( n χ − 1 ) + ≤ n supp χ 1 { n supp χ ≥ 2 } c v r − 3 E σ Q c_vr^{-3}\mathbb{E}_\sigma Q c v r − 3 E σ Q b 0 b_0 b 0 ζ \zeta ζ
Apply this augmented lower bound to G = λ Q G=\lambda Q G = λ Q 0 ≤ λ ≤ λ 0 0\le\lambda\le\lambda_0 0 ≤ λ ≤ λ 0 C ρ E g n U 0 C\rho\mathbb{E}_gn_{U_0} C ρ E g n U 0 H 0 H_0 H 0 2 T λ E g Q 2T\lambda\mathbb{E}_gQ 2 T λ E g Q c v r − 3 E g Q c_vr^{-3}\mathbb{E}_gQ c v r − 3 E g Q E g Q ≤ C v r 3 ρ 2 ∣ S ∣ = C v ζ 3 ρ ∣ S ∣ \mathbb{E}_gQ\le C_vr^3\rho^2|S|=C_v\zeta^3\rho|S| E g Q ≤ C v r 3 ρ 2 ∣ S ∣ = C v ζ 3 ρ ∣ S ∣ ∂ λ log E e 2 λ Q = 2 E λ Q \partial_\lambda\log\mathbb{E}e^{2\lambda Q}=2\mathbb{E}_\lambda Q ∂ λ log E e 2 λ Q = 2 E λ Q log E e 2 λ 0 Q ≤ C v λ 0 ζ 3 ρ ∣ S ∣ \log\mathbb{E}e^{2\lambda_0Q}\le C_v\lambda_0\zeta^3\rho|S| log E e 2 λ 0 Q ≤ C v λ 0 ζ 3 ρ ∣ S ∣ 2 Q ≥ # { nonisolated in S } 2Q\ge\#\{\text{nonisolated in }S\} 2 Q ≥ # { nonisolated in S }
A short imaginary-time slab
The next estimates describe the part of the Gibbs path measure that will be changed later. Set δ = s 2 \delta=s^2 δ = s 2 β > 2 δ \beta>2\delta β > 2 δ δ ≤ 4 K 2 ρ − 1 \delta\le4K^2\rho^{-1} δ ≤ 4 K 2 ρ − 1 β ≥ ρ − 2 \beta\ge\rho^{-2} β ≥ ρ − 2 P s y m P_{\mathrm{sym}} P sym F ( Y , S ) F(Y,S) F ( Y , S ) Y Y Y S S S e − ( β − δ ) H P s y m e^{-(\beta-\delta)H}P_{\mathrm{sym}} e − ( β − δ ) H P sym ω i \omega_i ω i S i S_i S i E i E_i E i K S i , E i K_{S_i,E_i} K S i , E i
ν i j ( ω ) = ∫ 0 δ v ( ω i ( t ) − ω j ( t ) ) d t , W ( S , E ; d ω ) = e − ∑ i < j ν i j ( ω ) ∏ i = 1 N K S i , E i ( d ω i ) . \nu_{ij}(\omega)=\int_0^\delta v(\omega_i(t)-\omega_j(t))\,dt,\qquad
W(S,E;d\omega)=e^{-\sum_{i<j}\nu_{ij}(\omega)}\prod_{i=1}^{N}K_{S_i,E_i}(d\omega_i). ν ij ( ω ) = ∫ 0 δ v ( ω i ( t ) − ω j ( t )) d t , W ( S , E ; d ω ) = e − ∑ i < j ν ij ( ω ) i = 1 ∏ N K S i , E i ( d ω i ) . In the closed measure E = Y E=Y E = Y
Z N − 1 F ( Y , S ) d Y d S W ( S , Y ; d ω ) . (14) Z_N^{-1}F(Y,S)\,dY\,dS\,W(S,Y;d\omega). \tag*{(14)} Z N − 1 F ( Y , S ) d Y d S W ( S , Y ; d ω ) . (14) Denote this closed probability law by P c l P_{\mathrm{cl}} P cl E c l \mathbb{E}_{\mathrm{cl}} E cl [35 ] . At each fixed time, permutation-invariant position tests have the equilibrium law P P P
For clarity, a torus bridge includes its lift choice. Fix a representative of its end, lift its start in every possible way, and assign a displacement d ℓ d_\ell d ℓ k ( d ℓ ) = ( 4 π δ ) − 3 / 2 e − ∣ d ℓ ∣ 2 / ( 4 δ ) k(d_\ell)=(4\pi\delta)^{-3/2}e^{-\lvert d_\ell\rvert^2/(4\delta)} k ( d ℓ ) = ( 4 π δ ) − 3/2 e − ∣ d ℓ ∣ 2 / ( 4 δ ) Δ \Delta Δ ℓ ∈ Z 3 \ell\in\mathbb{Z}^3 ℓ ∈ Z 3
Fix a large number D D D R = D + 30 R=D+30 R = D + 30 R A = 2 R + 20 R_A=2R+20 R A = 2 R + 20 0 < ε < 1 / 2 0<\varepsilon<1/2 0 < ε < 1/2 r = ζ ρ − 1 / 3 r=\zeta\rho^{-1/3} r = ζ ρ − 1/3 ζ \zeta ζ D , ε D,\varepsilon D , ε r → ∞ r\to\infty r → ∞ r / s → 0 r/s\to0 r / s → 0 2 r < δ 2r<\delta 2 r < δ a a a
A − ( a ; ω ) = ∫ 0 δ / 2 w ( t ) ∑ i e − d ( ω i ( t ) , a ) 2 / [ 1000 ( 1 + t ) ] 1 { d ( ω i ( t ) , a ) ≤ R A s } d t , (15) \mathcal{A}^{-}(a;\omega)=\int_{0}^{\delta/2}w(t)\sum_i e^{-d(\omega_i(t),a)^2/[1000(1+t)]}\mathbf{1}_{\{d(\omega_i(t),a)\leq R_As\}}\,dt, \tag*{(15)} A − ( a ; ω ) = ∫ 0 δ /2 w ( t ) i ∑ e − d ( ω i ( t ) , a ) 2 / [ 1000 ( 1 + t )] 1 { d ( ω i ( t ) , a ) ≤ R A s } d t , (15) w ( t ) = min ( 1 , t − 3 / 2 ) , w ( 0 ) = 1. w(t)=\min(1,t^{-3/2}),\qquad w(0)=1. w ( t ) = min ( 1 , t − 3/2 ) , w ( 0 ) = 1. Define A + \mathcal{A}^{+} A + ω i ( t ) \omega_i(t) ω i ( t ) ω i ( δ − t ) \omega_i(\delta-t) ω i ( δ − t ) ω \omega ω
A closed slot i i i eligible if all of the following hold: its start S i S_i S i Y i Y_i Y i r r r D s Ds D s r r r r / 10 r/10 r /10 A − ( S i ; ω ) \mathcal{A}^{-}(S_i;\omega) A − ( S i ; ω ) A + ( Y i ; ω ) \mathcal{A}^{+}(Y_i;\omega) A + ( Y i ; ω ) C 0 C_0 C 0 usable if its end count belongs to [ b − m , b + m ] [b-m,b+m] [ b − m , b + m ] 1 − ε 1-\varepsilon 1 − ε cut R \operatorname{cut}_R cut R R s R s R s r / 10 r/10 r /10
Lemma 5.1 (Single-bridge estimates). *Suppose a bridge’s endpoints have torus distance at most ( D + 10 ) s (D+10)s ( D + 10 ) s L L L R s R s R s cut R \operatorname{cut}_R cut R π S , E \pi_{S,E} π S , E p ∗ > 0 p_*>0 p ∗ > 0 π S , E \pi_{S,E} π S , E ω j \omega_j ω j C V C_V C V [ r , δ − r ] [r,\delta-r] [ r , δ − r ] C V / r C_V/\sqrt{r} C V / r ω f i x \omega_{\mathrm{fix}} ω fix
E π S , E [ 1 cut R ∑ j ∈ f i x ∫ 0 δ ( v ( ω ( t ) − ω j ( t ) ) ) d t ] ≤ C ( γ , D ) [ A − ( S ; ω f i x ) + A + ( E ; ω f i x ) ] . (16) \mathbb{E}_{\pi_{S,E}}\left[\mathbf{1}_{\operatorname{cut}_R}\sum_{j\in\mathrm{fix}}\int_0^\delta\bigl(v(\omega(t)-\omega_j(t))\bigr)\,dt\right]\leq C(\gamma,D)\left[\mathcal{A}^{-}(S;\omega_{\mathrm{fix}})+\mathcal{A}^{+}(E;\omega_{\mathrm{fix}})\right]. \tag*{(16)} E π S , E 1 cut R j ∈ fix ∑ ∫ 0 δ ( v ( ω ( t ) − ω j ( t )) ) d t ≤ C ( γ , D ) [ A − ( S ; ω fix ) + A + ( E ; ω fix ) ] . (16) Finally, for any fixed collection of at most N N N x x x E x A + ( x ; ω ) ≤ C K 2 \mathbb{E}_x\mathcal{A}^{+}(x;\omega)\leq CK^2 E x A + ( x ; ω ) ≤ C K 2 p ∗ p_* p ∗ D D D
Proof. Take L > 4 R s L>4Rs L > 4 R s R s Rs R s B t − ( t / δ ) B δ B_t-(t/\delta)B_\delta B t − ( t / δ ) B δ B B B Δ \Delta Δ s s s ( D + 11 ) s < R s (D+11)s<Rs ( D + 11 ) s < R s ( D + 10 ) r / s (D+10)r/s ( D + 10 ) r / s r / 20 r/20 r /20 s ≥ 20 ( D + 10 ) s\ge20(D+10) s ≥ 20 ( D + 10 ) B δ B_\delta B δ r / 20 r/20 r /20 C e − c r Ce^{-cr} C e − cr sup t ≤ r ∣ B t ∣ \sup_{t\le r}|B_t| sup t ≤ r ∣ B t ∣ r / 40 r/40 r /40 ∣ B δ ∣ |B_\delta| ∣ B δ ∣ δ / 40 \delta/40 δ /40 p ∗ p_\ast p ∗
At time t t t 2 t ( δ − t ) / δ 2t(\delta-t)/\delta 2 t ( δ − t ) / δ t ′ = min ( t , δ − t ) t'=\min(t,\delta-t) t ′ = min ( t , δ − t ) [ t ′ , 2 t ′ ] [t',2t'] [ t ′ , 2 t ′ ] C ( t ′ ) − 3 / 2 C(t')^{-3/2} C ( t ′ ) − 3/2 t ′ ≤ δ ≪ L 2 t'\le\delta\ll L^2 t ′ ≤ δ ≪ L 2 v v v C v w ( t ′ ) C_vw(t') C v w ( t ′ ) ∥ v ∥ ∞ \lVert v\rVert_\infty ∥ v ∥ ∞ t ′ ≤ 1 t'\le1 t ′ ≤ 1 ∥ v ∥ 1 \lVert v\rVert_1 ∥ v ∥ 1 t ′ ≥ 1 t'\ge1 t ′ ≥ 1 C v C_v C v C v ∫ r ∞ t − 3 / 2 d t ≤ C v / r C_v\int_r^\infty t^{-3/2}\,\mathrm{d}t\le C_v/\sqrt{r} C v ∫ r ∞ t − 3/2 d t ≤ C v / r
To retain spatial decay, first take t ≤ δ / 2 t\le\delta/2 t ≤ δ /2 S S S ( D + 10 ) t / s ≤ ( D + 10 ) t (D+10)t/s\le(D+10)\sqrt{t} ( D + 10 ) t / s ≤ ( D + 10 ) t S S S t ≥ 1 t\ge1 t ≥ 1 y y y
C ( v , D ) t − 3 / 2 ∑ ℓ ∈ Z 3 e − ∣ y + ℓ L − S ∣ 2 / ( 16 t ) . C(v,D)t^{-3/2}\sum_{\ell\in\mathbb{Z}^3}e^{-\lvert y+\ell L-S\rvert^2/(16t)}. C ( v , D ) t − 3/2 ℓ ∈ Z 3 ∑ e − ∣ y + ℓ L − S ∣ 2 / ( 16 t ) . Indeed ∣ u + b ∣ 2 ≥ ∣ u ∣ 2 / 2 − ∣ b ∣ 2 \lvert u+b\rvert^2\ge\lvert u\rvert^2/2-\lvert b\rvert^2 ∣ u + b ∣ 2 ≥ ∣ u ∣ 2 /2 − ∣ b ∣ 2 ∣ b ∣ ≤ R v + ( D + 10 ) t \lvert b\rvert\le R_v+(D+10)\sqrt{t} ∣ b ∣ ≤ R v + ( D + 10 ) t e − d ( y , S ) 2 / ( 32 t ) e^{-d(y,S)^2/(32t)} e − d ( y , S ) 2 / ( 32 t ) t ≤ δ ≪ L 2 t\le\delta\ll L^2 t ≤ δ ≪ L 2 0 < t ≤ 1 0<t\le1 0 < t ≤ 1 2 ( R v + D + 10 ) 2(R_v+D+10) 2 ( R v + D + 10 ) S S S d ( y , S ) / 2 d(y,S)/2 d ( y , S ) /2 C ( v , D ) C(v,D) C ( v , D ) C ( v , D ) w ( t ) e − d ( y , S ) 2 / [ 1000 ( 1 + t ) ] C(v,D)w(t)e^{-d(y,S)^2/[1000(1+t)]} C ( v , D ) w ( t ) e − d ( y , S ) 2 / [ 1000 ( 1 + t )]
On the cut, a possible hit also requires d ( y , S ) ≤ ( R + D + 10 ) s + R v < R A s d(y,S)\le(R+D+10)s+R_v<R_As d ( y , S ) ≤ ( R + D + 10 ) s + R v < R A s R s Rs R s ( D + 10 ) s (D+10)s ( D + 10 ) s (15) may be inserted in this upper bound. Reverse time for the other half, sum over the fixed trajectories, and integrate to obtain (16). Finally the integral over a uniform anchor of the Gaussian factor is at most C ( 1 + t ) 3 / 2 / V C(1+t)^{3/2}/V C ( 1 + t ) 3/2 / V w ( t ) ( 1 + t ) 3 / 2 ≤ C w(t)(1+t)^{3/2}\le C w ( t ) ( 1 + t ) 3/2 ≤ C C N δ / V ≤ C ρ s 2 ≤ C κ 2 CN\delta/V\le C\rho s^2\le C\kappa^2 C N δ / V ≤ C ρ s 2 ≤ C κ 2
A joint bound for atypical trajectories
The bridge criteria require a bound for many failures at once. A single-bridge estimate is insufficient here: the interacting closed measure does not make its bridges independent. We obtain a localized exponential moment by assigning an auxiliary color to each tested bridge, retaining its spatial confinement, and estimating the resulting partition function.
We use the free-energy increment bound (6) and the density bound (10). More explicitly, the two inputs needed in this section are
F j − F j − 1 ≤ C v ρ ( 1 ≤ j ≤ N ) , F_j-F_{j-1}\le C_v\rho\qquad(1\le j\le N), F j − F j − 1 ≤ C v ρ ( 1 ≤ j ≤ N ) , ⟨ n B ⟩ J , − W ≤ C v ρ ∣ B ∣ ( 0 ≤ J ≤ N , 0 ≤ W ≤ ρ ) , \langle n_B\rangle_{J,-W}\le C_v\rho|B|\qquad(0\le J\le N,\ 0\le W\le\rho), ⟨ n B ⟩ J , − W ≤ C v ρ ∣ B ∣ ( 0 ≤ J ≤ N , 0 ≤ W ≤ ρ ) , where B B B J J J − W -W − W D D D ζ \zeta ζ
Recall that δ = s 2 \delta=s^2 δ = s 2 r = ζ ρ − 1 / 3 r=\zeta\rho^{-1/3} r = ζ ρ − 1/3 U U U J U f a i l J_U^{\mathrm{fail}} J U fail U U U D s Ds D s r r r r / 10 r/10 r /10
Proposition 6.1 (Localized exponential moment for trajectory failures). Fix K K K D min D_{\min} D m i n C v C_v C v κ > 0 \kappa> 0 κ > 0 K K K D ≥ D min D \ge D_{\min} D ≥ D m i n 0 < ζ ≤ 1 0 < \zeta\le1 0 < ζ ≤ 1 ρ > 0 \rho> 0 ρ > 0 ρ / 2 ≤ N / V ≤ 2 ρ \rho/2 \le N/V \le2\rho ρ /2 ≤ N / V ≤ 2 ρ 0 < T ≤ ρ 2 0 < T \le\rho^2 0 < T ≤ ρ 2 U U U
log E c l exp ( J U f a i l ) ≤ [ C v ∑ j ≥ 0 D j 3 e − κ D j 2 + o ρ ( 1 ) ] ρ ∣ U ∣ , D j = 2 j D . (17) \log\mathbb{E}_{\mathrm{cl}}\exp(J_U^{\mathrm{fail}}) \le\left[C_v \sum_{j \ge0} D_j^3 e^{-\kappa D_j^2} + o_\rho(1)\right]\rho|U|,\qquad D_j = 2^jD. \tag*{(17)} log E cl exp ( J U fail ) ≤ [ C v j ≥ 0 ∑ D j 3 e − κ D j 2 + o ρ ( 1 ) ] ρ ∣ U ∣ , D j = 2 j D . (17) The constants C v , κ C_v,\kappa C v , κ D D D ζ \zeta ζ o ρ ( 1 ) o_\rho(1) o ρ ( 1 ) v , K , D , ζ v,K,D,\zeta v , K , D , ζ
A localized kernel for one marked bridge
Partition the failure event into disjoint bins. In bin j ≥ 0 j \ge0 j ≥ 0 ( D j s , 2 D j s ] (D_js,2D_js] ( D j s , 2 D j s ] ∗ * ∗
h j = D j , b j = D j 2 ( j ≥ 0 ) , h ∗ = D , b ∗ = r . h_j = D_j,\qquad b_j = D_j^2\quad(j \ge0),\qquad h_* = D,\quad b_* = r. h j = D j , b j = D j 2 ( j ≥ 0 ) , h ∗ = D , b ∗ = r . Here and below a sum over bins includes ∗ * ∗
Ω j = { x ∈ Λ L : d ( x , U ) < ( 2 h j + 4 ) s } . \Omega_j = \{x \in\Lambda_L : d(x,U) < (2h_j+4)s\}. Ω j = { x ∈ Λ L : d ( x , U ) < ( 2 h j + 4 ) s } . If this set is the whole torus, its Laplacian has periodic boundary conditions; otherwise we use the Dirichlet Laplacian. Choose a Lipschitz function 0 ≤ p j ≤ 1 0 \le p_j \le1 0 ≤ p j ≤ 1 d ( x , U ) ≤ 2 h j s d(x,U) \le2h_js d ( x , U ) ≤ 2 h j s d ( x , U ) ≥ ( 2 h j + 2 ) s d(x,U) \ge(2h_j+2)s d ( x , U ) ≥ ( 2 h j + 2 ) s ∣ ∇ p j ∣ ≤ C / s |\nabla p_j| \le C/s ∣∇ p j ∣ ≤ C / s Ω j \Omega_j Ω j Ω j \Omega_j Ω j
∣ Ω j ∣ ≤ C h j 3 ∣ U ∣ , Ω j ⊂ B j , ∣ B j ∣ ≤ C h j 3 ∣ U ∣ , |\Omega_j| \le Ch_j^3|U|,\qquad\Omega_j \subset B_j,\qquad|B_j| \le Ch_j^3|U|, ∣ Ω j ∣ ≤ C h j 3 ∣ U ∣ , Ω j ⊂ B j , ∣ B j ∣ ≤ C h j 3 ∣ U ∣ , for suitable coarse-cell unions B j B_j B j
Let K j ( E , S ) K_j(E,S) K j ( E , S ) E ∈ U E \in U E ∈ U j j j c , C > 0 c,C > 0 c , C > 0 c 1 > 1 c_1 > 1 c 1 > 1
K j ( E , S ) ≤ C e − c b j p j ( E ) p j ( S ) [ e c 1 δ Δ Ω j ] ( E , S ) . (18) K_j(E,S) \le Ce^{-cb_j}p_j(E)p_j(S)[e^{c_1\delta\Delta_{\Omega_j}}](E,S). \tag*{(18)} K j ( E , S ) ≤ C e − c b j p j ( E ) p j ( S ) [ e c 1 δ Δ Ω j ] ( E , S ) . (18) These constants are independent of D D D ζ \zeta ζ
To prove this, fix a lift of E E E S S S d d d E E E k t ( d ) = ( 4 π t ) − 3 / 2 e − ∣ d ∣ 2 / ( 4 t ) k_t(d)=(4\pi t)^{-3/2}e^{-|d|^2/(4t)} k t ( d ) = ( 4 π t ) − 3/2 e − ∣ d ∣ 2 / ( 4 t ) ∣ d ∣ ≤ 2 h j s |d| \le2h_js ∣ d ∣ ≤ 2 h j s ∣ d ∣ > h j s / 2 |d| > h_js/2 ∣ d ∣ > h j s /2
k δ ( d ) ≤ C e − c h j 2 k c 1 δ ( d ) . k_\delta(d) \le Ce^{-ch_j^2}k_{c_1\delta}(d). k δ ( d ) ≤ C e − c h j 2 k c 1 δ ( d ) . If ∣ d ∣ ≤ h j s / 2 |d| \le h_js/2 ∣ d ∣ ≤ h j s /2 h j s / 2 h_js/2 h j s /2 C e − c h j 2 Ce^{-ch_j^2} C e − c h j 2 k δ ≤ c 1 3 k c 1 δ k_\delta\le c_1^3 k_{c_1\delta} k δ ≤ c 1 3 k c 1 δ
For bin ∗ * ∗ ∣ d ∣ ≤ D s |d| \le Ds ∣ d ∣ ≤ D s s ≥ 40 D s \ge40D s ≥ 40 D r r r r / 40 r/40 r /40 B B B Δ \Delta Δ B t − ( t / δ ) B δ B_t - (t/\delta)B_\delta B t − ( t / δ ) B δ [ 0 , r ] [0,r] [ 0 , r ] r r r C e − c r Ce^{-cr} C e − cr C e − c δ Ce^{-c\delta} C e − cδ r r r ∣ B δ ∣ ≥ c δ |B_\delta| \ge c\delta ∣ B δ ∣ ≥ cδ r ≤ δ / 2 r \le\delta/2 r ≤ δ /2 ∗ * ∗ C e − c r k c 1 δ ( d ) Ce^{-cr}k_{c_1\delta}(d) C e − cr k c 1 δ ( d )
It remains to justify the Dirichlet kernel on the right of (18), rather than a free kernel without localization. For every retained lift, the straight segment from E E E S S S 2 h j s 2h_j s 2 h j s E E E s s s Ω j \Omega_j Ω j c 1 c_1 c 1 δ = s 2 \delta=s^2 δ = s 2 q ∗ > 0 q_* > 0 q ∗ > 0
q ∗ ∑ retained lifts of S k c 1 δ ( d ) . q_* \sum_{\text{retained lifts of }S} k_{c_1\delta}(d). q ∗ retained lifts of S ∑ k c 1 δ ( d ) . The terms are disjoint winding classes of torus paths. This remains valid for extents greater than L L L Ω j = Λ L \Omega_j=\Lambda_L Ω j = Λ L K j K_j K j p j ( E ) p_j(E) p j ( E ) p j ( S ) p_j(S) p j ( S )
The colored operator and its large powers
Choose κ > 0 \kappa>0 κ > 0
η j = e − κ b j , M j = κ b j / δ . \eta_j=e^{-\kappa b_j}, \qquad M_j=\kappa b_j/\delta. η j = e − κ b j , M j = κ b j / δ . By increasing the lower restrictions on D D D r r r e − 1 e-1 e − 1 η j e − δ M j = e − 2 κ b j \eta_j e^{-\delta M_j}=e^{-2\kappa b_j} η j e − δ M j = e − 2 κ b j κ \kappa κ D D D ζ \zeta ζ
We will represent marked bridges by auxiliary particle colors. The small coefficient η j \eta_j η j M j M_j M j
The enlarged one-particle Hilbert space is
h e = L 2 ( Λ L ) ⊕ ⨁ j L 2 ( Ω j ) , \mathfrak{h}_e=L^2(\Lambda_L)\oplus\bigoplus_j L^2(\Omega_j), h e = L 2 ( Λ L ) ⊕ j ⨁ L 2 ( Ω j ) , with the first summand called the original color and the others indexed by the failure bins. Define
v e f = ( f , ( η j p j f ) j ) , V e = v e ⊗ N . v_e f=(f,(\sqrt{\eta_j}p_jf)_j), \qquad V_e=v_e^{\otimes N}. v e f = ( f , ( η j p j f ) j ) , V e = v e ⊗ N . The restrictions to the relevant domains in each marked component are understood. On the labeled N N N H H H j j j − c 1 Δ Ω j + M j -c_1\Delta_{\Omega_j}+M_j − c 1 Δ Ω j + M j N N N
T = V e ∗ e − δ H V e . \mathcal{T}=V_e^*e^{-\delta H}V_e. T = V e ∗ e − δ H V e . Indeed, expanding
e J U f a i l = ∏ i = 1 N ( 1 + ( e − 1 ) 1 { E i ∈ U , i f a i l s } ) e^{J_U^{\mathrm{fail}}}=\prod_{i=1}^{N}\left(1+(e-1)\mathbf{1}_{\{E_i\in U,\ i\ \mathrm{fails}\}}\right) e J U fail = i = 1 ∏ N ( 1 + ( e − 1 ) 1 { E i ∈ U , i fails } ) marks subsets of labels. For each marked subset, decompose its failure events into bins and drop precisely the interactions incident to marked labels. This increases the nonnegative path weight. Applying (18) to each marked path leaves the original interacting kernel on the other labels. The sum over all color assignments is exactly the kernel of T \mathcal{T} T T \mathcal{T} T
E c l e J U f a i l ≤ Z N − 1 Tr s y m e − ( β − δ ) H T . \mathbb{E}_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}}\leq Z_N^{-1}\operatorname{Tr}_{\mathrm{sym}}e^{-(\beta-\delta)H}\mathcal{T}. E cl e J U fail ≤ Z N − 1 Tr sym e − ( β − δ ) H T . For β > δ \beta>\delta β > δ β / ( β − δ ) \beta/(\beta-\delta) β / ( β − δ ) β / δ \beta/\delta β / δ
E c l e J U f a i l ≤ ( Tr s y m T β / δ Z N ) δ / β . (19) \mathbb{E}_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}}\leq\left(\frac{\operatorname{Tr}_{\mathrm{sym}}\mathcal{T}^{\beta/\delta}}{Z_N}\right)^{\delta/\beta}. \tag*{(19)} E cl e J U fail ≤ ( Z N Tr sym T β / δ ) δ / β . (19) The embedding has norm larger than one. We next account for its effect on the high power in (19) without losing the spatial localization. Define the bounded Lipschitz function
a ( x ) = 1 2 log ( 1 + ∑ j η j p j ( x ) 2 ) , A = ∑ i = 1 N a ( x i ) . a(x)=\frac{1}{2}\log\left(1+\sum_j\eta_jp_j(x)^2\right),\qquad A=\sum_{i=1}^{N}a(x_i). a ( x ) = 2 1 log ( 1 + j ∑ η j p j ( x ) 2 ) , A = i = 1 ∑ N a ( x i ) . The same position multiplier acts in every color. One has V e = e A U e V_e=e^AU_e V e = e A U e U e U_e U e e 2 a ( x ) e^{2a(x)} e 2 a ( x )
T = U e ∗ X U e , X = e A e − δ H ~ e A . \mathcal{T}=U_e^*XU_e,\qquad X=e^Ae^{-\delta\widetilde{H}}e^A. T = U e ∗ X U e , X = e A e − δ H e A . Eigenvalue min–max for a positive compact compression gives λ k ( T ) ≤ λ k ( X ) \lambda_k(\mathcal{T})\leq\lambda_k(X) λ k ( T ) ≤ λ k ( X ) k k k T \mathcal{T} T X X X
Lemma 6.2 (Trace bound for a Lipschitz sandwich). For the operators just defined, put
W 0 = 2 a / δ + c 1 ∣ ∇ a ∣ 2 , H ~ = H − ∑ i = 1 N W 0 ( x i ) . W_0=2a/\delta+c_1|\nabla a|^2,\qquad\widetilde{H}=H-\sum_{i=1}^{N}W_0(x_i). W 0 = 2 a / δ + c 1 ∣∇ a ∣ 2 , H = H − i = 1 ∑ N W 0 ( x i ) . Then
Tr X β / δ ≤ Tr e − β H ~ . (20) \operatorname{Tr}X^{\beta/\delta}\leq\operatorname{Tr}e^{-\beta\widetilde{H}}. \tag*{(20)} Tr X β / δ ≤ Tr e − β H . (20) In fact the analogous inequality holds with every positive power q q q q δ q\delta q δ
Proof. We interpolate from the identity to X X X H ~ \widetilde{H} H
First, the heat traces below are finite and X X X N N N L L L ρ \rho ρ M j M_j M j t > 0 t>0 t > 0
∑ j e − t M j Tr e c 1 t Δ Ω j ≤ C V ( V − 1 + t − 3 / 2 ) ∑ j e − t M j < ∞ . \sum_j e^{-tM_j}\operatorname{Tr}e^{c_1t\Delta_{\Omega_j}}\leq CV\left(V^{-1}+t^{-3/2}\right)\sum_j e^{-tM_j}<\infty. j ∑ e − t M j Tr e c 1 t Δ Ω j ≤ C V ( V − 1 + t − 3/2 ) j ∑ e − t M j < ∞. Including the original color, the free labeled N N N N N N H \mathrm{H} H ∑ i W 0 ( x i ) \sum_i W_0(x_i) ∑ i W 0 ( x i ) e A e^A e A X X X
For 0 ≤ t ≤ δ 0 \le t \le\delta 0 ≤ t ≤ δ
U t = e f ( t ) A e − t H ~ e A , f ( t ) = − 1 + 2 t / δ . U_t=e^{f(t)A}e^{-t\widetilde{\mathrm{H}}}e^A,\qquad f(t)=-1+2t/\delta. U t = e f ( t ) A e − t H e A , f ( t ) = − 1 + 2 t / δ . Thus U 0 = 1 U_0=1 U 0 = 1 U δ = X U_\delta=X U δ = X H \mathrm{H} H x x x t > 0 t>0 t > 0
e − f ( t ) A U t x = e − t H e A x ∈ D ( H ) . e^{-f(t)A}U_tx=e^{-t\mathrm{H}}e^Ax\in D(\mathrm{H}). e − f ( t ) A U t x = e − t H e A x ∈ D ( H ) . The vector y ( t ) = U t x y(t)=U_tx y ( t ) = U t x t > 0 t>0 t > 0
y ′ ( t ) = 2 A δ y ( t ) − e f ( t ) A H e − f ( t ) A y ( t ) . y'(t)=\frac{2A}{\delta}y(t)-e^{f(t)A}\mathrm{H}e^{-f(t)A}y(t). y ′ ( t ) = δ 2 A y ( t ) − e f ( t ) A H e − f ( t ) A y ( t ) . The last expression is well-defined on this evolving vector. Its real inner product with y y y e f A y e^{fA}y e f A y e − f A y e^{-fA}y e − f A y D ( H ) D(\mathrm{H}) D ( H )
In a fixed color component, the diffusion coefficient c i c_i c i i i i 1 1 1 c 1 c_1 c 1
Re [ ∇ i ( e f A y ) ⋅ ∇ i ( e − f A y ) ] = ∣ ∇ i y ∣ 2 − f 2 ∣ ∇ a ( x i ) ∣ 2 ∣ y ∣ 2 . \operatorname{Re}\left[\nabla_i(e^{fA}y)\cdot\nabla_i(e^{-fA}y)\right]=|\nabla_i y|^2-f^2|\nabla a(x_i)|^2|y|^2. Re [ ∇ i ( e f A y ) ⋅ ∇ i ( e − f A y ) ] = ∣ ∇ i y ∣ 2 − f 2 ∣∇ a ( x i ) ∣ 2 ∣ y ∣ 2 . The cross terms are purely imaginary and all potentials commute with these multipliers. Since ∣ f ∣ ≤ 1 |f|\le1 ∣ f ∣ ≤ 1 c i ≤ c 1 c_i\le c_1 c i ≤ c 1
1 2 d d t ∥ U t x ∥ 2 ≤ − q ~ [ U t x ] , \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\lVert U_tx\rVert^2\le-\widetilde{q}[U_tx], 2 1 d t d ∥ U t x ∥ 2 ≤ − q [ U t x ] , where q ~ \widetilde{q} q H ~ \widetilde{\mathrm{H}} H
Let E 1 ≤ E 2 ≤ ⋯ E_1\le E_2\le\cdots E 1 ≤ E 2 ≤ ⋯ H ~ \widetilde{\mathrm{H}} H U t ⊗ k U_t^{\otimes k} U t ⊗ k A A A H ~ \widetilde{\mathrm{H}} H k k k H ~ \widetilde{\mathrm{H}} H E 1 + ⋯ + E k E_1+\cdots+E_k E 1 + ⋯ + E k
1 2 d d t ∥ U t ⊗ k x ∥ 2 ≤ − ( E 1 + ⋯ + E k ) ∥ U t ⊗ k x ∥ 2 \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\lVert U_t^{\otimes k}x\rVert^2\le-(E_1+\cdots+E_k)\lVert U_t^{\otimes k}x\rVert^2 2 1 d t d ∥ U t ⊗ k x ∥ 2 ≤ − ( E 1 + ⋯ + E k ) ∥ U t ⊗ k x ∥ 2 for every antisymmetric x x x t > 0 t>0 t > 0 U 0 = 1 U_0=1 U 0 = 1 k k k U δ U_\delta U δ e − δ ( E 1 + ⋯ + E k ) e^{-\delta(E_1+\cdots+E_k)} e − δ ( E 1 + ⋯ + E k )
The exterior-power norm is the product of the largest k k k U δ = X U_\delta=X U δ = X x i > 0 x_i>0 x i > 0
∑ i = 1 k log x i ≤ − δ ∑ i = 1 k E i ( k ≥ 1 ) . \sum_{i=1}^{k}\log x_i\le-\delta\sum_{i=1}^{k}E_i\qquad(k\ge1). i = 1 ∑ k log x i ≤ − δ i = 1 ∑ k E i ( k ≥ 1 ) . These partial-sum inequalities imply, for every real t t t
∑ i ( log x i − t ) + ≤ ∑ i ( − δ E i − t ) + . \sum_i(\log x_i-t)_+\le\sum_i(-\delta E_i-t)_+. i ∑ ( log x i − t ) + ≤ i ∑ ( − δ E i − t ) + . Indeed each side is the supremum over k ≥ 0 k \ge0 k ≥ 0 t t t q > 0 q > 0 q > 0
e q u = ∫ R ( u − t ) + q 2 e q t d t e^{qu} = \int_{\mathbb{R}} (u-t)_{+}q^{2}e^{qt}\,\mathrm{d}t e q u = ∫ R ( u − t ) + q 2 e q t d t and Tonelli’s theorem now give ∑ i x i q ≤ ∑ i e − q δ E i \sum_i x_i^q \le\sum_i e^{-q\delta E_i} ∑ i x i q ≤ ∑ i e − q δ E i q = β / δ q = \beta/\delta q = β / δ
The partition-function estimate with confined colors
The preceding lemma has converted the large power of the marked-slab operator into an ordinary heat trace. It remains to compare that trace with the original partition function. Spatial confinement is essential here: it makes the final cost proportional to ∣ U ∣ |U| ∣ U ∣
Set
Σ = ∑ j η j h j 3 = ∑ j ≥ 0 D j 3 e − κ D j 2 + D 3 e − κ r . \Sigma= \sum_j \eta_j h_j^3 = \sum_{j \ge0} D_j^3 e^{-\kappa D_j^2} + D^3 e^{-\kappa r}. Σ = j ∑ η j h j 3 = j ≥ 0 ∑ D j 3 e − κ D j 2 + D 3 e − κ r . From the definition of a a a
0 ≤ W 0 ≤ C δ ∑ j η j 1 Ω j . 0 \le W_0 \le\frac{C}{\delta}\sum_j \eta_j 1_{\Omega_j}. 0 ≤ W 0 ≤ δ C j ∑ η j 1 Ω j . For example,
∣ ∇ a ∣ 2 = ∣ ∑ j η j p j ∇ p j ∣ 2 ( 1 + ∑ j η j p j 2 ) 2 ≤ C δ ∑ j η j 1 Ω j . |\nabla a|^2 = \frac{\left|\sum_j \eta_j p_j \nabla p_j\right|^2}{\left(1+\sum_j \eta_j p_j^2\right)^2} \le\frac{C}{\delta}\sum_j \eta_j 1_{\Omega_j}. ∣∇ a ∣ 2 = ( 1 + ∑ j η j p j 2 ) 2 ∑ j η j p j ∇ p j 2 ≤ δ C j ∑ η j 1 Ω j . All constants are independent of the number or sizes of the domains. Taking D D D ρ \rho ρ ∥ W 0 ∥ ∞ ≤ ρ \|W_0\|_\infty\le\rho ∥ W 0 ∥ ∞ ≤ ρ δ ≥ K 2 ρ − 1 \delta\ge K^2\rho^{-1} δ ≥ K 2 ρ − 1 r → ∞ r \to\infty r → ∞
For 0 ≤ J ≤ N 0 \le J \le N 0 ≤ J ≤ N − W 0 -W_0 − W 0 B j B_j B j − t W 0 -tW_0 − t W 0 0 ≤ t ≤ 1 0 \le t \le1 0 ≤ t ≤ 1
log Z J ( − W 0 ) − log Z J = β ∫ 0 1 ⟨ n W 0 ⟩ J , − t W 0 d t ≤ β C v ρ ∣ U ∣ δ Σ . \log Z_J^{(-W_0)}-\log Z_J=\beta\int_0^1\langle nW_0\rangle_{J,-tW_0}\,\mathrm{d}t\le\frac{\beta C v_\rho|U|}{\delta}\Sigma. log Z J ( − W 0 ) − log Z J = β ∫ 0 1 ⟨ n W 0 ⟩ J , − t W 0 d t ≤ δ β C v ρ ∣ U ∣ Σ. The free-energy increment input also gives Z J / Z N ≤ e β C v ρ ( N − J ) Z_J/Z_N \le e^{\beta C v_\rho(N-J)} Z J / Z N ≤ e β C v ρ ( N − J )
To use these inequalities, decompose the enlarged symmetric space according to the occupation of each color:
Sym N h e ≃ ⨁ J + ∑ j k j = N Sym J L 2 ( Λ L ) ⊗ ⨂ j Sym k j L 2 ( Ω j ) . \operatorname{Sym}^N \mathfrak{h}_e \simeq\bigoplus_{J+\sum_j k_j=N}\operatorname{Sym}^J L^2(\Lambda_L)\otimes\bigotimes_j\operatorname{Sym}^{k_j}L^2(\Omega_j). Sym N h e ≃ J + ∑ j k j = N ⨁ Sym J L 2 ( Λ L ) ⊗ j ⨂ Sym k j L 2 ( Ω j ) . Each number sector is one tensor product. There is no multinomial multiplicity in its trace: the normalized symmetrization over color assignments is the unitary realizing this decomposition. In that sector, the cost e β C v ρ ( N − J ) e^{\beta C v_\rho(N-J)} e β C v ρ ( N − J ) j j j W 0 W_0 W 0
− c 1 Δ Ω j + M j − ∥ W 0 ∥ ∞ − C v ρ ≥ − c 1 Δ Ω j + M j / 2. -c_1\Delta_{\Omega_j}+M_j-\|W_0\|_\infty-Cv_\rho\ge-c_1\Delta_{\Omega_j}+M_j/2. − c 1 Δ Ω j + M j − ∥ W 0 ∥ ∞ − C v ρ ≥ − c 1 Δ Ω j + M j /2. For this last inequality use M j ≥ κ b j ρ / ( 4 K 2 ) M_j \ge\kappa b_j\rho/(4K^2) M j ≥ κ b j ρ / ( 4 K 2 ) D D D v v v K K K r r r β M j / 2 ≥ 1 \beta M_j/2 \ge1 β M j /2 ≥ 1
Sum the marked occupation numbers and drop their total-number constraint. This bounds the sum by a product of ideal Bose grand partition functions with gaps M j / 2 M_j/2 M j /2
∑ ℓ ≥ 1 e − ℓ β M j / 2 ℓ Tr e c 1 ℓ β Δ Ω j ≤ C ∣ Ω j ∣ ( V − 1 + T 3 / 2 ) e − β M j / 2 . \sum_{\ell\ge1}\frac{e^{-\ell\beta M_j/2}}{\ell}\operatorname{Tr}e^{c_1\ell\beta\Delta_{\Omega_j}}\le C|\Omega_j|(V^{-1}+T^{3/2})e^{-\beta M_j/2}. ℓ ≥ 1 ∑ ℓ e − ℓ β M j /2 Tr e c 1 ℓ β Δ Ω j ≤ C ∣ Ω j ∣ ( V − 1 + T 3/2 ) e − β M j /2 . The estimate uses the killed heat-kernel bound by the periodic one and the geometric series with β M j / 2 ≥ 1 \beta M_j/2\ge1 β M j /2 ≥ 1 (19) , the compression inequality, and (20) , we obtain the quantitative estimate
log E c l e J U f a i l ≤ C v ρ ∣ U ∣ Σ + C δ T ∣ U ∣ ( V − 1 + T 3 / 2 ) ∑ j h j 3 e − β M j / 2 . \log\mathbb{E}_{\mathrm{cl}}e^{J_U^{\mathrm{fail}}}\le C_{v}\rho|U|\Sigma+C\delta T|U|(V^{-1}+T^{3/2})\sum_j h_j^3e^{-\beta M_j/2}. log E cl e J U fail ≤ C v ρ ∣ U ∣Σ + C δ T ∣ U ∣ ( V − 1 + T 3/2 ) j ∑ h j 3 e − β M j /2 . This also proves the required uniformity. Since β ≥ δ \beta\ge\delta β ≥ δ D D D ρ \rho ρ D j 3 e − κ D j 2 / 2 D_j^3e^{-\kappa D_j^2/2} D j 3 e − κ D j 2 /2 D 3 e − κ r / 2 D^3e^{-\kappa r/2} D 3 e − κ r /2 V ≥ s 3 V\ge s^3 V ≥ s 3 δ ≤ 4 K 2 / ρ \delta\le4K^2/\rho δ ≤ 4 K 2 / ρ T ≤ ρ 2 T\le\rho^2 T ≤ ρ 2
δ T ρ ( V − 1 + T 3 / 2 ) ≤ C K ( ρ 3 / 2 + ρ 3 ) . \frac{\delta T}{\rho}(V^{-1}+T^{3/2})\le C_K(\rho^{3/2}+\rho^3). ρ δ T ( V − 1 + T 3/2 ) ≤ C K ( ρ 3/2 + ρ 3 ) . The micro contribution D 3 e − κ r D^3e^{-\kappa r} D 3 e − κ r Σ \Sigma Σ D , ζ D,\zeta D , ζ (17) , with all errors uniform in the temperature and the thermodynamic volume, as asserted.
Defective cells and an insertion comparison
We combine occupation, isolation, and trajectory estimates to control an arbitrary specified set of unusable cells. The final comparison of this section transfers that control from a probability measure on closed paths to the finite measure obtained by opening one end.
Fix Δ A = 1000 ( 2 R A + 2 R + 20 ) 3 \Delta_A=1000(2R_A+2R+20)^3 Δ A = 1000 ( 2 R A + 2 R + 20 ) 3 defective if its end count is below 2 b − m 2b_{-m} 2 b − m b + m / 2 b_{+m/2} b + m /2 ϵ / 4 \epsilon/4 ϵ /4 C 0 − Δ A C_0-\Delta_A C 0 − Δ A C 0 C_0 C 0
Lemma 7.1 (Scores at separated anchors). Let U U U D \mathcal{D} D r r r D s Ds D s U U U S S S ( D + R A ) s (D+R_A)s ( D + R A ) s U U U ∣ S ∣ ≤ C ( D + R A + 2 ) 3 ∣ U ∣ |S|\le C(D+R_A+2)^3|U| ∣ S ∣ ≤ C ( D + R A + 2 ) 3 ∣ U ∣
∑ a ∈ D A − ( a ; ω ) ≤ C ∫ 0 δ / 2 ( w ( t ) + r − 3 ) n S ( ω ( t ) ) d t . (21) \sum_{a\in\mathcal{D}}A^{-}(a;\omega)\le C\int_0^{\delta/2}(w(t)+r^{-3})n_S(\omega(t))\,\mathrm{d}t. \tag*{(21)} a ∈ D ∑ A − ( a ; ω ) ≤ C ∫ 0 δ /2 ( w ( t ) + r − 3 ) n S ( ω ( t )) d t . (21) The analogous inequality holds for the end scores after time reversal. The assertion holds also when the anchors are chosen from the paths by an arbitrary measurable rule, provided their separation and location hold for each realization. There is B 0 = C ( 1 + K 2 / ζ 3 ) B_0=C(1+K^2/\zeta^3) B 0 = C ( 1 + K 2 / ζ 3 ) (21) , divided by B 0 B_0 B 0 exp ( C v ρ ∣ S ∣ ) \exp(C_v\rho|S|) exp ( C v ρ ∣ S ∣ )
Proof. Lift the anchors periodically to R 3 \mathbb{R}^3 R 3 r r r L > 2 r L>2r L > 2 r y y y a t = 1 + t a_t=\sqrt{1+t} a t = 1 + t r / 2 r/2 r /2 u u u C ( 1 + u / r ) 3 C(1+u/r)^3 C ( 1 + u / r ) 3 max ( r , a t ) \max(r,a_t) max ( r , a t )
∑ a ∈ D e − d ( y , a ) 2 / [ 1000 ( 1 + t ) ] ≤ C ( 1 + ( 1 + t ) 3 / 2 r − 3 ) . \sum_{a\in\mathcal{D}} e^{-d(y,a)^2/[1000(1+t)]} \le C(1+(1+t)^{3/2}r^{-3}). a ∈ D ∑ e − d ( y , a ) 2 / [ 1000 ( 1 + t )] ≤ C ( 1 + ( 1 + t ) 3/2 r − 3 ) . For the torus expression, select a nearest lift of each anchor; it is bounded by the full lifted Gaussian sum just estimated. Only y y y S S S (15) , because the anchors are within D s Ds D s U U U w ( t ) w(t) w ( t ) (21) , since w ( t ) ( 1 + t ) 3 / 2 ≤ C w(t)(1+t)^{3/2}\le C w ( t ) ( 1 + t ) 3/2 ≤ C
The integral of the scalar weight on the right of (21) is at most C ( 1 + δ r − 3 ) ≤ B 0 C(1+\delta r^{-3})\le B_0 C ( 1 + δ r − 3 ) ≤ B 0 B 0 B_0 B 0 n S ( ω ( t ) ) n_S(\omega(t)) n S ( ω ( t )) (11) at λ = 1 / 2 \lambda=1/2 λ = 1/2
E c l exp [ B 0 − 1 C ∫ 0 δ / 2 ( w ( t ) + r − 3 ) n S ( ω ( t ) ) d t ] ≤ E P e n S ≤ E P e n ϕ S ≤ e C v ρ ∣ S ∣ . \mathbb{E}_{\mathrm{cl}}\exp\left[B_0^{-1}C\int_0^{\delta/2}(w(t)+r^{-3})n_S(\omega(t))\,\mathrm{d}t\right]\le\mathbb{E}_{\mathrm{P}}e^{n_S}\le\mathbb{E}_{\mathrm{P}}e^{n_{\phi_S}}\le e^{C_v\rho|S|}. E cl exp [ B 0 − 1 C ∫ 0 δ /2 ( w ( t ) + r − 3 ) n S ( ω ( t )) d t ] ≤ E P e n S ≤ E P e n ϕ S ≤ e C v ρ ∣ S ∣ . Time reversal gives the other half of the assertion.
Proposition 7.2 (Specified defective cells). Choose D D D ε \varepsilon ε C v ∑ j ≥ 0 D j 3 e − κ D j 2 ≤ 10 − 4 b − ε C_v\sum_{j\ge0}D_j^3e^{-\kappa D_j^2}\le10^{-4}b_{-\varepsilon} C v ∑ j ≥ 0 D j 3 e − κ D j 2 ≤ 1 0 − 4 b − ε (17) . Then choose ζ > 0 \zeta>0 ζ > 0 C 0 > Δ A C_0>\Delta_A C 0 > Δ A q c l = q c l ( ρ ) → 0 q_{\mathrm{cl}}=q_{\mathrm{cl}}(\rho)\to0 q cl = q cl ( ρ ) → 0 ρ ↓ 0 \rho\downarrow0 ρ ↓ 0 U U U
P c l ( every cell of U is defective ) ≤ q c l ∣ U ∣ . (22) P_{\mathrm{cl}}(\text{every cell of }U\text{ is defective})\le q_{\mathrm{cl}}^{|U|}. \tag*{(22)} P cl ( every cell of U is defective ) ≤ q cl ∣ U ∣ . (22) The bound and the convergence of q c l q_{\mathrm{cl}} q cl C 0 C_0 C 0
Proof. Let u = ∣ U ∣ ≥ 1 u=|U|\ge1 u = ∣ U ∣ ≥ 1 u / 3 u/3 u /3 ε / 4 \varepsilon/4 ε /4 [ 2 b − m , b + m / 2 ] [2b_{-m},b_{+m/2}] [ 2 b − m , b + m /2 ] (12) applies to the union of that subcollection.
Fix a subcollection of u ′ u' u ′ U ′ U' U ′ ε b − m u ′ / 2 \varepsilon b_{-m}u'/2 ε b − m u ′ /2 k ′ = ε b − m u ′ / 20 k'=\varepsilon b_{-m}u'/20 k ′ = ε b − m u ′ /20
For trajectories, exponential Markov and (17) give probability at most
exp { [ 10 − 4 b − ε + o ρ ( 1 ) ] ρ ∣ U ′ ∣ − k ′ } , \exp\{[10^{-4}b_{-\varepsilon}+o_\rho(1)]\rho|U'|-k'\}, exp {[ 1 0 − 4 b − ε + o ρ ( 1 )] ρ ∣ U ′ ∣ − k ′ } , which is at most e − c k ′ e^{-ck'} e − c k ′ ρ ∣ U ′ ∣ ≤ 2 m u ′ \rho|U'|\le2mu' ρ ∣ U ′ ∣ ≤ 2 m u ′ (13) with S = U ′ S=U' S = U ′ D s Ds D s U ′ U' U ′ C ( D + 2 ) 3 ∣ U ′ ∣ C(D+2)^3|U'| C ( D + 2 ) 3 ∣ U ′ ∣ (13) applies to it at time zero. Fix ζ \zeta ζ C v ζ 3 ρ ∣ S ∣ ≤ k ′ / 2 C_v\zeta^3\rho|S|\le k'/2 C v ζ 3 ρ ∣ S ∣ ≤ k ′ /2 e − λ 0 k ′ / 2 e^{-\lambda_0 k'/2} e − λ 0 k ′ /2
For either score reason the failed anchors are mutually separated by more than r r r U ′ U' U ′ D s Ds D s U ′ U' U ′ k ′ k' k ′ C 0 − Δ A C_0-\Delta_A C 0 − Δ A ( C 0 − Δ A ) k ′ (C_0-\Delta_A)k' ( C 0 − Δ A ) k ′
exp { C v ρ ∣ S ∣ − ( C 0 − Δ A ) k ′ / B 0 } . \exp\{C_v\rho|S|-(C_0-\Delta_A)k'/B_0\}. exp { C v ρ ∣ S ∣ − ( C 0 − Δ A ) k ′ / B 0 } . The ratio ρ ∣ S ∣ / k ′ \rho|S|/k' ρ ∣ S ∣/ k ′ C 0 C_0 C 0 e − k ′ e^{-k'} e − k ′
For small density, λ 0 = ζ ρ 1 / 6 ≤ 1 \lambda_0=\zeta\rho^{1/6}\le1 λ 0 = ζ ρ 1/6 ≤ 1 e − c λ 0 m u ′ e^{-c\lambda_0mu'} e − c λ 0 m u ′ 2 u 2^u 2 u u ′ ≥ u / 3 u'\ge u/3 u ′ ≥ u /3 C 1 u e − c λ 0 m u / 3 C_1^u e^{-c\lambda_0mu/3} C 1 u e − c λ 0 m u /3 C 1 C_1 C 1
λ 0 m ≥ 1 2 ζ K 3 ρ − 1 / 3 ⟶ ∞ . \lambda_0m\ge\frac{1}{2}\zeta K^3\rho^{-1/3}\longrightarrow\infty. λ 0 m ≥ 2 1 ζ K 3 ρ − 1/3 ⟶ ∞. Absorbing C 1 u C_1^u C 1 u (22) , for example with q c l = C 1 e − c λ 0 m / 3 q_{\mathrm{cl}}=C_1e^{-c\lambda_0m/3} q cl = C 1 e − c λ 0 m /3 m m m (22) .
We have obtained a joint defect bound under the closed probability law. The second-moment comparison will instead produce a finite measure with one endpoint opened. The following transfer retains both the small defect parameter and the cell-pair mass factor M − 2 M^{-2} M − 2
For cells a , b a,b a , b μ a b \mu_{ab} μ ab i i i Y i Y_i Y i b b b z ∈ a z\in a z ∈ a Y j Y_j Y j 1 / ( N V ) 1/(NV) 1/ ( N V ) Z N Z_N Z N symmetric when it is invariant under simultaneous relabeling of all slots; it must not single out the opened slot.
Proposition 7.3 (Transfer to an open insertion). For any measurable symmetric slab event E \mathcal{E} E
μ a b ( E ) ≤ C M 2 P c l ( E ) 1 / 4 . (23) \mu_{ab}(\mathcal{E})\le\frac{C}{M^2}P_{\mathrm{cl}}(\mathcal{E})^{1/4}. \tag*{(23)} μ ab ( E ) ≤ M 2 C P cl ( E ) 1/4 . (23) In particular, the insertion mass is at most C / M 2 C/M^2 C / M 2 (22) transfers to this measure with exponent q c l ∣ U ∣ / 4 q_{\mathrm{cl}}^{|U|/4} q cl ∣ U ∣/4 C / M 2 C/M^2 C / M 2
Proof. Two applications of Cauchy–Schwarz give the fourth root: the first separates the two insertion factors, and the second bounds a local particle count on the event E \mathcal{E} E K E ( E , S ) K_{\mathcal{E}}(E,S) K E ( E , S ) W ( S , E ; d ω ) 1 E W(S,E;d\omega)1_{\mathcal{E}} W ( S , E ; d ω ) 1 E U E = K E e − ( β / 2 − δ ) H U_{\mathcal{E}}=K_{\mathcal{E}}e^{-(\beta/2-\delta)H} U E = K E e − ( β /2 − δ ) H
For a one-particle function f f f a ( f ) a(f) a ( f ) N N N ( N − 1 ) (N-1) ( N − 1 ) a ∗ ( 1 b ) a ( 1 a ) = ∑ i ( ∣ 1 b ⟩ ⟨ 1 a ∣ ) i a^*(\mathbf{1}_b)a(\mathbf{1}_a)=\sum_i(\lvert\mathbf{1}_b\rangle\langle\mathbf{1}_a\rvert)_i a ∗ ( 1 b ) a ( 1 a ) = ∑ i (∣ 1 b ⟩ ⟨ 1 a ∣ ) i μ a b \mu_{ab} μ ab
μ a b ( E ) = 1 N V Z N Tr s y m e − β H / 2 a ∗ ( 1 b ) a ( 1 a ) U E . \mu_{ab}(\mathcal{E})=\frac{1}{NVZ_N}\operatorname{Tr}_{\mathrm{sym}}e^{-\beta H/2}a^*(\mathbf{1}_b)a(\mathbf{1}_a)U_{\mathcal{E}}. μ ab ( E ) = N V Z N 1 Tr sym e − β H /2 a ∗ ( 1 b ) a ( 1 a ) U E . Apply Hilbert–Schmidt Cauchy–Schwarz to a ( 1 b ) e − β H / 2 a(\mathbf{1}_b)e^{-\beta H/2} a ( 1 b ) e − β H /2 a ( 1 a ) U E a(\mathbf{1}_a)U_{\mathcal{E}} a ( 1 a ) U E s 3 s^3 s 3 a ∗ ( 1 a ) a ( 1 a ) ≤ s 3 n a a^*(\mathbf{1}_a)a(\mathbf{1}_a)\le s^3n_a a ∗ ( 1 a ) a ( 1 a ) ≤ s 3 n a b b b Tr e − β H / 2 n b = Z N m \operatorname{Tr}e^{-\beta H/2}n_b=Z_Nm Tr e − β H /2 n b = Z N m s 3 Z N m s^3Z_Nm s 3 Z N m
For the second norm use Tr U E ∗ n a U E \operatorname{Tr}U_{\mathcal{E}}^*n_aU_{\mathcal{E}} Tr U E ∗ n a U E U E P s y m U_{\mathcal{E}}P_{\mathrm{sym}} U E P sym
0 ≤ U E P s y m ≤ e − β H / 2 P s y m pointwise in kernels. 0\le U_{\mathcal{E}}P_{\mathrm{sym}}\le e^{-\beta H/2}P_{\mathrm{sym}}\quad\text{pointwise in kernels.} 0 ≤ U E P sym ≤ e − β H /2 P sym pointwise in kernels. Replace only the adjoint kernel in that trace by the untested one. This increases the integral and, by cyclicity, gives
Tr s y m U ε ∗ n a U ε ≤ Tr s y m e − ( β − δ ) H n a K ε = Z N E c l [ n a 1 E ] . \operatorname{Tr}_{\mathrm{sym}} U_\varepsilon^* n_a U_\varepsilon\le\operatorname{Tr}_{\mathrm{sym}} e^{-(\beta-\delta)H} n_a K_\varepsilon= Z_N \mathbb{E}_{\mathrm{cl}}[n_a \mathbf{1}_{\mathcal{E}}]. Tr sym U ε ∗ n a U ε ≤ Tr sym e − ( β − δ ) H n a K ε = Z N E cl [ n a 1 E ] . Here n a n_a n a C m 2 Cm^2 C m 2 C Z N m P c l ( E ) 1 / 2 CZ_NmP_{\mathrm{cl}}(\mathcal{E})^{1/2} C Z N m P cl ( E ) 1/2 C s 3 Z N m P c l ( E ) 1 / 2 Cs^3Z_NmP_{\mathrm{cl}}(\mathcal{E})^{1/2} C s 3 Z N m P cl ( E ) 1/2 N V Z N NVZ_N N V Z N C s 3 m / ( N V ) Cs^3m/(NV) C s 3 m / ( N V ) m = N / M m=N/M m = N / M s 3 = V / M s^3=V/M s 3 = V / M C / M 2 C/M^2 C / M 2
Cell paths
We now construct paths between distant cells. Their purpose is to keep the second moment of an averaged endpoint change bounded as the volume grows. The first step controls encounters of two random skeletons; the second step adapts each skeleton to the actual unusable cells.
Random skeletons and their encounter moments
The use of unpredictable paths and intersection moments follows Benjamini–Pemantle–Peres [5 ] (Theorem 1.3 and Lemma 3.1). The independent block velocities below are a continuous dyadic version of the aligned block-drift construction of Häggström and Mossel [24 ] (Section 3, Proposition 3.1). Finite-endpoint path averaging appears in the synchronization argument of Abbe, Massoulié, Montanari, Sly, and Srivastava [1 ] (Section 6, Lemma 6.1) and in the symmetry-breaking argument of Garban and Spencer [23 ] (Section 2, Lemma 2.5). We give an explicit dyadic construction and prove the thickened and soft-distance estimates needed here.
The construction in this subsection is purely geometric. Let G n = ( Z / n Z ) 3 \mathcal{G}_n=(\mathbb{Z}/n\mathbb{Z})^3 G n = ( Z / n Z ) 3 a , b a,b a , b ( w , d 2 , d 3 ) (w,d_2,d_3) ( w , d 2 , d 3 )
⌈ n / 10 ⌉ ≤ w ≤ ⌊ n / 8 ⌋ , ∣ d 2 ∣ , ∣ d 3 ∣ ≤ w . \lceil n/10\rceil\le w \le\lfloor n/8\rfloor,\qquad|d_2|,|d_3|\le w. ⌈ n /10 ⌉ ≤ w ≤ ⌊ n /8 ⌋ , ∣ d 2 ∣ , ∣ d 3 ∣ ≤ w . Call such a pair admissible. Admissible pairs form a fraction at least an absolute υ > 0 \upsilon>0 υ > 0 n n n a a a G n \mathcal{G}_n G n
The first coordinate will advance from 0 0 0 w w w
For q = 2 , 3 q=2,3 q = 2 , 3 ( U ℓ , j ( q ) ) ℓ , j ≥ 0 (U_{\ell,j}^{(q)})_{\ell,j\ge0} ( U ℓ , j ( q ) ) ℓ , j ≥ 0 [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] c v e l = ( 1 − 2 − 1 / 4 ) / 2 c_{\mathrm{vel}}=(1-2^{-1/4})/2 c vel = ( 1 − 2 − 1/4 ) /2
v i ( q ) = c v e l ∑ ℓ ≥ 0 2 − ℓ / 4 U ℓ , ⌊ ( i − 1 ) / 2 ℓ ⌋ ( q ) , Z t ( q ) = ∑ i = 1 t v i ( q ) , Z 0 ( q ) = 0. v_i^{(q)}=c_{\mathrm{vel}}\sum_{\ell\ge0}2^{-\ell/4}U_{\ell,\lfloor(i-1)/2^\ell\rfloor}^{(q)},\qquad Z_t^{(q)}=\sum_{i=1}^{t}v_i^{(q)},\qquad Z_0^{(q)}=0. v i ( q ) = c vel ℓ ≥ 0 ∑ 2 − ℓ /4 U ℓ , ⌊( i − 1 ) / 2 ℓ ⌋ ( q ) , Z t ( q ) = i = 1 ∑ t v i ( q ) , Z 0 ( q ) = 0. The series converges absolutely, and ∣ v i ( q ) ∣ ≤ 1 / 2 |v_i^{(q)}|\le1/2 ∣ v i ( q ) ∣ ≤ 1/2 t = 0 , … , w t=0,\ldots,w t = 0 , … , w
( t , ⌊ t d 2 / w + Z min ( t , w − t ) ( 2 ) ⌋ , ⌊ t d 3 / w + Z min ( t , w − t ) ( 3 ) ⌋ ) . \left(t,\left\lfloor td_2/w+Z_{\min(t,w-t)}^{(2)}\right\rfloor,\left\lfloor td_3/w+Z_{\min(t,w-t)}^{(3)}\right\rfloor\right). ( t , ⌊ t d 2 / w + Z m i n ( t , w − t ) ( 2 ) ⌋ , ⌊ t d 3 / w + Z m i n ( t , w − t ) ( 3 ) ⌋ ) . Starting from the base vertex at time t t t t + 1 t+1 t + 1 a a a γ = ( z 0 , … , z l ) \gamma=(z_0,\ldots,z_l) γ = ( z 0 , … , z l ) t ≥ 1 t\ge1 t ≥ 1 U U U k k k k 3 / 4 k^{3/4} k 3/4 C k − 3 / 2 Ck^{-3/2} C k − 3/2
For vertices on the torus write
d ∞ ( x , y ) = min k ∈ Z 3 ∥ x ~ − y ~ + n k ∥ ∞ , d ∞ ( x , γ ′ ) = min z ′ ∈ γ ′ d ∞ ( x , z ′ ) . d_\infty(x,y)=\min_{k\in\mathbb{Z}^3}\|\widetilde{x}-\widetilde{y}+nk\|_\infty,\qquad d_\infty(x,\gamma')=\min_{z'\in\gamma'}d_\infty(x,z'). d ∞ ( x , y ) = k ∈ Z 3 min ∥ x − y + nk ∥ ∞ , d ∞ ( x , γ ′ ) = z ′ ∈ γ ′ min d ∞ ( x , z ′ ) . where the tildes denote arbitrary lifts. For two independent skeletons γ , γ ′ \gamma,\gamma' γ , γ ′ R ∗ ≥ 1 R_* \ge1 R ∗ ≥ 1
J 0 = ∑ z ∈ γ 1 { d ∞ ( z , γ ′ ) ≤ R ∗ } + ∑ z ′ ∈ γ ′ 1 { d ∞ ( z ′ , γ ) ≤ R ∗ } , S B = ∑ z ∈ γ e − d ∞ ( z , γ ′ ) / B ( B > 0 ) . J_0=\sum_{z\in\gamma}\mathbf{1}_{\{d_\infty(z,\gamma')\le R_*\}}+\sum_{z'\in\gamma'}\mathbf{1}_{\{d_\infty(z',\gamma)\le R_*\}},\qquad S_B=\sum_{z\in\gamma}e^{-d_\infty(z,\gamma')/B}\quad(B>0). J 0 = z ∈ γ ∑ 1 { d ∞ ( z , γ ′ ) ≤ R ∗ } + z ′ ∈ γ ′ ∑ 1 { d ∞ ( z ′ , γ ) ≤ R ∗ } , S B = z ∈ γ ∑ e − d ∞ ( z , γ ′ ) / B ( B > 0 ) . Every sum uses the listed vertex occurrences, including the endpoints. An occurrence contributes once to J 0 J_0 J 0
Lemma 8.1 (Skeleton encounters). For the independent skeletons just constructed, there are absolute constants c h i t > 0 c_{\mathrm{hit}}>0 c hit > 0 C < ∞ C<\infty C < ∞ R ∗ ≥ 1 R_*\ge1 R ∗ ≥ 1
E e α J 0 ≤ C , α = c h i t ( 1 + R ∗ ) 2 . (24) \mathbb{E}e^{\alpha J_0}\le C,\qquad\alpha=\frac{c_{\mathrm{hit}}}{(1+R_*)^2}. \tag*{(24)} E e α J 0 ≤ C , α = ( 1 + R ∗ ) 2 c hit . (24) For every fixed B > 0 B>0 B > 0 ξ ( B ) > 0 \xi(B)>0 ξ ( B ) > 0 C ( B ) < ∞ C(B)<\infty C ( B ) < ∞
E e ξ ( B ) S B ≤ C ( B ) . (25) \mathbb{E}e^{\xi(B)S_B}\le C(B). \tag*{(25)} E e ξ ( B ) S B ≤ C ( B ) . (25) Both estimates are uniform in n n n
Proof. Conditional density bound. In this proof write Z t = ( Z t ( 2 ) , Z t ( 3 ) ) Z_t=(Z_t^{(2)},Z_t^{(3)}) Z t = ( Z t ( 2 ) , Z t ( 3 ) ) Z t ′ Z'_t Z t ′ h = ⌊ w / 2 ⌋ h=\lfloor w/2\rfloor h = ⌊ w /2 ⌋ γ ′ \gamma' γ ′
I ℓ , j = { j 2 ℓ + 1 , … , ( j + 1 ) 2 ℓ } , I_{\ell,j}=\{j2^\ell+1,\ldots,(j+1)2^\ell\}, I ℓ , j = { j 2 ℓ + 1 , … , ( j + 1 ) 2 ℓ } , and let F t \mathcal{F}_t F t U ℓ , j ( q ) U_{\ell,j}^{(q)} U ℓ , j ( q ) I ℓ , j ∩ { 1 , … , t } ≠ ∅ I_{\ell,j}\cap\{1,\ldots,t\}\ne\varnothing I ℓ , j ∩ { 1 , … , t } = ∅ q = 2 , 3 q=2,3 q = 2 , 3 t t t t + k ≤ h t+k\le h t + k ≤ h k ≥ 1 k\ge1 k ≥ 1 k / 4 k/4 k /4 k / 4 k/4 k /4 k / 2 k/2 k /2 k = 1 k=1 k = 1 c k 3 / 4 ck^{3/4} c k 3/4 ℓ \ell ℓ c 2 3 ℓ / 4 c2^{3\ell/4} c 2 3 ℓ /4 Z t + k Z_{t+k} Z t + k F t \mathcal{F}_t F t C k − 3 / 2 Ck^{-3/2} C k − 3/2
Same-time encounter tests. Write D t = ∣ Z t − Z t ′ ∣ ∞ D_t=|Z_t-Z'_t|_\infty D t = ∣ Z t − Z t ′ ∣ ∞ 0 ≤ t ≤ h 0\le t\le h 0 ≤ t ≤ h t t t b t , b t ′ b_t,b'_t b t , b t ′ z z z t t t b t b_t b t z ′ z' z ′ t ′ t' t ′ ∣ t − t ′ ∣ ≤ d ∞ ( z , z ′ ) |t-t'|\le d_\infty(z,z') ∣ t − t ′ ∣ ≤ d ∞ ( z , z ′ ) w ≤ n / 8 w \le n/8 w ≤ n /8
d ∞ ( b t , b t ′ ) ≤ 3 d ∞ ( z , z ′ ) + 4. d_{\infty}(b_t,b_t') \le3d_{\infty}(z,z') + 4. d ∞ ( b t , b t ′ ) ≤ 3 d ∞ ( z , z ′ ) + 4. The same-time lifted transverse differences also do not wrap. The linear drifts are common, each random deviation has absolute value at most w / 4 w/4 w /4 w / 2 + 1 < n / 2 w/2+1<n/2 w /2 + 1 < n /2 u = min ( t , w − t ) u=\min(t,w-t) u = min ( t , w − t )
D u ≤ 3 d ∞ ( z , z ′ ) + 5. D_u \le3d_{\infty}(z,z') + 5. D u ≤ 3 d ∞ ( z , z ′ ) + 5. Consequently
J 0 ≤ C ∑ t = 0 h 1 { D t ≤ C ( R ∗ + 1 ) } . J_0 \le C\sum_{t=0}^{h}\mathbf{1}_{\{D_t\le C(R_*+1)\}}. J 0 ≤ C t = 0 ∑ h 1 { D t ≤ C ( R ∗ + 1 )} . The reflected times t t t w − t w-t w − t Z t Z_t Z t
For the distance-weighted sum, take z ′ z' z ′ γ ′ \gamma' γ ′
e − d ∞ ( z , γ ′ ) / B ≤ e 5 / ( 3 B ) e − D u / ( 3 B ) , S B ≤ C ( B ) ∑ t = 0 h e − D t / ( C B ) . e^{-d_{\infty}(z,\gamma')/B}\le e^{5/(3B)}e^{-D_u/(3B)},\qquad S_B\le C(B)\sum_{t=0}^{h}e^{-D_t/(CB)}. e − d ∞ ( z , γ ′ ) / B ≤ e 5/ ( 3 B ) e − D u / ( 3 B ) , S B ≤ C ( B ) t = 0 ∑ h e − D t / ( C B ) . These are the two deterministic reductions used in the moment estimate.
Exponential moment. Let X t ∈ [ 0 , 1 ] X_t\in[0,1] X t ∈ [ 0 , 1 ] F t \mathcal{F}_t F t γ ′ \gamma' γ ′
E [ ∑ u = t + 1 h X u ∣ F t ] ≤ b 0 , b 0 = { C ( R ∗ + 1 ) 2 , X u = 1 { ∣ Z u − Z u ′ ∣ ∞ ≤ C ( R ∗ + 1 ) } , C ( B ) , X u = e − ∣ Z u − Z u ′ ∣ ∞ / ( C B ) . \mathbb{E}\left[\left.\sum_{u=t+1}^{h}X_u\right|\mathcal{F}_t\right]\le b_0,\qquad
b_0=
\begin{cases}
C(R_*+1)^2,&X_u=\mathbf{1}_{\{|Z_u-Z'_u|_{\infty}\le C(R_*+1)\}},\\
C(B),&X_u=e^{-|Z_u-Z'_u|_{\infty}/(CB)}.
\end{cases} E [ u = t + 1 ∑ h X u F t ] ≤ b 0 , b 0 = { C ( R ∗ + 1 ) 2 , C ( B ) , X u = 1 { ∣ Z u − Z u ′ ∣ ∞ ≤ C ( R ∗ + 1 )} , X u = e − ∣ Z u − Z u ′ ∣ ∞ / ( C B ) . For every j ≥ 1 j\ge1 j ≥ 1 j j j ( b 0 + 1 ) j (b_0+1)^j ( b 0 + 1 ) j X 0 ≤ 1 X_0\le1 X 0 ≤ 1
e θ ∑ t = 0 h X t = ∏ t = 0 h ( 1 + ( e θ X t − 1 ) ) . e^{\theta\sum_{t=0}^{h}X_t}=\prod_{t=0}^{h}\left(1+(e^{\theta X_t}-1)\right). e θ ∑ t = 0 h X t = t = 0 ∏ h ( 1 + ( e θ X t − 1 ) ) . The inequality e θ x − 1 ≤ ( e θ − 1 ) x e^{\theta x}-1\le(e^\theta-1)x e θ x − 1 ≤ ( e θ − 1 ) x 0 ≤ x ≤ 1 0\le x\le1 0 ≤ x ≤ 1 ∑ j ≥ 0 [ ( e θ − 1 ) ( b 0 + 1 ) ] j \sum_{j\ge0}[(e^\theta-1)(b_0+1)]^j ∑ j ≥ 0 [( e θ − 1 ) ( b 0 + 1 ) ] j θ \theta θ 1 / ( b 0 + 1 ) 1/(b_0+1) 1/ ( b 0 + 1 )
Detours around unusable components
We now apply the geometric construction to the coarse grid of Section 2. This is the point at which the physical cutoffs enter: set
R ∗ = ⌈ 20 ( R + R A + D + 20 ) ⌉ , ε = α / 20 , R_*=\lceil20(R+R_A+D+20)\rceil,\qquad\varepsilon=\alpha/20, R ∗ = ⌈ 20 ( R + R A + D + 20 )⌉ , ε = α /20 , decreasing the absolute constant in α \alpha α D D D D D D ζ \zeta ζ C 0 C_0 C 0 C 0 C_0 C 0 C 0 C_0 C 0 ν / 8 \nu/8 ν /8
The skeleton law does not see the unusable cells. We next adapt each sampled skeleton to those cells by a deterministic detour procedure. This step uses nearest-neighbor components of bad cells but allows sup-norm-one steps on the resulting path. Here and below, bad means unusable in the sense of Section 5. Reject the configuration if any nearest-neighbor component of bad cells has size ≥ n / 4 \ge n/4 ≥ n /4
A smaller component T T T n / 4 n/4 n /4 T T T n / 4 n/4 n /4 T T T
The detour procedure scans an accepted skeleton. On entering an outermost fill, meaning one not contained in another fill, it replaces the portion through its last visit there by an exterior-boundary path. The next proof establishes the boundary connectivity that permits this replacement with sup-norm-one steps. Fix all tie-breaking rules for boundary paths in advance, and use chronological loop erasure for the resulting walk.
Lemma 8.2 (Deterministic detours). Apply the rejection rules just specified to a nearest-neighbor skeleton from a a a b b b ( Z / n Z ) 3 (\mathbb{Z}/n\mathbb{Z})^3 ( Z / n Z ) 3 a a a b b b T T T 6 ∣ T ∣ 6|T| 6∣ T ∣
Proof. Fills and projection. We first establish the boundary facts needed by the procedure. In the lattice the fill is connected, since each hole neighbors the obstacle. Its exterior is connected, since a finite set has just one infinite complementary component. An edge from the fill to its exterior starts in the obstacle itself.
For disjoint non-neighboring connected obstacles, their fills are nested or disjoint and non-neighboring. Indeed, each obstacle is in a single complementary component of the other. If it is in a hole, its fill stays there: outside that hole one connects to infinity avoiding the obstacle in it, either in the exterior of the enclosing obstacle or by using that connected enclosing obstacle, also to escape from any of its other holes. If the two obstacles are mutually in the exteriors, each connected fill avoids the other obstacle and thus stays in the other’s exterior. Adjacency would imply adjacency of the obstacles themselves by the crossing-edge property.
These facts survive projection to the torus. An intersection or adjacency can be lifted with corresponding translates of the two obstacles and their fills. The small box size avoids self-overlap and preserves cardinality, so proper nesting stays proper. We therefore use the outermost fills in the construction below.
Connected exterior boundary. The exterior vertex boundary consists of the exterior neighbors of the obstacle. It has at most 6 ∣ T ∣ 6|T| 6∣ T ∣
We give a self-contained cycle-space proof of this boundary connectivity fact; see Timár [34 ] , arXiv version 2, Lemmas 1–2. Consider the edge cut from the lattice fill to its exterior. Link two cut edges when they belong to a common elementary square. All cut edges belong to one link component. Otherwise choose one link component. Every square contains an even number of its edges: it contains an even number of edges of the full cut, and all those edges are linked. The same parity holds for every lattice cycle, since squares generate cycles modulo two. Explicitly, in a closed walk commute steps in different axes, each swap changing the walk by an elementary square modulo two, and then cancel backtracks. Counting the selected edges modulo two along paths from a fixed vertex now defines a vertex potential whose differing-neighbor edges are exactly the selected edges. This potential is constant on the connected fill and on its connected exterior. Its cut must therefore be either empty or the full cut, contradicting the choice of a proper nonempty link component. This proves the claim about cut edges. Their linked exterior endpoints are star neighbors or identical, proving boundary connectivity.
Project this connected boundary back to the torus. Any edge leaving the projected fill lifts from within the fill to outside and hence goes from the projected obstacle to this projected boundary. The boundary vertices are good: the small box size places them outside the projected fill, and if one were bad it would belong to the adjacent bad component. Thus each outermost fill has a connected star boundary of at most 6 ∣ T ∣ 6|T| 6∣ T ∣
Detour construction. Scan the nearest-neighbor skeleton starting outside the outermost fills. If the next vertex enters one, replace the portion through the last skeleton vertex in that same fill by a star-boundary path from the exterior predecessor to the exterior successor. These are boundary vertices by nearest-neighbor crossing, and the successor is outside all outermost fills by non-adjacency. Choose a deterministic simple path on the projected boundary, with fixed tie-breaking depending only on that component and the two vertices. Projection permits this connection regardless of the endpoint lifts used. Continue scanning from the successor.
Finally apply chronological loop erasure to obtain a simple star path P P P a a a b b b 6 ∣ T ∣ 6|T| 6∣ T ∣
Deletion invariance. Suppose the original configuration is accepted. Any fill that intersects the skeleton must have its obstacle hit by the skeleton: the start is exterior and the skeleton has nearest-neighbor steps. Enclosing fills of such obstacles are hit as well. The construction is therefore determined by the actual bad components intersecting the skeleton. Deleting other whole bad components in this accepted configuration preserves acceptance and leaves both the detours and the final loop-erased path unchanged. □
Rejection by large components has closed probability tending to zero at large n n n [22 ] : count connected sets of size ⌈ n / 4 ⌉ \lceil n/4\rceil ⌈ n /4 ⌉ M C ⌈ n / 4 ⌉ MC^{\lceil n/4\rceil} M C ⌈ n /4 ⌉ ∑ j ≥ 1 C ( j + 1 ) 3 C j q c l j \sum_{j\ge1} C(j+1)^3 C^j q_{\rm cl}^j ∑ j ≥ 1 C ( j + 1 ) 3 C j q cl j j j j j j j
Opening the slab
The local estimates and cell paths now define a change from the closed Gibbs law to an open path measure. We specify the target measure, the accepted choices, and the interacting proposal in that order. Keeping its normalization exact will let the two copies of the change cancel in the next section.
Let q 0 = ⟨ u 0 , γ ( 1 ) u 0 ⟩ / N q_0=\langle u_0,\gamma^{(1)}u_0\rangle/N q 0 = ⟨ u 0 , γ ( 1 ) u 0 ⟩ / N i 0 i_0 i 0 1 / N 1/N 1/ N E i 0 = z E_{i_0}=z E i 0 = z d z / V dz/V d z / V Y i Y_i Y i Y i Y_i Y i q 0 q_0 q 0 N − 1 ∑ i ( ∣ u 0 ⟩ ⟨ u 0 ∣ ) i N^{-1}\sum_i(|u_0\rangle\langle u_0|)_i N − 1 ∑ i ( ∣ u 0 ⟩ ⟨ u 0 ∣ ) i x x x Y i 0 Y_{i_0} Y i 0 i 0 i_0 i 0 μ \mu μ
The base law and acceptance
Our base law is the closed probability (14), denoted by the configuration C C C x , z x,z x , z d x d z / V 2 dx\,dz/V^2 d x d z / V 2 S C S^C S C
Set a = cell ( x ) a=\operatorname{cell}(x) a = cell ( x ) b = cell ( z ) b=\operatorname{cell}(z) b = cell ( z ) μ a b \mu_{ab} μ ab
Table 1.
The reverse slot e e e C C C A + ( x ; C ) , A + ( z ; C ) ≤ C 0 A^+(x;C),A^+(z;C)\le C_0 A + ( x ; C ) , A + ( z ; C ) ≤ C 0 acc ( C , γ ) \operatorname{acc}(C,\gamma) acc ( C , γ ) υ / 2 \upsilon/2 υ /2 υ \upsilon υ
Selecting one eligible slot in each path cell
On acceptance let P ( C , γ ) P(C,\gamma) P ( C , γ ) Q Q Q
g Q ( C ) = ∑ i : Y i ∈ Q 1 { i eligible in C } . g_Q(C)=\sum_{i:Y_i\in Q}\mathbf{1}_{\{i\text{ eligible in }C\}}. g Q ( C ) = i : Y i ∈ Q ∑ 1 { i eligible in C } . Independently in each path cell choose a uniform eligible slot. This gives an ordered tuple I = ( i 0 , … , i k ) I=(i_0,\ldots,i_k) I = ( i 0 , … , i k )
h γ ( C , I ) = 1 { ( cell ( Y i 0 ) , … , cell ( Y i k ) ) = P ( C , γ ) } ∏ j = 0 k 1 { i j eligible in C } g cell ( Y i j ) ( C ) . h_\gamma(C,I)=\mathbf{1}_{\{(\operatorname{cell}(Y_{i_0}),\ldots,\operatorname{cell}(Y_{i_k}))=P(C,\gamma)\}}
\prod_{j=0}^{k}\frac{\mathbf{1}_{\{i_j\text{ eligible in }C\}}}{g_{\operatorname{cell}(Y_{i_j})}(C)}. h γ ( C , I ) = 1 {( cell ( Y i 0 ) , … , cell ( Y i k )) = P ( C , γ )} j = 0 ∏ k g cell ( Y i j ) ( C ) 1 { i j eligible in C } . Set the weight to zero off acceptance, and interpret a factor with zero eligible count as zero. The cells of P P P ∑ I h γ ( C , I ) = 1 \sum_I h_\gamma(C,I)=1 ∑ I h γ ( C , I ) = 1 a a a b b b Y Y Y a , b , Y a,b,Y a , b , Y C C C
The conditional interacting proposal
For a tuple I I I ( p I S ) i = S pred I ( i ) (p_I S)_i=S_{\operatorname{pred}_I(i)} ( p I S ) i = S pred I ( i ) S O = p I S C S^O=p_I S^C S O = p I S C E O E^O E O Y Y Y i 0 i_0 i 0 z z z I I I I I I cut R \operatorname{cut}_R cut R
More precisely, for a state X X X W G X W^X_G W G X G G G G G G Z G ( X ) = ∫ 1 cut R ( G ) W G X Z_G(X)=\int\mathbf{1}_{\operatorname{cut}_R(G)}W^X_G Z G ( X ) = ∫ 1 cut R ( G ) W G X I I I O O O Z I ( O ) Z_I(O) Z I ( O ) ( D + 10 ) s (D+10)s ( D + 10 ) s z z z
The density and its second moment
Averaging a path-dependent likelihood before estimating its second moment also appears in random-trail detection and perturbed-lattice reassignment [2 , 6 , 32 ] . In the present setting the conditional interacting measures and both proposal normalizers must be retained explicitly.
Write ν \nu ν υ / 2 \upsilon/2 υ /2 μ \mu μ
f = m E γ ∑ I : first ( I ) = i 0 1 cut R ( O I ) Z I ( O ) ∫ W I C ^ h γ ( C ^ , I ) . (26) f=m\mathbb{E}_{\gamma}\sum_{I:\,\operatorname{first}(I)=i_0}\frac{\mathbf{1}_{\operatorname{cut}_R(O_I)}}{Z_I(O)}\int W_I^{\widehat{C}}h_\gamma(\widehat{C},I). \tag*{(26)} f = m E γ I : first ( I ) = i 0 ∑ Z I ( O ) 1 cut R ( O I ) ∫ W I C h γ ( C , I ) . (26) where the hypothetical closed state C ^ \widehat{C} C S C ^ = p I − 1 S O S^{\widehat{C}}=p_I^{-1}S^O S C = p I − 1 S O E C ^ = Y E^{\widehat{C}}=Y E C = Y I I I O O O
To verify the density, change start variables for each candidate tuple. The Jacobian is one and the outer kernel is invariant under this permutation. Write W I ‾ int W_{\overline{I}}^{\operatorname{int}} W I int I I I
W I ‾ int [ ∫ W I C ^ h γ ( C ^ , I ) ] 1 cut R ( O I ) W I O Z I ( O ) W_{\overline{I}}^{\operatorname{int}}\left[\int W_I^{\widehat{C}}h_\gamma(\widehat{C},I)\right]\mathbf{1}_{\operatorname{cut}_R(O_I)}\frac{W_I^O}{Z_I(O)} W I int [ ∫ W I C h γ ( C , I ) ] 1 cut R ( O I ) Z I ( O ) W I O (all incident weights and normalizers use the displayed fixed complement). The corresponding factor in the open measure is W I ‾ int W I O W_{\overline{I}}^{\operatorname{int}}W_I^O W I int W I O
base density of x target index factor × target density of x = V − 1 N − 1 s − 3 = N M = m . \frac{\text{base density of }x}{\text{target index factor}\times\text{target density of }x}
=\frac{V^{-1}}{N^{-1}s^{-3}}=\frac{N}{M}=m. target index factor × target density of x base density of x = N − 1 s − 3 V − 1 = M N = m . Here both measures restrict x x x Y i 0 Y_{i_0} Y i 0 d z / V dz/V d z / V
Cauchy–Schwarz gives ( υ / 2 ) 2 ≤ q 0 ∫ f 2 d μ (\upsilon/2)^2\leq q_0\int f^2\,d\mu ( υ /2 ) 2 ≤ q 0 ∫ f 2 d μ ∫ f 2 d μ = ∫ f d ν \int f^2\,d\mu=\int f\,d\nu ∫ f 2 d μ = ∫ f d ν γ ′ \gamma' γ ′ I ′ I' I ′
m E x , z , γ , γ ′ Z N − 1 ∫ F ( Y , S C ) d Y d S C ∑ I ′ , I : first ( I ′ ) = first ( I ) ∫ W ( C ) h γ ′ ( C , I ′ ) 1 cut R ( O I ′ ) W I ′ O Z I ′ ( O ) 1 cut R ( O I ) Z I ( O ) W I C ^ h γ ( C ^ , I ) . (27) m\mathbb{E}_{x,z,\gamma,\gamma'}Z_N^{-1}\int F(Y,S^C)\,dY\,dS^C
\sum_{I',I:\,\operatorname{first}(I')=\operatorname{first}(I)}
\int W(C)h_{\gamma'}(C,I')
\frac{\mathbf{1}_{\operatorname{cut}_R(O_{I'})}W_{I'}^O}{Z_{I'}(O)}
\frac{\mathbf{1}_{\operatorname{cut}_R(O_I)}}{Z_I(O)}W_I^{\widehat{C}}h_\gamma(\widehat{C},I). \tag*{(27)} m E x , z , γ , γ ′ Z N − 1 ∫ F ( Y , S C ) d Y d S C I ′ , I : first ( I ′ ) = first ( I ) ∑ ∫ W ( C ) h γ ′ ( C , I ′ ) Z I ′ ( O ) 1 cut R ( O I ′ ) W I ′ O Z I ( O ) 1 cut R ( O I ) W I C h γ ( C , I ) . (27) The path integrations are over all closed paths, then proposed paths on I ′ I' I ′ I I I W ( C ) W(C) W ( C ) S O = p I S C S^{O}=p_I S^{C} S O = p I S C x , z x,z x , z
Cross comparison in the slab
The construction of Section 9 reduces the condensate bound to the following estimate.
Proposition 10.1 (Uniform second moment). After fixing the cutoffs as above, there is a constant C v < ∞ C_v < \infty C v < ∞
∫ f 2 d μ ≤ C v . \int f^2\,\mathrm{d}\mu\le C_v. ∫ f 2 d μ ≤ C v . The constant is uniform for ρ / 2 ≤ N / V ≤ 2 ρ \rho/2 \le N/V \le2\rho ρ /2 ≤ N / V ≤ 2 ρ 0 < T ≤ ρ 2 0 < T \le\rho^2 0 < T ≤ ρ 2
We prove this by comparing the two changes in (27) with a single reverse insertion. Interactions away from encounters must cancel: a fixed cost at every path cell would grow with the volume. The analytic comparison below leaves only a fixed cost for shared labels and a vanishing cost for nearby distinct bridges. To retain this gain after averaging, we must also preserve the configuration-dependent path and label choices. We establish that preservation before removing their constraints from any conditional integral.
The reverse insertion and the slot partition
Fix a term of (27) . The tuple I ′ I' I ′ P ′ = P ( C , γ ′ ) P'=P(C,\gamma') P ′ = P ( C , γ ′ ) O O O O O O I I I P = P ( C , γ ) P=P(C,\gamma) P = P ( C , γ ) i 0 i_0 i 0 e e e I I I B B B
Table 1 (PDF p. 29).
The outer tuple Y Y Y Figure 1 shows the two single-tuple assignments. In the cross term, the forward state uses I ′ I' I ′ I I I
Figure 1. Endpoint assignments relative to the same original closed-state starts S C S^C S C i 0 i_0 i 0 z ∈ b z \in b z ∈ b i 2 i_2 i 2 x ∈ a x \in a x ∈ a
Identify a tuple with its slot set when taking intersections and differences, and partition the selected slots as
h = I ∩ I ′ , d = h ∪ pred I ( h ) , j = I ∖ d , l = I ′ ∖ h , L = ( I ∪ I ′ ) c . h = I \cap I', \qquad d = h \cup\operatorname{pred}_{I}(h), \qquad j = I \setminus d, \qquad l = I' \setminus h, \qquad L = (I \cup I')^{c}. h = I ∩ I ′ , d = h ∪ pred I ( h ) , j = I ∖ d , l = I ′ ∖ h , L = ( I ∪ I ′ ) c . Thus h h h d d d I I I ∣ d ∣ ≤ 2 ∣ h ∣ |d| \le2|h| ∣ d ∣ ≤ 2∣ h ∣ e ∈ d e \in d e ∈ d j j j l l l L L L J J J P P P R ∗ R_* R ∗ P ′ P' P ′
All L L L j j j C = O C = O C = O I ′ I' I ′ B B B C ^ \widehat{C} C I I I I ′ I' I ′ S i B = S succ I ( i ) C = S succ I ( i ) O = S i C ^ S_i^B = S_{\operatorname{succ}_{I}(i)}^{C} = S_{\operatorname{succ}_{I}(i)}^{O} = S_i^{\widehat{C}} S i B = S succ I ( i ) C = S succ I ( i ) O = S i C l l l B B B C C C C ^ = O \widehat{C} = O C = O L L L
Table 1 (PDF p. 30).
We will also reintegrate some of j j j O O O d ∖ h d \setminus h d ∖ h C C C
We work first on the domain of the original nonzero weights and cuts in (27). All paths there on I ∪ I ′ I \cup I' I ∪ I ′ C , O , C ^ C,O,\widehat{C} C , O , C cut R \operatorname{cut}_R cut R ( D + 10 ) s (D+10)s ( D + 10 ) s O O O C C C I ′ I' I ′ C ^ \widehat{C} C I I I S O = p I S C ^ S^O = p_I S^{\widehat{C}} S O = p I S C C , C ^ C,\widehat{C} C , C O O O B B B d d d C C C e e e x x x C ^ \widehat{C} C
For the beginning and end anchors of the integrated rows where (16) is needed the scores against L L L ≤ C 0 \le C_0 ≤ C 0 O O O I ′ I' I ′ C C C I I I C ^ \widehat{C} C z z z B j , B l B_j,B_l B j , B l C ^ , C \widehat{C},C C , C d d d C ^ \widehat{C} C x x x L L L j j j l l l O O O B B B r r r O , B O,B O , B Y Y Y C ^ , C \widehat{C}, C C , C
All confined bridge measures use nearest lifts. Write π i X \pi_i^X π i X k k k K X ( G ) K_X(G) K X ( G ) G G G Z ‾ \overline{Z} Z L L L
K C ( I ∪ I ′ ) K C ^ ( I ) K B ( I ∪ I ′ ) K O ( I ) K C ( d ) K C ^ ( d ) K B ( d ) K O ( d ) ≤ e C ∣ h ∣ . (28) \frac{K_C(I\cup I')K_{\widehat{C}}(I)}{K_B(I\cup I')K_O(I)}
\frac{K_C(\mathrm{d})K_{\widehat{C}}(\mathrm{d})}{K_B(\mathrm{d})K_O(\mathrm{d})}
\le e^{C|h|}. \tag*{(28)} K B ( I ∪ I ′ ) K O ( I ) K C ( I ∪ I ′ ) K C ( I ) K B ( d ) K O ( d ) K C ( d ) K C ( d ) ≤ e C ∣ h ∣ . (28) Conditional laws and the two denominator bounds
For G = j G=\mathrm{j} G = j l \mathrm{l} l Φ G X \Phi_G^X Φ G X G G G L L L Z [ G , X ] Z[G,X] Z [ G , X ] R R R L L L
A one-path marginal bound. In these normalized group distributions an individual path marginal is bounded by C 4 π i X C_4\pi_i^X C 4 π i X p ∗ p_* p ∗ L L L p ∗ p_* p ∗ 2 R + 5 2R+5 2 R + 5 R R R p ∗ p_* p ∗ p ∗ e − C / p ∗ p_*e^{-C/p_*} p ∗ e − C / p ∗ v ≥ 0 v\ge0 v ≥ 0 C 4 C_4 C 4 π i X \pi_i^X π i X j \mathrm{j} j C C C O O O B B B l \mathrm{l} l O O O
The needed lower normalizations are
Z ‾ I ′ ( O ) ≥ Z [ l , O ] exp [ − C ∣ h ∣ − o ρ ( 1 ) J ] , \overline{Z}_{I'}(O)\ge Z[\mathrm{l},O]\exp[-C|h|-o_\rho(1)J], Z I ′ ( O ) ≥ Z [ l , O ] exp [ − C ∣ h ∣ − o ρ ( 1 ) J ] , Z ‾ I ( O ) ≥ Z [ j , C ] exp [ − C ∣ h ∣ − o ρ ( 1 ) J ] . (29) \overline{Z}_{I}(O)\ge Z[\mathrm{j},C]\exp[-C|h|-o_\rho(1)J]. \tag*{(29)} Z I ( O ) ≥ Z [ j , C ] exp [ − C ∣ h ∣ − o ρ ( 1 ) J ] . (29) Here the small coefficients tend to zero after all cut and eligibility constants have been fixed. For the first, sample l \mathrm{l} l h \mathrm{h} h V G , G ′ \mathcal V_{G,G'} V G , G ′ V G , G \mathcal V_{G,G} V G , G Z [ l , O ] Z[\mathrm{l},O] Z [ l , O ] h \mathrm{h} h
E exp [ − V l , j − V l , d \ h − V h , L ∪ j ( d \ h ) ∪ l − V h , h ] \mathbb{E}\exp[-\mathcal V_{\mathrm{l},\mathrm{j}}-\mathcal V_{\mathrm{l},\mathrm{d}\backslash\mathrm{h}}-\mathcal V_{\mathrm{h},L\cup\mathrm{j}(\mathrm{d}\backslash\mathrm{h})\cup\mathrm{l}}-\mathcal V_{\mathrm{h},\mathrm{h}}] E exp [ − V l , j − V l , d \ h − V h , L ∪ j ( d \ h ) ∪ l − V h , h ] with the complement frozen in O O O l \mathrm{l} l j \mathrm{j} j C / r C/\sqrt{r} C / r r − 2 r / 10 > R ν r-2r/10>R_\nu r − 2 r /10 > R ν C ( R ) J C(R)J C ( R ) J 2 R + 5 2R+5 2 R + 5 d \ h \mathrm{d}\backslash\mathrm{h} d \ h C ∣ d ∣ C|\mathrm{d}| C ∣ d ∣ C ∣ h ∣ C|\mathrm{h}| C ∣ h ∣ L L L
For the second denominator, sample j j j C C C O O O d d d l l l
V j , l , V d , L ∪ j ∪ l , V d , d . \mathcal{V}_{j,l},\quad\mathcal{V}_{d,L\cup j\cup l},\quad\mathcal{V}_{d,d}. V j , l , V d , L ∪ j ∪ l , V d , d . The first sum costs C r − 1 / 2 J Cr^{-1/2}J C r − 1/2 J O O O j j j C ∣ d ∣ ≤ 2 C ∣ h ∣ C|d|\le2C|h| C ∣ d ∣ ≤ 2 C ∣ h ∣ (29) .
The remaining normalized factors. Using (28) and (29) , drop interaction costs from the numerator weights by positivity except for: internal L L L Φ j C , Φ l C \Phi_j^C,\Phi_l^C Φ j C , Φ l C Φ O \Phi^O Φ O Φ j C ^ \Phi_j^{\widehat{C}} Φ j C
To spell out the remaining integrations, let π G X \pi_G^X π G X c G X = 1 cut R ( X G ) c_G^X=1_{\operatorname{cut}_R(X_G)} c G X = 1 cut R ( X G ) d ν G X = c G X Φ G X d π G X / Z [ G , X ] d\nu_G^X=c_G^X\Phi_G^X d\pi_G^X/Z[G,X] d ν G X = c G X Φ G X d π G X / Z [ G , X ] K B ( I ∪ I ′ ) K_B(I\cup I') K B ( I ∪ I ′ ) L L L
( c l C Φ l C d π l C ) ( c j C ^ Φ j C ^ d π j C ^ ) d ν j C d ν l O d π d C d π d C ^ d π h O . (c_l^C\Phi_l^C\,d\pi_l^C)(c_j^{\widehat{C}}\Phi_j^{\widehat{C}}\,d\pi_j^{\widehat{C}})\,d\nu_j^C\,d\nu_l^O\,d\pi_d^C\,d\pi_d^{\widehat{C}}\,d\pi_h^O. ( c l C Φ l C d π l C ) ( c j C Φ j C d π j C ) d ν j C d ν l O d π d C d π d C d π h O . In particular the extras C j , O l C_j,O_l C j , O l L L L Q Q Q [ ( 1 − ε ) n Q ] − 1 [(1-\varepsilon)n_Q]^{-1} [( 1 − ε ) n Q ] − 1 n Q n_Q n Q
Restoring the baseline interaction weight
We describe the weight restoration in B B B j n ⊆ j j_n\subseteq j j n ⊆ j 2 R + 5 2R+5 2 R + 5 l l l j j j l l l
Fixed data and feasible fibers. Throughout this step, all endpoints, tuples, cell paths and skeletons are fixed. The path integrations divide as follows:
Call the fixed data feasible if some choice of the remaining original paths satisfies both nonzero selection weights, their specified adaptations, and the cuts. Call such a choice a witness . We only need an upper comparison on feasible data: otherwise the original inner integral vanishes. The probability laws of the extra original paths depend only on the fixed endpoints and L L L C ^ j n \widehat{C}_{j_n} C j n
Restoring the missing interactions. First consider interaction weights alone. The bound needed for replacing the C ^ j n \widehat{C}_{j_n} C j n
Φ l B ∫ c j n B Φ j B d π j n B ≤ e C ∣ h ∣ + o ρ ( 1 ) J ∫ c j n B c d B exp [ − ∑ { i , i ′ } ⊆ L V i i ′ ( B ) ] d π j n B d π d B . (30) \Phi_l^B\int c_{j_n}^B\Phi_j^B\,d\pi_{j_n}^B
\le e^{C|h|+o_{\rho}(1)J}\int c_{j_n}^B c_d^B\exp\left[-\sum_{\{i,i'\}\subseteq L}\mathcal{V}_{ii'}(B)\right]d\pi_{j_n}^B d\pi_d^B. \tag*{(30)} Φ l B ∫ c j n B Φ j B d π j n B ≤ e C ∣ h ∣ + o ρ ( 1 ) J ∫ c j n B c d B exp − { i , i ′ } ⊆ L ∑ V i i ′ ( B ) d π j n B d π d B . (30) where the sum is over unordered pairs of distinct slots indicated.
Indeed, normalize c j n B Φ j n B d π j n B c_{jn}^{B}\Phi_{jn}^{B}\,d\pi_{jn}^{B} c j n B Φ j n B d π j n B B j \ j n B_{j\backslash jn} B j \ j n d d d L L L j j j d d d
The j n j_{n} j n l l l o ρ ( 1 ) J o_{\rho}(1)J o ρ ( 1 ) J j j j l l l d d d C ∣ d ∣ C|d| C ∣ d ∣ (16) . Jensen and the cut probabilities prove (29a). Its endpoint and frozen-path hypotheses follow from original feasibility, independently of the reintegrated paths.
This inequality concerns interaction weights alone. To apply it to the original integral, we must still account for the far selection factors and the path-adaptation constraints. The next step proves that their needed properties hold for every baseline cut replacement.
Tracking the constraints in this comparison
The original choices depend on which cells are usable and which slots are eligible. We now compare those tests between any original witness and every replacement of B j n , B d B_{jn},B_d B j n , B d
In B B B Y Y Y Y e Y_e Y e e e e S 0 ( B ) S_0(B) S 0 ( B ) S 1 ( B ) S_1(B) S 1 ( B ) C 0 − Δ A C_0-\Delta_A C 0 − Δ A 1 − ε / 2 1-\varepsilon/2 1 − ε /2 S 0 ( B ) ⊆ S 1 ( B ) S_0(B)\subseteq S_1(B) S 0 ( B ) ⊆ S 1 ( B ) S ( C ) , S ( C ^ ) S(C),S(\widehat{C}) S ( C ) , S ( C )
Table 1 (PDF p. 33).
Counts outside the displayed interval also cause failure. The symmetric defective test uses actual inner endpoints, whereas S 0 ( B ) , S 1 ( B ) S_0(B),S_1(B) S 0 ( B ) , S 1 ( B ) C C C C ^ \widehat{C} C Y Y Y
Lemma 10.2 (Stability under every cut replacement ). Fix feasible endpoint and path data as in the preceding subsection. For every replacement of B j n , B d B_{jn},B_d B j n , B d C , C ^ C,\widehat{C} C , C
S ( C ) , S ( C ^ ) ⊆ S 1 ( B ) , S ( C ) = S 0 ( B ) pointwise outside distance R ∗ of P , S ( C ^ ) = S 0 ( B ) pointwise outside distance R ∗ of P ′ . \begin{equation}
\begin{gathered}
S(C),S(\widehat C)\ \subseteq S_1(B),\\
S(C)=S_0(B)\ \text{pointwise outside distance }R_*\text{ of }P,\\
S(\widehat C)=S_0(B)\ \text{pointwise outside distance }R_*\text{ of }P'.
\end{gathered}
\end{equation} S ( C ) , S ( C ) ⊆ S 1 ( B ) , S ( C ) = S 0 ( B ) pointwise outside distance R ∗ of P , S ( C ) = S 0 ( B ) pointwise outside distance R ∗ of P ′ . In the far cells specified by the last two lines, all individual eligibility indicators and eligible counts agree as well.
Proof. The state C C C B B B I I I C ^ \widehat{C} C B B B I , d , j n I,d,j_n I , d , j n B B B a a a Δ A \Delta_A Δ A ( R + R A ) s (R+R_A)s ( R + R A ) s 2 ( 2 R + 2 R A + 4 ) 3 + 1 2(2R+2R_A+4)^3+1 2 ( 2 R + 2 R A + 4 ) 3 + 1 ∫ 0 δ / 2 w ≤ 3 \int_0^{\delta/2}w\le3 ∫ 0 δ /2 w ≤ 3 B B B S 1 ( B ) S_1(B) S 1 ( B ) ( 1 − ε / 2 ) n Q − 2 ≥ ( 1 − ε ) n Q (1-\varepsilon/2)n_Q-2\ge(1-\varepsilon)n_Q ( 1 − ε /2 ) n Q − 2 ≥ ( 1 − ε ) n Q m m m (30) by contraposition.
For exact equality, changed paths comparing with C C C P P P C ^ \widehat{C} C 2 R + 5 2R+5 2 R + 5 P ′ P' P ′ R ∗ R_* R ∗ R ∗ R_* R ∗
Lists containing every feasible pair. Define an explicit path predicate given B B B γ \gamma γ γ ′ \gamma' γ ′ S 1 ( B ) S_1(B) S 1 ( B ) K b = 2 R ∗ + 10 K_b=2R_*+10 K b = 2 R ∗ + 10 R ∗ + 2 R_*+2 R ∗ + 2 b t o t b_{\mathrm{tot}} b tot P P P γ \gamma γ S 0 ( B ) S_0(B) S 0 ( B ) P ′ P' P ′ γ ′ \gamma' γ ′ G G G
The lists depend only on B , γ , γ ′ B,\gamma,\gamma' B , γ , γ ′ G G G
Lemma 10.3 (Path lists and their encounter bound). For every original witness and every baseline cut replacement in Lemma 10.2, the witness paths P , P ′ P,P' P , P ′ G G G B B B 4 b t o t 4^{b_{\mathrm{tot}}} 4 b tot
J ≤ J 0 + C ( R ∗ ) b t o t . (31) J\le J_0+C(R_*)b_{\mathrm{tot}}. \tag*{(31)} J ≤ J 0 + C ( R ∗ ) b tot . (31) Here J 0 J_0 J 0
Proof. First fix a witness and any cut replacement. A nonbridge component of S 1 ( B ) S_1(B) S 1 ( B ) γ \gamma γ R ∗ R_* R ∗ P ′ P' P ′ P ′ P' P ′ K b K_b K b γ ′ \gamma' γ ′ C ^ \widehat{C} C S 0 ( B ) S_0(B) S 0 ( B )
On bridge components, choose the guessed subsets to be the actual bad cells of C ^ \widehat{C} C K b K_b K b S 1 ( B ) S_1(B) S 1 ( B ) γ \gamma γ γ \gamma γ P P P C , γ ′ C,\gamma' C , γ ′ P ′ P' P ′
There are at most 2 b t o t 2^{b_{\mathrm{tot}}} 2 b tot 4 b t o t 4^{b_{\mathrm{tot}}} 4 b tot R ∗ R_* R ∗ J 0 J_0 J 0 K b K_b K b 12 b t o t 12b_{\mathrm{tot}} 12 b tot C ( R ∗ ) b t o t C(R_*)b_{\mathrm{tot}} C ( R ∗ ) b tot J 0 J_0 J 0 S 1 ( B ) S_1(B) S 1 ( B )
The common far-choice product. For a slot, its choice factor in B B B S 0 ( B ) S_0(B) S 0 ( B ) H F ( B ) H_F(B) H F ( B ) P P P C ^ \widehat{C} C P ′ P' P ′ C C C H ∗ H_* H ∗ any witness and every replacement, and the replacement cuts are nonempty. This is the selection property needed to apply (29a).
Explicitly, at those fixed variables let D \mathcal{D} D H o l d H_{\mathrm{old}} H old
d P = d ν j C d ν l O d π d C d π d C ^ d π h O d\mathcal{P}=d\nu_j^C\,d\nu_l^O\,d\pi_d^C\,d\pi_d^{\widehat{C}}\,d\pi_h^O d P = d ν j C d ν l O d π d C d π d C d π h O on the extras (its laws do not depend on the integrated C ^ j n \widehat{C}_{jn} C j n j , l j,l j , l
Φ l B ∫ C ^ j n C Φ ^ j C 1 D H o l d d π j n C ^ d P ≤ H ∗ Φ l B ∫ C j n B Φ j B d π j n B ≤ e C ∣ h ∣ + o ρ ( 1 ) J ∫ C j n B C d B exp [ − ∑ { i , i ′ } ⊈ L V i i ′ ( B ) ] 1 G H F ( B ) d π j n B d π d B . \begin{aligned}
\Phi^B_l\int\widehat{C}_{jn}^{\,C}\widehat{\Phi}_j^{\,C}\mathbf{1}_{\mathcal{D}}H_{\mathrm{old}}\,d\pi_{jn}^{\widehat{C}}\,d\mathcal{P} \\
&\le H_*\Phi_l^B\int C_{jn}^B\Phi_j^B\,d\pi_{jn}^B \\
&\le e^{C|h|+o_{\rho}(1)J}\int C_{jn}^B C_{d}^{B}\exp\left[-\sum_{\{i,i'\}\not\subseteq L}V_{ii'}(B)\right]\mathbf{1}_{G}H_F(B)\,d\pi_{jn}^B d\pi_d^B.
\end{aligned} Φ l B ∫ C j n C Φ j C 1 D H old d π j n C d P ≤ H ∗ Φ l B ∫ C j n B Φ j B d π j n B ≤ e C ∣ h ∣ + o ρ ( 1 ) J ∫ C j n B C d B exp − { i , i ′ } ⊆ L ∑ V i i ′ ( B ) 1 G H F ( B ) d π j n B d π d B . The first inequality removes the original constraints only after factoring out their common far product. The second uses (29a) and Lemma 10.3: 1 G H F ( B ) = H ∗ \mathbf{1}_{G}H_F(B)=H_* 1 G H F ( B ) = H ∗
For precision, denote the original conditional path integral by
I I , I ′ = ∫ W ( C ) h γ ′ ( C , I ′ ) 1 c u t R ( O I ′ ) W I ′ O Z I ′ ( O ) ⋅ 1 c u t R ( O I ) Z I ( O ) W I C ^ h γ ( C ^ , I ) . \begin{aligned}
\mathcal{I}_{I,I'}=\int W(C)h_{\gamma'}(C,I')\frac{\mathbf{1}_{\mathrm{cut}_R}(O_{I'})W_{I'}^O}{Z_{I'}(O)} \\
&\cdot\frac{\mathbf{1}_{\mathrm{cut}_R}(O_I)}{Z_I(O)}W_I^{\widehat{C}}h_{\gamma}(\widehat{C},I).
\end{aligned} I I , I ′ = ∫ W ( C ) h γ ′ ( C , I ′ ) Z I ′ ( O ) 1 cut R ( O I ′ ) W I ′ O ⋅ Z I ( O ) 1 cut R ( O I ) W I C h γ ( C , I ) . The integrations are over the original closed paths, proposed paths on I ′ I' I ′ I I I 1 o c c \mathbf{1}_{\mathrm{occ}} 1 occ [ b − m , b + m ] [b-m,b+m] [ b − m , b + m ]
Lemma 10.4 (Integrated cross comparison). Fix two simple star cell paths from a a a b b b Y Y Y Y , S C , x , z Y,S^{C},x,z Y , S C , x , z (27) . With B B B G G G H F ( B ) H_{\mathrm{F}}(B) H F ( B )
I I , I ′ ≤ e C ∣ h ∣ + σ ρ J 1 o c c ∏ near occurrences in cell Q [ ( 1 − ε ) n Q ( Y ) ] − 1 ⋅ ∫ W ( S B , E B ; d ω ) 1 G H F ( B ) . (32) \mathcal{I}_{I,I'} \le e^{C|h|+\sigma_{\rho}}J_{1\mathrm{occ}}\prod_{\substack{\text{near occurrences}\\\text{in cell }Q}}\left[(1-\varepsilon)n_Q(Y)\right]^{-1}
\cdot\int W(S^{B},E^{B};\mathrm{d}\omega)\,\mathbf{1}_{G H_{\mathrm{F}}(B)}. \tag*{(32)} I I , I ′ ≤ e C ∣ h ∣ + σ ρ J 1 occ near occurrences in cell Q ∏ [ ( 1 − ε ) n Q ( Y ) ] − 1 ⋅ ∫ W ( S B , E B ; d ω ) 1 G H F ( B ) . (32) Here C < ∞ C<\infty C < ∞ σ ρ ≥ 0 \sigma_{\rho}\ge0 σ ρ ≥ 0 ρ ↓ 0 \rho\downarrow0 ρ ↓ 0
Proof. The preceding inner-integral comparison proves the displayed bound on every feasible fiber. On an infeasible fiber the original contribution is zero. We explain the remaining bookkeeping to specify precisely how the full integral is recovered. Keep L L L B l B_l B l B j \ j n B_{j\backslash j_n} B j \ j n j n , d j_n,d j n , d (28) , then integrate the fixed paths. Finally drop the remaining cut and nearest-lift restrictions from the baseline integral by positivity, obtaining (32). Its predicates are measurable functions of the baseline paths, specified by finite cell algorithms. This use of the conditional comparison is valid pointwise at the fixed data: a positive original inner integral has a witness, and otherwise the required upper bound is automatic.
The conditional comparison is now established. To average its path-list cost, we need the defect bound under the reverse insertion. Under μ a b \mu_{ab} μ ab e e e U \mathcal{U} U
μ a b ( U ⊆ S 1 ( B ) ) ≤ C M − 2 q ∣ U ∣ , q = q c l 1 / 4 , (33) \mu_{ab}(\mathcal{U}\subseteq S_1(B))\le CM^{-2q|U|},\qquad q=q_{\mathrm{cl}}^{1/4}, \tag*{(33)} μ ab ( U ⊆ S 1 ( B )) ≤ C M − 2 q ∣ U ∣ , q = q cl 1/4 , (33) including the empty set bound. To compare to the symmetric defective test replace the exceptional outer endpoint by its actual inner end for counting and isolation. Counts change by at most one; only a bounded number of other end-isolation flags can change: those isolated before but not after are mutually separated by more than r r r r r r n Q ( Y ) n_Q(Y) n Q ( Y ) ( ε / 2 ) n Q ( Y ) − C (\varepsilon/2)n_Q(Y)-C ( ε /2 ) n Q ( Y ) − C ε / 4 \varepsilon/4 ε /4 m m m (22) and (23) .
Summing the comparison
The conditional comparison is now complete, including its selection predicates. We sum first over labels at fixed baseline paths, then over the finite lists of possible cell paths. The remaining environment average will be controlled by the insertion defect bound and the two skeleton moments.
Summing label choices. Sum (32) over candidate cell-path pairs and their tuples; this candidate range depends on a a a b b b Y Y Y S C S^{C} S C S B S^{B} S B e e e I I I b b b e e e S 0 S_0 S 0 S 1 S_1 S 1
Fix a listed pair satisfying the occupancy bound. A far cell occurs on only one path, and summing its choice factor from H F ( B ) H_F(B) H F ( B ) ( 1 − ε ) − 1 (1-\varepsilon)^{-1} ( 1 − ε ) − 1
Table 1 (PDF p. 37).
For example, at an ordinary common cell, the n Q n_Q n Q e C e^C e C n Q ( n Q − 1 ) n_Q(n_Q-1) n Q ( n Q − 1 ) n Q 2 n_Q^2 n Q 2 a a a b b b n Q ≥ b − m n_Q\ge b-m n Q ≥ b − m e σ ρ J e^{\sigma_\rho J} e σ ρ J
C m 2 e κ 0 J 0 , κ 0 = 2 ε + o ρ ( 1 ) . \frac{C}{m^2}e^{\kappa_0J_0},\qquad\kappa_0=2\varepsilon+o_\rho(1). m 2 C e κ 0 J 0 , κ 0 = 2 ε + o ρ ( 1 ) . Lemma 10.3 now sums the cell-path pairs, giving C m − 2 exp ( κ 0 J 0 + C ′ b t o t ) Cm^{-2}\exp(\kappa_0J_0+C'b_{\mathrm{tot}}) C m − 2 exp ( κ 0 J 0 + C ′ b tot ) C ′ C' C ′
Recovering the reverse insertion. There is no remaining dependence on the dummy-for-B B B z z z b b b 1 / M 1/M 1/ M m / ( m 2 M ) = 1 / N m/(m^2M)=1/N m / ( m 2 M ) = 1/ N e e e Y e Y_e Y e b b b x x x a a a d x / V dx/V d x / V μ a b \mu_{ab} μ ab b b b a a a
∫ f 2 d μ ≤ C ∑ a , b v a l i d E γ , γ ′ [ e κ 0 J 0 ∫ e C ′ b t o t d μ a b ] (34) \int f^2\,d\mu\le C\sum_{a,b\ \mathrm{valid}}\mathbb{E}_{\gamma,\gamma'}\left[e^{\kappa_0J_0}\int e^{C'b_{\mathrm{tot}}}\,d\mu_{ab}\right] \tag*{(34)} ∫ f 2 d μ ≤ C a , b valid ∑ E γ , γ ′ [ e κ 0 J 0 ∫ e C ′ b tot d μ ab ] (34) The exponential moment of rare components. For fixed skeletons, we claim
∫ e C ′ b t o t d μ a b ≤ C M 2 exp [ ϑ ( q ) S K b ] , ϑ ( q ) ⟶ 0 ( q ↓ 0 ) . \int e^{C'b_{\mathrm{tot}}}\,d\mu_{ab}\le\frac{C}{M^2}\exp\left[\vartheta(q)S_{K_b}\right],\qquad\vartheta(q)\longrightarrow0\quad(q\downarrow0). ∫ e C ′ b tot d μ ab ≤ M 2 C exp [ ϑ ( q ) S K b ] , ϑ ( q ) ⟶ 0 ( q ↓ 0 ) . where S K b S_{K_b} S K b
Expand the product over realized bridge components of S 1 ( B ) S_1(B) S 1 ( B ) 1 + ( e C ′ ∣ T ∣ − 1 ) 1+(e^{C'|T|}-1) 1 + ( e C ′ ∣ T ∣ − 1 ) T T T K b K_b K b R ∗ + 2 R_*+2 R ∗ + 2 S 1 ( B ) S_1(B) S 1 ( B )
C M − 2 exp ( ∑ T ( e C ′ q ∣ T ∣ ) ) CM^{-2}\exp\left(\sum_T(e^{C' q}|T|)\right) C M − 2 exp ( T ∑ ( e C ′ q ∣ T ∣ ) ) (the sum in the exponent is over individual qualifying sets).
Root their counting near vertex occurrences z ∈ γ z\in\gamma z ∈ γ C ( K b ) j C(K_b)^j C ( K b ) j j j j d ∞ ( z , γ ′ ) ≤ 2 ( R ∗ + 2 ) + K b j d_{\infty}(z,\gamma') \le2(R_*+2)+K_bj d ∞ ( z , γ ′ ) ≤ 2 ( R ∗ + 2 ) + K b j
o q ( 1 ) ∑ z ∈ γ e − d ∞ ( z , γ ′ ) / K b o_q(1)\sum_{z\in\gamma} e^{-d_{\infty}(z,\gamma')/K_b} o q ( 1 ) z ∈ γ ∑ e − d ∞ ( z , γ ′ ) / K b by a geometric series: per root sum ( C ( K b ) e C ′ j ) (C(K_b)e^{C'j}) ( C ( K b ) e C ′ j ) d ∞ ( z , γ ′ ) / K b ≤ j + 2 ( R ∗ + 2 ) / K b d_{\infty}(z,\gamma')/K_b \le j+2(R_*+2)/K_b d ∞ ( z , γ ′ ) / K b ≤ j + 2 ( R ∗ + 2 ) / K b
The geometric-series bound proves the claim with a coefficient ϑ ( q ) → 0 \vartheta(q)\to0 ϑ ( q ) → 0 2 κ 0 ≤ α 2\kappa_0\le\alpha 2 κ 0 ≤ α 2 ϑ ( q ) ≤ ξ ( K b ) 2\vartheta(q)\le\xi(K_b) 2 ϑ ( q ) ≤ ξ ( K b ) (24) –(25) then give, for each valid pair,
E γ , γ ′ [ e κ 0 J 0 ∫ e C ′ b t o t d μ a b ] ≤ C M 2 ( E e 2 κ 0 J 0 ) 1 / 2 ( E e 2 ϑ ( q ) S K b ) 1 / 2 ≤ C M 2 . \mathbb{E}_{\gamma,\gamma'}\left[e^{\kappa_0J_0}\int e^{C' b_{\mathrm{tot}}}\,d\mu_{ab}\right]\le\frac{C}{M^2}\left(\mathbb{E}e^{2\kappa_0J_0}\right)^{1/2}\left(\mathbb{E}e^{2\vartheta(q)SK_b}\right)^{1/2}\le\frac{C}{M^2}. E γ , γ ′ [ e κ 0 J 0 ∫ e C ′ b tot d μ ab ] ≤ M 2 C ( E e 2 κ 0 J 0 ) 1/2 ( E e 2 ϑ ( q ) S K b ) 1/2 ≤ M 2 C . There are at most M 2 M^2 M 2 (34) proves Proposition 10.1.
Choice of constants and the limits
We assemble the finite-volume estimate and make its uniformity explicit. The cutoffs are fixed from the potential before the density is reduced; the volume threshold is chosen afterward at each fixed density. This order also permits the zero-temperature limit at fixed volume.
Proof of Theorem 1.1. The change of measure in Section 9 has mass at least υ / 2 \upsilon/2 υ /2 ∫ f 2 d μ ≤ C υ \int f^2\,d\mu\le C_\upsilon ∫ f 2 d μ ≤ C υ C υ C_\upsilon C υ ρ , L , N , T \rho,L,N,T ρ , L , N , T μ \mu μ q 0 q_0 q 0
q 0 ≥ υ 2 4 C υ = : c ∗ ( υ ) > 0. q_0\ge\frac{\upsilon^2}{4C_\upsilon}=:c_*(\upsilon)>0. q 0 ≥ 4 C υ υ 2 =: c ∗ ( υ ) > 0. We specify why all choices can be made without introducing density or temperature dependence into this constant.
Choose the parameters in this order.
Fix the localization multipliers in the upper density estimates, reserving the positive energy fraction used for isolation in (13). Fix the small particle-addition fraction η \eta η K K K b − , b + b_-,b_+ b − , b +
Choose the macro cutoff D D D ε = α ( D ) / 20 \varepsilon=\alpha(D)/20 ε = α ( D ) /20 α ( D ) \alpha(D) α ( D ) D D D
Choose ζ > 0 \zeta>0 ζ > 0 C 0 C_0 C 0 υ \upsilon υ
Decrease ρ ∗ ( υ ) \rho_*(\upsilon) ρ ∗ ( υ ) r = ζ ρ − 1 / 3 r=\zeta\rho^{-1/3} r = ζ ρ − 1/3 m ≍ ρ − 1 / 2 m\asymp\rho^{-1/2} m ≍ ρ − 1/2 λ 0 m = r ρ m ≍ ρ − 1 / 3 \lambda_0m=r\sqrt{\rho}\,m\asymp\rho^{-1/3} λ 0 m = r ρ m ≍ ρ − 1/3 r − 1 / 2 r^{-1/2} r − 1/2 m − 1 m^{-1} m − 1 2 κ 0 ≤ α 2\kappa_0\le\alpha 2 κ 0 ≤ α
(5) At each ρ < ρ ∗ ( v ) \rho< \rho_{*}(v) ρ < ρ ∗ ( v ) L 0 ( ρ , v ) L_{0}(\rho,v) L 0 ( ρ , v ) 0 < T ≤ ρ 2 0 < T \le\rho^{2} 0 < T ≤ ρ 2
This gives the claimed finite-volume lower bound at positive temperature.
At fixed such N , L N,L N , L β → ∞ \beta\to\infty β → ∞ N − 1 ∑ i ( ∣ u 0 ⟩ ⟨ u 0 ∣ ) i N^{-1}\sum_i (|u_0\rangle\langle u_0|)_i N − 1 ∑ i ( ∣ u 0 ⟩ ⟨ u 0 ∣ ) i β ≥ ρ − 2 \beta\ge\rho^{-2} β ≥ ρ − 2 T = 0 T=0 T = 0
Finally fix ρ \rho ρ T T T L → ∞ L \to\infty L → ∞ N / L 3 → ρ N/L^3 \to\rho N / L 3 → ρ c ∗ ( v ) c_{*}(v) c ∗ ( v )
