Introduction
Loop-erased random walk (LERW) is the random simple path obtained by chronologically erasing the loops of a simple random walk. Introduced by Lawler [25 ] as a model of self-avoiding walk, it is one of the most tractable non-Gaussian critical models of statistical physics. In dimensions four and higher, its scaling limit is Brownian motion; see [26 ] and the references therein. In dimension three, Kozma established the existence of the scaling limit in [24 ] (see also [10 , 12 ] for a new proof and characterization of the scaling limit). Many aspects of this limit have since been understood, including its Hausdorff dimension, geometric properties and convergence in natural parametrization; see [50 , 51 , 35 , 36 , 37 , 21 ] . In dimension two, Lawler, Schramm and Werner [30 ] proved that LERW has a conformally invariant scaling limit, described by Schramm–Loewner evolution with parameter 2, abbreviated as SLE2. Subsequently, Yadin and Yehudayoff [52 ] established universality, and Lawler and Viklund [32 , 33 ] proved convergence in natural parametrization in the chordal and radial cases.
As for other critical lattice models, it is natural to study near-critical versions of LERW, obtained by perturbing it away from criticality. As part of a program to study near-critical planar lattice models, Makarov and Smirnov [40 ] proposed massive LERW (mLERW). Given a mesh size δ > 0 \delta> 0 δ > 0 m > 0 m > 0 m > 0 m 2 δ 2 m^{2}\delta^{2} m 2 δ 2 [40 Theorem 2.1] and Chelkak–Wan [15 ] for a detailed proof. Camia [13 ] introduced massive loop soups and explained their connection with mLERW.
Near-critical behavior has also been studied for other planar lattice models. Garban, Pete and Schramm [19 ] constructed near-critical percolation from pivotal points. For Ising, Park [45 ] studied thermally perturbed spin correlations, and Camia, Garban and Newman [14 ] determined the magnetization exponent in a small external field. More recently, Papon [44 ] proved convergence of the magnetically perturbed interface to a law absolutely continuous with respect to S L E 3 \mathrm{SLE}_3 SLE 3 [22 ] proved quenched convergence of the interface to an almost surely absolutely continuous tilt of S L E 3 \mathrm{SLE}_3 SLE 3 C L E 3 \mathrm{CLE}_3 CLE 3 [18 ] proved local singularity from the critical limit in the FK quad-crossing topology. Mahfouf [39 ] studied random-bond perturbations through deformations of s s s [49 ] and Papon [43 ] studied the massive harmonic explorer and its convergence to massive S L E 4 \mathrm{SLE}_4 SLE 4 [5 ] related near-critical dimers to massive S L E 2 \mathrm{SLE}_2 SLE 2 [47 ] extended the convergence of the associated massive LERW and tree models to isoradial graphs. Berestycki, Mason and Rey [6 ] identified the near-critical dimer height limit with an electromagnetically tilted sine-Gordon field. A related bosonization identity for near-critical Ising correlations was proved by Park, Virtanen and Webb [46 ] .
In this paper we introduce a different near-critical perturbation, boosted LERW (bLERW): before the loops are erased, longer random-walk paths are rewarded rather than penalized. Our main interest is the geometry of bLERW as the boost increases. This is also motivated by a question of Makarov and Smirnov in [40 ] , which we briefly recall.
The planar O ( N ) O(N) O ( N ) S L E κ \mathrm{SLE}_{\kappa} SLE κ S L E κ ~ \mathrm{SLE}_{\widetilde{\kappa}} SLE κ
1 κ + 1 κ ~ = 1 2 , κ ∈ [ 8 / 3 , 4 ] , κ ~ ∈ [ 4 , 8 ] . \frac{1}{\kappa}+\frac{1}{\widetilde{\kappa}}=\frac{1}{2},\qquad\kappa\in[8/3,4],\quad\widetilde{\kappa}\in[4,8]. κ 1 + κ 1 = 2 1 , κ ∈ [ 8/3 , 4 ] , κ ∈ [ 4 , 8 ] . For self-avoiding walk, the pair is κ = 8 / 3 \kappa=8/3 κ = 8/3 κ ~ = 8 \widetilde{\kappa}=8 κ = 8 [16 ] . In [40 ] , Question 4.12, Makarov and Smirnov ask whether near-critical models provide transition states between these regimes. Although LERW, with κ = 2 \kappa=2 κ = 2
The model
We first define the boosted walk in the range where it can be normalized, and then introduce the random-walk loop measure used to extend the model beyond this range.
Let d ∈ { 2 , 3 } d\in\{2,3\} d ∈ { 2 , 3 } D = B ( 0 , 1 ) ⊂ R d D=B(0,1)\subset\mathbb{R}^d D = B ( 0 , 1 ) ⊂ R d D δ = D ∩ δ Z d D_\delta=D\cap\delta\mathbb{Z}^d D δ = D ∩ δ Z d S S S δ Z d \delta\mathbb{Z}^d δ Z d 0 0 0 T δ = inf { j ≥ 0 : S j ∉ D δ } T_\delta=\inf\{j\ge0:S_j\notin D_\delta\} T δ = inf { j ≥ 0 : S j ∈ / D δ } γ δ = LE ( S [ 0 , T δ ] ) \gamma^\delta=\operatorname{LE}(S[0,T_\delta]) γ δ = LE ( S [ 0 , T δ ]) η δ \eta^\delta η δ K δ K^\delta K δ P δ \mathbb{P}_\delta P δ η δ \eta^\delta η δ P \mathbb{P} P K K K
For m ≥ 0 m\ge0 m ≥ 0 a δ = 1 + m 2 δ 2 a_\delta=1+m^2\delta^2 a δ = 1 + m 2 δ 2 ω \omega ω ( a δ / ( 2 d ) ) ∣ ω ∣ (a_\delta/(2d))^{|\omega|} ( a δ / ( 2 d ) ) ∣ ω ∣
Z δ , m R W = ∑ ω : 0 → ∂ D δ ( a δ 2 d ) ∣ ω ∣ = E 0 [ a δ T δ ] . (1) Z^{\mathrm{RW}}_{\delta,m}=\sum_{\omega:0\to\partial D_\delta}\left(\frac{a_\delta}{2d}\right)^{|\omega|}=\mathbb{E}^{0}\left[a_\delta^{T_\delta}\right]. \tag*{(1)} Z δ , m RW = ω : 0 → ∂ D δ ∑ ( 2 d a δ ) ∣ ω ∣ = E 0 [ a δ T δ ] . (1) where the sum is over walks stopped at their first exit. When this sum is finite, bLERW is the chronological loop-erasure of the walk sampled with these normalized weights. At m = 0 m=0 m = 0 δ 2 \delta^2 δ 2 d d d
L = 1 2 d Δ , Δ f = ∑ j = 1 d ∂ j 2 f . (2) L = \frac{1}{2d}\Delta,\qquad\Delta f = \sum_{j=1}^{d}\partial_j^2 f. \tag*{(2)} L = 2 d 1 Δ , Δ f = j = 1 ∑ d ∂ j 2 f . (2) For an open set U U U A U = − 1 2 d Δ U A_U=-\frac{1}{2d}\Delta_U A U = − 2 d 1 Δ U λ 1 ( U ) \lambda_1(U) λ 1 ( U ) § 2.4 .
For a finite lattice domain U U U p δ , U p_{\delta,U} p δ , U
p δ , U ( x , y ) = 1 2 d 1 { ∣ x − y ∣ = δ } , x , y ∈ U . (3) p_{\delta,U}(x,y)=\frac{1}{2d}\mathbf{1}\{|x-y|=\delta\},\qquad x,y\in U. \tag*{(3)} p δ , U ( x , y ) = 2 d 1 1 { ∣ x − y ∣ = δ } , x , y ∈ U . (3) Write ρ ( p δ , U ) \rho(p_{\delta,U}) ρ ( p δ , U ) a δ ρ ( p δ , D δ ) < 1 a_\delta\rho(p_{\delta,D_\delta})<1 a δ ρ ( p δ , D δ ) < 1 E e m 2 T D \mathbb{E}e^{m^2T_D} E e m 2 T D m 2 < λ 1 ( D ) m^2<\lambda_1(D) m 2 < λ 1 ( D ) T D T_D T D D D D
Random-walk loops enter because the loops erased from the walk contribute to the weight of LERW. Their total weight is expressed through the return traces
Tr ( p δ , U n ) = ∑ x ∈ U P x ( S n = x , n < T U ) . (4) \operatorname{Tr}(p_{\delta,U}^{n})=\sum_{x\in U}\mathbb{P}^{x}(S_n=x,\ n<T_U). \tag*{(4)} Tr ( p δ , U n ) = x ∈ U ∑ P x ( S n = x , n < T U ) . (4) The diagonal entry at x x x n n n x x x x x x U U U μ δ , U \mu_{\delta,U} μ δ , U n n n n n n n n n Tr ( p δ , U n ) / n \operatorname{Tr}(p_{\delta,U}^{n})/n Tr ( p δ , U n ) / n
A loop of n n n t ℓ = n δ 2 t_\ell=n\delta^2 t ℓ = n δ 2 a δ n a_\delta^n a δ n
M δ , m ( U ) : = ∫ ( a δ ∣ ℓ ∣ − 1 ) μ δ , U ( d ℓ ) = ∑ n ≥ 1 a δ n − 1 n Tr ( p δ , U n ) ∈ [ 0 , ∞ ] . (5) \mathcal{M}_{\delta,m}(U):=\int(a_\delta^{|\ell|}-1)\,\mu_{\delta,U}(\mathrm{d}\ell)=\sum_{n\geq1}\frac{a_\delta^n-1}{n}\operatorname{Tr}(p_{\delta,U}^{n})\in[0,\infty]. \tag*{(5)} M δ , m ( U ) := ∫ ( a δ ∣ ℓ ∣ − 1 ) μ δ , U ( d ℓ ) = n ≥ 1 ∑ n a δ n − 1 Tr ( p δ , U n ) ∈ [ 0 , ∞ ] . (5) Here ∣ ℓ ∣ |\ell| ∣ ℓ ∣ U U U K K K U ∖ K U\setminus K U ∖ K
Renormalization and the continuum weight
First suppose a δ ρ ( p δ , D δ ) < 1 a_\delta\rho(p_{\delta,D_\delta})<1 a δ ρ ( p δ , D δ ) < 1 γ \gamma γ K K K
w δ , m ( γ ) : = E [ a δ δ 0 ∣ γ δ ∣ ∣ γ δ = γ ] = a δ ∣ γ ∣ exp { M δ , m ( D δ ) − M δ , m ( D δ ∖ K ) } . (6) w_{\delta,m}(\gamma):=\mathbb{E}\left[a_\delta^{\delta^0|\gamma^\delta|}\mid\gamma^\delta=\gamma\right]=a_\delta^{|\gamma|}\exp\{\mathcal{M}_{\delta,m}(D_\delta)-\mathcal{M}_{\delta,m}(D_\delta\setminus K)\}. \tag*{(6)} w δ , m ( γ ) := E [ a δ δ 0 ∣ γ δ ∣ ∣ γ δ = γ ] = a δ ∣ γ ∣ exp { M δ , m ( D δ ) − M δ , m ( D δ ∖ K )} . (6) The exponent is the excess mass of loops intersecting K K K ∣ γ ∣ |\gamma| ∣ γ ∣ Proposition 3.1 .
The factor exp { M δ , m ( D δ ) } \exp\{\mathcal{M}_{\delta,m}(D_\delta)\} exp { M δ , m ( D δ )} γ \gamma γ (6) , makes sense even when the loop mass in D δ D_\delta D δ
Definition 1.1 (Extended discrete model). For a simple first-exit path γ \gamma γ K K K
W δ , m ( γ ) = a δ ∣ γ ∣ exp { − M δ , m ( D δ ∖ K ) } , Z δ , m = E P δ W δ , m ( γ δ ) . W_{\delta,m}(\gamma)=a_\delta^{|\gamma|}\exp\{-\mathcal{M}_{\delta,m}(D_\delta\setminus K)\},\qquad Z_{\delta,m}=\mathbb{E}_{\mathbb{P}_\delta}W_{\delta,m}(\gamma^\delta). W δ , m ( γ ) = a δ ∣ γ ∣ exp { − M δ , m ( D δ ∖ K )} , Z δ , m = E P δ W δ , m ( γ δ ) . Use exp ( − ∞ ) = 0 \exp(-\infty)=0 exp ( − ∞ ) = 0 Z δ , m > 0 Z_{\delta,m}>0 Z δ , m > 0 W δ , m / Z δ , m W_{\delta,m}/Z_{\delta,m} W δ , m / Z δ , m P δ ( m ) \mathbb{P}_{\delta}^{(m)} P δ ( m )
Since there are finitely many simple paths, Z δ , m < ∞ Z_{\delta,m}<\infty Z δ , m < ∞ Z δ , m Z_{\delta,m} Z δ , m δ \delta δ Theorem 1.2 .
This is the first change of normalization: it extends the discrete loop-erased model beyond the range in which the walk itself can be normalized. However, it does not yet produce an expression with a nontrivial continuum limit.
Indeed, formally replacing random-walk loop measure in M δ , m ( D δ ∖ K ) \mathcal{M}_{\delta,m}(D_\delta\setminus K) M δ , m ( D δ ∖ K )
∫ 0 ∞ ( e m 2 t − 1 ) H D ∖ K ( t ) d t t , H U ( t ) : = ∫ U p U ( t , x , x ) d x , \int_{0}^{\infty}(e^{m^{2}t}-1)H_{D\setminus K}(t)\frac{\mathrm{d}t}{t},\qquad H_U(t):=\int_U p_U(t,x,x)\,\mathrm{d}x, ∫ 0 ∞ ( e m 2 t − 1 ) H D ∖ K ( t ) t d t , H U ( t ) := ∫ U p U ( t , x , x ) d x , where p U p_U p U (2) . Thus H U H_U H U (4) . For the classical limiting trace, D ∖ K D\setminus K D ∖ K D D D
( e m 2 t − 1 ) H D ∖ K ( t ) d t t ≍ m 2 t − d / 2 d t , t ↓ 0 , m > 0. (e^{m^{2}t}-1)H_{D\setminus K}(t)\frac{\mathrm{d}t}{t}\asymp m^{2}t^{-d/2}\,\mathrm{d}t,\qquad t\downarrow0,\qquad m>0. ( e m 2 t − 1 ) H D ∖ K ( t ) t d t ≍ m 2 t − d /2 d t , t ↓ 0 , m > 0. This is not integrable at zero for d ∈ { 2 , 3 } d\in\{2,3\} d ∈ { 2 , 3 } W δ , m W_{\delta,m} W δ , m
Fix T > 0 T>0 T > 0 M δ , m ≤ T \mathcal{M}_{\delta,m}^{\leq T} M δ , m ≤ T M δ , m > T \mathcal{M}_{\delta,m}^{>T} M δ , m > T W δ , m W_{\delta,m} W δ , m
Y δ , m , T ( γ ) : = e M δ , m ≤ T ( D δ ) W δ , m ( γ ) = a δ ∣ γ ∣ exp { M δ , m ≤ T ( D δ ) − M δ , m ≤ T ( D δ ∖ K ) − M δ , m > T ( D δ ∖ K ) } . (7) \begin{aligned}
Y_{\delta,m,T}(\gamma)&:=e^{\mathcal{M}_{\delta,m}^{\leq T}(D_\delta)}W_{\delta,m}(\gamma)\\
&=a_\delta^{|\gamma|}\exp\left\{\mathcal{M}_{\delta,m}^{\leq T}(D_\delta)-\mathcal{M}_{\delta,m}^{\leq T}(D_\delta\setminus K)-\mathcal{M}_{\delta,m}^{>T}(D_\delta\setminus K)\right\}. \tag*{(7)}
\end{aligned} Y δ , m , T ( γ ) := e M δ , m ≤ T ( D δ ) W δ , m ( γ ) = a δ ∣ γ ∣ exp { M δ , m ≤ T ( D δ ) − M δ , m ≤ T ( D δ ∖ K ) − M δ , m > T ( D δ ∖ K ) } . (7) This multiplication leaves the normalized discrete law unchanged. The first two terms in the exponent count short loops intersecting K K K K K K
We now take the continuum counterpart of this rewritten expression. Under diffusive scaling,
a δ ∣ ℓ ∣ = ( 1 + m 2 δ 2 ) t ℓ / δ 2 ⟶ e m 2 t ℓ , a_\delta^{|\ell|}=(1+m^{2}\delta^{2})^{t_\ell/\delta^{2}}\longrightarrow e^{m^{2}t_\ell}, a δ ∣ ℓ ∣ = ( 1 + m 2 δ 2 ) t ℓ / δ 2 ⟶ e m 2 t ℓ , and random-walk loop measure approaches Brownian loop measure. Moreover, since the growth exponent of LERW is below two, a δ ∣ γ ∣ → 1 a_\delta^{|\gamma|}\to1 a δ ∣ γ ∣ → 1 μ D \mu_D μ D (2) , with the same diffusive duration t ℓ t_\ell t ℓ
I m , T ( K ) = ∫ { t ℓ ≤ T , ℓ ∩ K ≠ ∅ } ( e m 2 t ℓ − 1 ) μ D ( d ℓ ) , (8) I_{m,T}(K)=\int_{\{t_\ell\leq T,\ \ell\cap K\ne\varnothing\}}(e^{m^{2}t_\ell}-1)\,\mu_D(\mathrm{d}\ell), \tag*{(8)} I m , T ( K ) = ∫ { t ℓ ≤ T , ℓ ∩ K = ∅ } ( e m 2 t ℓ − 1 ) μ D ( d ℓ ) , (8) J m , T ( K ) = ∫ { t ℓ > T , ℓ ∩ K = ∅ } ( e m 2 t ℓ − 1 ) μ D ( d ℓ ) , (9) J_{m,T}(K)=\int_{\{t_\ell>T,\ \ell\cap K=\varnothing\}}(e^{m^{2}t_\ell}-1)\,\mu_D(\mathrm{d}\ell), \tag*{(9)} J m , T ( K ) = ∫ { t ℓ > T , ℓ ∩ K = ∅ } ( e m 2 t ℓ − 1 ) μ D ( d ℓ ) , (9) Y m , T ( K ) = exp { I m , T ( K ) − J m , T ( K ) } . (10) Y_{m,T}(K)=\exp\{I_{m,T}(K)-J_{m,T}(K)\}. \tag*{(10)} Y m , T ( K ) = exp { I m , T ( K ) − J m , T ( K )} . (10) Here K K K T T T I m , T ( K ) I_{m,T}(K) I m , T ( K ) J m , T ( K ) = ∞ J_{m,T}(K)=\infty J m , T ( K ) = ∞ Y m , T ( K ) = 0 Y_{m,T}(K)=0 Y m , T ( K ) = 0 m = 0 m=0 m = 0 T T T T ′ > T T'>T T ′ > T
Y m , T ′ ( K ) = Y m , T ( K ) exp { ∫ T T ′ ( e m 2 t − 1 ) H D ( t ) d t t } . (11) Y_{m,T'}(K)=Y_{m,T}(K)\exp\left\{\int_T^{T'}(e^{m^{2}t}-1)H_D(t)\frac{\mathrm{d}t}{t}\right\}. \tag*{(11)} Y m , T ′ ( K ) = Y m , T ( K ) exp { ∫ T T ′ ( e m 2 t − 1 ) H D ( t ) t d t } . (11) The multiplier is finite and independent of the curve, so any normalized law obtained from this weight is independent of T T T
It remains to justify these formal correspondences: to control arbitrarily short and long loops, to preserve their intersections with the random curve, and to show that the limiting weight has a finite and strictly positive mean. The main theorem establishes these facts and identifies the candidate as the scaling limit of the extended discrete law.
Main results
We now state our results for the radial models defined above, in which the exit point is not prescribed. The first theorem shows that the weight (10) defines a probability law at every finite boost, and that this law is the scaling limit of the extended discrete model in natural parametrization.
Theorem 1.2. In either dimension, for every fixed m ≥ 0 m \ge0 m ≥ 0 δ \delta δ
P ( m ) ( d η ) = Y m , T ( K ) E P Y m , T ( K ) P ( d η ) , 0 < E P Y m , T < ∞ . \mathbb{P}^{(m)}(\mathrm{d}\eta)=\frac{Y_{m,T}(K)}{\mathbb{E}_{\mathbb{P}}Y_{m,T}(K)}\mathbb{P}(\mathrm{d}\eta), \qquad0<\mathbb{E}_{\mathbb{P}}Y_{m,T}<\infty. P ( m ) ( d η ) = E P Y m , T ( K ) Y m , T ( K ) P ( d η ) , 0 < E P Y m , T < ∞. The law is independent of T > 0 T>0 T > 0
Y m , T ( K ) > 0 ⟺ λ 1 ( D ∖ K ) > m 2 P -almost surely . Y_{m,T}(K)>0 \quad\Longleftrightarrow\quad\lambda_1(D\setminus K)>m^2 \qquad\mathbb{P}\text{-almost surely}. Y m , T ( K ) > 0 ⟺ λ 1 ( D ∖ K ) > m 2 P -almost surely . For m 2 < λ 1 ( D ) m^2<\lambda_1(D) m 2 < λ 1 ( D )
When m 2 < λ 1 ( D ) m^2<\lambda_1(D) m 2 < λ 1 ( D )
Theorem 1.3. For 0 ≤ m 2 < λ 1 ( D ) 0\le m^2<\lambda_1(D) 0 ≤ m 2 < λ 1 ( D )
d P ( m ) d P ( η ) = exp { Φ m ( K ) } E 0 e m 2 T D , Φ m ( K ) = ∫ ( e m 2 t ℓ − 1 ) 1 { ℓ ∩ K ≠ ∅ } μ D ( d ℓ ) , (12) \frac{\mathrm{d}\mathbb{P}^{(m)}}{\mathrm{d}\mathbb{P}}(\eta)=\frac{\exp\{\Phi_m(K)\}}{\mathbb{E}^{0}e^{m^2T_D}}, \qquad\Phi_m(K)=\int(e^{m^2t_\ell}-1)1\{\ell\cap K\ne\varnothing\}\,\mu_D(\mathrm{d}\ell), \tag*{(12)} d P d P ( m ) ( η ) = E 0 e m 2 T D exp { Φ m ( K )} , Φ m ( K ) = ∫ ( e m 2 t ℓ − 1 ) 1 { ℓ ∩ K = ∅ } μ D ( d ℓ ) , (12) where T D T_D T D D D D 0 0 0 L p ( P ) L^p(\mathbb{P}) L p ( P ) p > 1 p>1 p > 1 p m 2 < λ 1 ( D ) pm^2<\lambda_1(D) p m 2 < λ 1 ( D )
Our next result describes the phase transition in mutual absolute continuity, in relation to [40 Question 4.5] on absolute continuity under near-critical perturbations. For massive LERW in the plane, the perturbed and ordinary laws are equivalent for every mass [15 Lemma 2.9] ; boosting behaves differently.
Corollary 1.4. For every m ≥ 0 m\ge0 m ≥ 0 P ( m ) ≪ P \mathbb{P}^{(m)}\ll\mathbb{P} P ( m ) ≪ P
P ≪ P ( m ) ⟺ m 2 ≤ λ 1 ( D ) . \mathbb{P}\ll\mathbb{P}^{(m)}\quad\Longleftrightarrow\quad m^2\le\lambda_1(D). P ≪ P ( m ) ⟺ m 2 ≤ λ 1 ( D ) . In dimension two,
P ≪ P ( m ) ⟺ m 2 ≤ ess inf K λ 1 ( D ∖ K ) = π 2 4 . \mathbb{P}\ll\mathbb{P}^{(m)}\quad\Longleftrightarrow\quad m^2\le\underset{K}{\operatorname{ess\,inf}}\lambda_1(D\setminus K)=\frac{\pi^2}{4}. P ≪ P ( m ) ⟺ m 2 ≤ K ess inf λ 1 ( D ∖ K ) = 4 π 2 . Thus, in dimension two and for m 2 ∈ [ λ 1 ( D ) , π 2 / 4 ] m^2\in[\lambda_1(D),\pi^2/4] m 2 ∈ [ λ 1 ( D ) , π 2 /4 ] λ 1 ( D ) ≈ 1.45 \lambda_1(D)\approx1.45 λ 1 ( D ) ≈ 1.45 π 2 / 4 ≈ 2.47 \pi^2/4\approx2.47 π 2 /4 ≈ 2.47 E 0 e m 2 T D \mathbb{E}^{0}e^{m^2T_D} E 0 e m 2 T D 2 _2 2
As m m m
Corollary 1.5. In either dimension, for every m > 0 m > 0 m > 0 P ( m ) \mathbb{P}^{(m)} P ( m ) Q ⊂ D ∖ K Q \subset D \setminus K Q ⊂ D ∖ K
side ( Q ) < π 2 m . (13) \operatorname{side}(Q) < \frac{\pi}{\sqrt{2}m}. \tag*{(13)} side ( Q ) < 2 m π . (13) Consequently, for every ℓ > 0 \ell> 0 ℓ > 0 m > M ( ℓ ) : = π / ( 2 ℓ ) m > M(\ell) := \pi/(\sqrt{2}\ell) m > M ( ℓ ) := π / ( 2 ℓ ) D ∖ K D \setminus K D ∖ K ℓ \ell ℓ P ( m ) \mathbb{P}^{(m)} P ( m )
Since P ( m ) ≪ P \mathbb{P}^{(m)} \ll\mathbb{P} P ( m ) ≪ P 5 / 4 5/4 5/4 β ∈ [ 3 / 2 , 5 / 3 ] \beta\in[3/2,5/3] β ∈ [ 3/2 , 5/3 ] 5 β \beta β [50 , 51 ] . Thus macroscopic holes disappear as m → ∞ m \to\infty m → ∞
To compare the two perturbations further, in § 6 we construct a continuum chordal bLERW law for m 2 < λ 1 ( D ) m^2 < \lambda_1(D) m 2 < λ 1 ( D ) [7 ] . For earlier work on reconstructing planar Brownian motion from SLE2 and Brownian loop-soups, see also Ambrosio [1 ] . The driving-function computation follows the strategy of [40 , 15 ] , using a Green-kernel comparison that is uniform over the growing slits throughout the subcritical range.
Our method also yields convergence of three-dimensional massive LERW in the ball in natural parametrization. Here the per-step factor is 1 − m 2 δ 2 1-m^2\delta^2 1 − m 2 δ 2 m 2 m^2 m 2 − m 2 -m^2 − m 2 m ≥ 0 m \ge0 m ≥ 0
Proof sketch
After the two changes of normalization, it suffices to prove convergence and uniform integrability of the weights, whose logarithms split as
log Y δ , m , T = ∣ γ δ ∣ log a δ + I δ , m , T − J δ , m , T , ∣ γ δ ∣ log a δ ⟶ 0. \log Y_{\delta,m,T} = |\gamma^\delta| \log a_\delta+ I_{\delta,m,T} - J_{\delta,m,T}, \qquad|\gamma^\delta| \log a_\delta\longrightarrow0. log Y δ , m , T = ∣ γ δ ∣ log a δ + I δ , m , T − J δ , m , T , ∣ γ δ ∣ log a δ ⟶ 0. We split the loop terms according to duration: short loops, loops with duration in a fixed interval, and long loops. On a fixed duration interval bounded away from zero, the coupling and hittability results of [48 ] preserve whether a loop intersects the path. For short loops, the covering estimate controls I δ , m , T I_{\delta,m,T} I δ , m , T J δ , m , T J_{\delta,m,T} J δ , m , T E P Y m , T > 0 \mathbb{E}P Y_{m,T} > 0 E P Y m , T > 0
A technical input of independent interest is the fixed-scale box-count estimate of Theorem 3.4 , which is uniform in both the mesh and the box size and has the sharp covering powers 5 / 4 5/4 5/4 β \beta β
The proof follows the moment method of Barlow–Masson [3 ] and Shiraishi [50 ] , with points replaced by boxes. The new difficulty is that a visit of the walk to a box may later be erased. We therefore mark, in each box, the last visit of the walk to the first vertex of the box on the final LERW. The resulting decomposition retains the condition that the incoming erased arm avoids the box, so that the sharp hitting estimates of [32 ] in dimension two and of [36 , 21 ] in dimension three can be applied at each box.
Further questions and organization
We now briefly discuss two future directions and the organization of this work.
What can be said about boosted SLE2 as m → ∞ m \to\infty m → ∞
In dimensions four and higher the question is different. There ordinary LERW has a Brownian scaling limit, and Brownian traces have zero Newtonian capacity [42 Chapters 8–9] . Above the spectral threshold of the domain, however, the extended discrete weight forces the path to raise the lowest eigenvalue of its complement. For D = B ( 0 , 1 ) ⊂ R d D = B(0,1) \subset\mathbb{R}^{d} D = B ( 0 , 1 ) ⊂ R d a δ = 1 + m 2 δ 2 a_{\delta} = 1 + m^{2}\delta^{2} a δ = 1 + m 2 δ 2 m 2 > λ 1 ( D ) m^{2} > \lambda_{1}(D) m 2 > λ 1 ( D ) d ≥ 5 d \ge5 d ≥ 5 δ − ( d − 2 ) + o ( 1 ) \delta^{-(d-2)+o(1)} δ − ( d − 2 ) + o ( 1 )
Section 2 fixes the standard path, operator and loop-measure conventions. Section 3 proves the loop-erasure and compensation identities used in the introduction and states the inputs for convergence. Sections 4 and 5 prove convergence and the geometric consequences. Section 6 treats the continuum chordal law and its driving function. The box-count estimate is proved in Section 7, and the spectral support construction is given in Appendix A.
AI statement. The main body of this work was developed in 2022. Commercial large language models were used to assist with language polishing and to simplify the proof of Theorem 3.4.
Acknowledgments. X.L. thanks Nathanael Berestycki, Dmitry Chelkak, Nina Holden, Xin Sun, Yilin Wang and Yizheng Yuan for inspiring discussions. G.C. and X.L. are supported by Beijing Natural Science Foundation (JQ26001) and the National Key R&D Program of China (No. 2021YFA1002700). D.S. was supported in part by JSPS KAKENHI Grant Numbers JP26K00609 and JP23K22399.
Notation and preliminaries
This section collects standard background: path notation and loop erasure, natural parametrization and the classical scaling limits, Dirichlet operators, and the conventions for loop measures.
Write B ( x , r ) = { y : ∣ y − x ∣ < r } B(x,r)=\{y:|y-x|<r\} B ( x , r ) = { y : ∣ y − x ∣ < r } Q ( x , r ) = { y : ∣ y − x ∣ ∞ < r } Q(x,r)=\{y:|y-x|_{\infty}<r\} Q ( x , r ) = { y : ∣ y − x ∣ ∞ < r } U ⊂ δ Z d U \subset\delta\mathbb{Z}^{d} U ⊂ δ Z d
∂ U = { y ∉ U : ∣ x − y ∣ = δ for some x ∈ U } , ∂ i n U = { x ∈ U : ∣ x − y ∣ = δ for some y ∉ U } . \partial U=\{y\notin U:|x-y|=\delta\text{ for some }x\in U\},\qquad\partial_{\mathrm{in}}U=\{x\in U:|x-y|=\delta\text{ for some }y\notin U\}. ∂ U = { y ∈ / U : ∣ x − y ∣ = δ for some x ∈ U } , ∂ in U = { x ∈ U : ∣ x − y ∣ = δ for some y ∈ / U } . The symbols P x , E x \mathbb{P}^{x},\mathbb{E}^{x} P x , E x x x x c , C c,C c , C
Paths and stopping indices
We fix the notation for path segments.
For ω = ( ω 0 , … , ω n ) \omega= (\omega_0,\ldots,\omega_n) ω = ( ω 0 , … , ω n ) [ i , j ) [i,j) [ i , j ) ⊕ \oplus ⊕
τ B = inf { 0 ≤ j ≤ n : ω j ∈ B } , τ B + = inf { 1 ≤ j ≤ n : ω j ∈ B } , (14) \tau_B = \inf\{0 \le j \le n : \omega_j \in B\}, \qquad\tau_B^{+} = \inf\{1 \le j \le n : \omega_j \in B\}, \tag*{(14)} τ B = inf { 0 ≤ j ≤ n : ω j ∈ B } , τ B + = inf { 1 ≤ j ≤ n : ω j ∈ B } , (14) T U = inf { 0 ≤ j ≤ n : ω j ∉ U } , L B = sup { 0 ≤ j ≤ n : ω j ∈ B } . (15) T_U = \inf\{0 \le j \le n : \omega_j \notin U\}, \qquad L_B = \sup\{0 \le j \le n : \omega_j \in B\}. \tag*{(15)} T U = inf { 0 ≤ j ≤ n : ω j ∈ / U } , L B = sup { 0 ≤ j ≤ n : ω j ∈ B } . (15) Here inf ∅ = ∞ \inf\varnothing= \infty inf ∅ = ∞ sup ∅ = − ∞ \sup\varnothing= -\infty sup ∅ = − ∞
Loop erasure
We recall chronological loop erasure in terms of last visits and state the domain Markov property used below.
We use S , Z S,Z S , Z ω \omega ω η , γ \eta,\gamma η , γ ω = ( ω 0 , … , ω n ) \omega= (\omega_0,\ldots,\omega_n) ω = ( ω 0 , … , ω n )
p ( ω ) = ∏ j = 0 n − 1 p ( ω j , ω j + 1 ) , Range ( ω ) = { ω 0 , … , ω n } , p(\omega) = \prod_{j=0}^{n-1} p(\omega_j,\omega_{j+1}), \qquad\operatorname{Range}(\omega) = \{\omega_0,\ldots,\omega_n\}, p ( ω ) = j = 0 ∏ n − 1 p ( ω j , ω j + 1 ) , Range ( ω ) = { ω 0 , … , ω n } , where p ( u , v ) p(u,v) p ( u , v ) γ 0 = ω 0 \gamma_0 = \omega_0 γ 0 = ω 0
s 0 = max { 0 ≤ j ≤ n : ω j = γ 0 } , s_0 = \max\{0 \le j \le n : \omega_j = \gamma_0\}, s 0 = max { 0 ≤ j ≤ n : ω j = γ 0 } , γ k + 1 = ω s k + 1 , s k + 1 = max { s k + 1 ≤ j ≤ n : ω j = γ k + 1 } ( s k < n ) . (16) \gamma_{k+1} = \omega_{s_{k+1}}, \qquad s_{k+1} = \max\{s_k + 1 \le j \le n : \omega_j = \gamma_{k+1}\} \quad(s_k < n). \tag*{(16)} γ k + 1 = ω s k + 1 , s k + 1 = max { s k + 1 ≤ j ≤ n : ω j = γ k + 1 } ( s k < n ) . (16) Then LE ( ω ) = ( γ 0 , … , γ ℓ ) \operatorname{LE}(\omega) = (\gamma_0,\ldots,\gamma_\ell) LE ( ω ) = ( γ 0 , … , γ ℓ ) ℓ = inf { k : s k = n } \ell= \inf\{k : s_k = n\} ℓ = inf { k : s k = n } γ ℓ \gamma_\ell γ ℓ ℓ \ell ℓ
The first entrance of the final loop erasure into B B B
j B = inf { j : γ j ∈ B } , y B = γ j B , t B = s j B on { j B < ∞ } . (17) j_B = \inf\{j : \gamma_j \in B\}, \qquad y_B = \gamma_{j_B}, \qquad t_B = s_{j_B} \quad\text{on } \{j_B < \infty\}. \tag*{(17)} j B = inf { j : γ j ∈ B } , y B = γ j B , t B = s j B on { j B < ∞ } . (17) Thus t B t_B t B y B y_B y B τ B \tau_B τ B L B L_B L B t B t_B t B [29 Section 11.2] .
We recall the domain Markov property of γ = LE ( S [ 0 , T U ] ) \gamma= \operatorname{LE}(S[0,T_U]) γ = LE ( S [ 0 , T U ]) γ [ 0 , k ] = λ = ( v 0 , … , v k ) \gamma[0,k] = \lambda= (v_0,\ldots,v_k) γ [ 0 , k ] = λ = ( v 0 , … , v k ) v k ∈ U v_k \in U v k ∈ U S ~ \widetilde{S} S v k v_k v k T ~ U = inf { j : S ~ j ∉ U } \widetilde{T}_U = \inf\{j : \widetilde{S}_j \notin U\} T U = inf { j : S j ∈ / U } σ λ = inf { j ≥ 1 : S ~ j ∈ λ [ 0 , k ) } \sigma_\lambda= \inf\{j \ge1 : \widetilde{S}_j \in\lambda[0,k)\} σ λ = inf { j ≥ 1 : S j ∈ λ [ 0 , k )} [29 Section 11.2]
L ( γ [ k , ∣ γ ∣ ] ∣ γ [ 0 , k ] = λ ) = L ( LE ( S ~ [ 0 , T ~ U ] ) ∣ T ~ U < σ λ ) . (18) \mathcal{L}(\gamma[k,|\gamma|] \mid\gamma[0,k] = \lambda) = \mathcal{L}\left(\operatorname{LE}(\widetilde{S}[0,\widetilde{T}_U]) \mid\widetilde{T}_U < \sigma_\lambda\right). \tag*{(18)} L ( γ [ k , ∣ γ ∣ ] ∣ γ [ 0 , k ] = λ ) = L ( LE ( S [ 0 , T U ]) ∣ T U < σ λ ) . (18) In the conditioning the walk must avoid the retained path before its tip, but may return to the tip. Equivalently, one may forbid positive-time returns to the tip: the loops at the tip contribute the common factor G δ , U ∖ λ [ 0 , k ) ( v k , v k ) G_{\delta,U\setminus\lambda[0,k)}(v_k,v_k) G δ , U ∖ λ [ 0 , k ) ( v k , v k )
G δ , U ( x , y ) = E x ∑ j = 0 T U − 1 1 { S j = y } = ∑ j ≥ 0 P x ( S j = y , j < T U ) . (19) G_{\delta,U}(x,y) = \mathbb{E}^{x}\sum_{j=0}^{T_U-1} 1\{S_j = y\} = \sum_{j \ge0} \mathbb{P}^{x}(S_j = y,\ j < T_U). \tag*{(19)} G δ , U ( x , y ) = E x j = 0 ∑ T U − 1 1 { S j = y } = j ≥ 0 ∑ P x ( S j = y , j < T U ) . (19) It includes the time-zero visit; on the unit lattice we abbreviate it as G U G_U G U § 7 are path measures normalized by this sum.
Natural parametrization and the classical limit
We now specify the natural time assigned to each LERW edge and recall the convergence results for our two radial settings.
The raw walk runs on the diffusive clock, with time δ 2 \delta^2 δ 2
c δ = { c ∗ − 1 δ 5 / 4 , d = 2 , δ β , d = 3 , η δ ( j c δ ) = γ j δ , t η δ = c δ ∣ γ δ ∣ , c_\delta=
\begin{cases}
c_*^{-1}\delta^{5/4}, & d=2,\\
\delta^\beta, & d=3,
\end{cases}
\qquad
\eta^\delta(jc_\delta)=\gamma_j^\delta,
\qquad
t_{\eta^\delta}=c_\delta|\gamma^\delta|, c δ = { c ∗ − 1 δ 5/4 , δ β , d = 2 , d = 3 , η δ ( j c δ ) = γ j δ , t η δ = c δ ∣ γ δ ∣ , with linear interpolation. Here c ∗ c_* c ∗ [33 ] , and β ∈ [ 3 / 2 , 5 / 3 ] \beta\in[3/2,5/3] β ∈ [ 3/2 , 5/3 ]
ρ ( η , η ~ ) = ∣ t η − t η ~ ∣ + sup 0 ≤ s ≤ 1 ∣ η ( s t η ) − η ~ ( s t η ~ ) ∣ . \rho(\eta,\widetilde{\eta})=|t_\eta-t_{\widetilde{\eta}}|+\sup_{0\le s\le1}|\eta(st_\eta)-\widetilde{\eta}(st_{\widetilde{\eta}})|. ρ ( η , η ) = ∣ t η − t η ∣ + 0 ≤ s ≤ 1 sup ∣ η ( s t η ) − η ( s t η ) ∣. This metric has the same convergent sequences as the Lawler–Viklund metric, which allows time changes uniformly close to the identity.
In dimension two we need the following version of the radial convergence of [33 ] , in which the exit point is not prescribed.
Lemma 2.1. In the unit disk, η δ \eta^\delta η δ 2 _2 2 0 0 0 5 / 4 5/4 5/4
Proof. We first apply the known natural-parametrization convergence theorem to the standard inner approximation of the disk, and then show that replacing this approximation by D δ D_\delta D δ [33 Theorem 1.1] ; convergence is uniform over the starting edges by compactness of the circle and rotation-continuity of radial SLE2 _2 2 D δ D_\delta D δ O ( δ ) O(\delta) O ( δ ) [23 , 28 ] confines the added walk segment to an ϵ \epsilon ϵ ϵ ↓ 0 \epsilon\downarrow0 ϵ ↓ 0 [3 Theorem 5.6(1) and Lemma 3.13] gives expected natural LERW time at most C θ ϵ 5 / 4 − θ C_\theta\epsilon^{5/4-\theta} C θ ϵ 5/4 − θ 0 < θ < 5 / 4 0<\theta<5/4 0 < θ < 5/4 N = δ − 1 N=\delta^{-1} N = δ − 1 N 2 E s ( N ) = O ( N 5 / 4 ) N^2\mathop{\rm Es}(N)=O(N^{5/4}) N 2 Es ( N ) = O ( N 5/4 ) [3 Theorem 1.2 and Proposition 6.2(2)] . The common prefixes and the vanishing terminal pieces give convergence in the natural curve metric, also for arbitrary meshes. □
In dimension three we use the following result. Convergence in natural parametrization along dyadic meshes was proved by Li–Shiraishi [36 ] ; for arbitrary meshes, in the present setting (unit ball, start at 0 0 0 [21 Theorem 1.7] .
Proposition 2.2. In the unit ball, η δ \eta^\delta η δ δ ↓ 0 \delta\downarrow0 δ ↓ 0
Dirichlet operators and heat traces
We define the discrete operators, the Dirichlet operators on curve complements, and the heat traces used for loop masses. Recall that the generator is normalized as in (1.2).
Extending f : U → R f: U \to\mathbb{R} f : U → R f ~ \tilde f f ~
L δ , U f ( x ) = 1 2 d δ 2 ∑ ∣ y − x ∣ = δ [ f ~ ( y ) − f ( x ) ] , A δ , U = − L δ , U = δ − 2 ( I − p δ , U ) . (20) L_{\delta,U}f(x)=\frac{1}{2d\delta^2}\sum_{|y-x|=\delta}[\tilde f(y)-f(x)],\qquad A_{\delta,U}=-L_{\delta,U}=\delta^{-2}(I-p_{\delta,U}). \tag*{(20)} L δ , U f ( x ) = 2 d δ 2 1 ∣ y − x ∣ = δ ∑ [ f ~ ( y ) − f ( x )] , A δ , U = − L δ , U = δ − 2 ( I − p δ , U ) . (20) For any open U ⊂ D U \subset D U ⊂ D A U = − Δ U / ( 2 d ) A_U=-\Delta_U/(2d) A U = − Δ U / ( 2 d )
E U ( f , g ) = 1 2 d ∫ U ∇ f ⋅ ∇ g d x , f , g ∈ H 0 1 ( U ) . (21) \mathcal{E}_U(f,g)=\frac{1}{2d}\int_U \nabla f\cdot\nabla g\,\mathrm{d}x,\qquad f,g\in H_0^1(U). \tag*{(21)} E U ( f , g ) = 2 d 1 ∫ U ∇ f ⋅ ∇ g d x , f , g ∈ H 0 1 ( U ) . (21) Here H 0 1 ( U ) H_0^1(U) H 0 1 ( U ) H 1 H^1 H 1 C c ∞ ( U ) C_c^\infty(U) C c ∞ ( U ) D D D [9 Section 2] for the capacitary formulation. Compact embedding gives eigenvalues 0 < λ 1 ( U ) ≤ λ 2 ( U ) ≤ ⋯ 0<\lambda_1(U)\leq\lambda_2(U)\leq\cdots 0 < λ 1 ( U ) ≤ λ 2 ( U ) ≤ ⋯
λ 1 ( U ) = inf f ∈ H 0 1 ( U ) ∥ f ∥ 2 = 1 1 2 d ∫ U ∣ ∇ f ∣ 2 d x . (22) \lambda_1(U)=\inf_{\substack{f\in H_0^1(U)\\ \lVert f\rVert_2=1}}\frac{1}{2d}\int_U|\nabla f|^2\,\mathrm{d}x. \tag*{(22)} λ 1 ( U ) = f ∈ H 0 1 ( U ) ∥ f ∥ 2 = 1 inf 2 d 1 ∫ U ∣∇ f ∣ 2 d x . (22) The killed Brownian density p U ( t , x , y ) p_U(t,x,y) p U ( t , x , y ) e − t A U e^{-tA_U} e − t A U
H U ( t ) = Tr ( e − t A U ) = ∫ U p U ( t , x , x ) d x = ∑ j ≥ 1 e − t λ j ( U ) . (23) H_U(t)=\operatorname{Tr}(e^{-tA_U})=\int_U p_U(t,x,x)\,\mathrm{d}x=\sum_{j\geq1}e^{-t\lambda_j(U)}. \tag*{(23)} H U ( t ) = Tr ( e − t A U ) = ∫ U p U ( t , x , x ) d x = j ≥ 1 ∑ e − t λ j ( U ) . (23) We put H ∅ = 0 H_{\varnothing}=0 H ∅ = 0 λ 1 ( ∅ ) = ∞ \lambda_1(\varnothing)=\infty λ 1 ( ∅ ) = ∞
Loop measure conventions
We fix the conventions for rooted and unrooted loops, for both the random-walk and the Brownian loop measure, with the diffusive clock used throughout.
A rooted loop ω = ( x 0 , … , x n = x 0 ) \omega=(x_0,\ldots,x_n=x_0) ω = ( x 0 , … , x n = x 0 ) n − 1 ∏ j = 0 n − 1 p δ , U ( x j , x j + 1 ) n^{-1}\prod_{j=0}^{n-1}p_{\delta,U}(x_j,x_{j+1}) n − 1 ∏ j = 0 n − 1 p δ , U ( x j , x j + 1 ) n n n Tr ( p δ , U n ) / n \operatorname{Tr}(p_{\delta,U}^n)/n Tr ( p δ , U n ) / n k k k n / k n/k n / k k k k [29 Sections 9.2–9.5] . Its diffusive duration is t = n δ 2 t=n\delta^2 t = n δ 2 a δ n = e ν δ t a_\delta^n=e^{\nu_\delta t} a δ n = e ν δ t ν δ = δ − 2 log a δ → m 2 \nu_\delta=\delta^{-2}\log a_\delta\to m^2 ν δ = δ − 2 log a δ → m 2 O ( δ ) O(\delta) O ( δ )
Brownian loop measure, with the same clock, is defined by
∫ F ( ℓ ) μ U ( d ℓ ) = ∫ 0 ∞ d t t ∫ U p U ( t , x , x ) E x , x U , t [ F ( ℓ ) ] d x , (24) \int F(\ell)\,\mu_U(\mathrm{d}\ell)=\int_0^\infty\frac{\mathrm{d}t}{t}\int_U p_U(t,x,x)\mathbb{E}^{U,t}_{x,x}[F(\ell)]\,\mathrm{d}x, \tag*{(24)} ∫ F ( ℓ ) μ U ( d ℓ ) = ∫ 0 ∞ t d t ∫ U p U ( t , x , x ) E x , x U , t [ F ( ℓ )] d x , (24) for nonnegative root-independent functionals F F F E x , x U , t \mathbb{E}^{U,t}_{x,x} E x , x U , t [34 ] and [48 Section 2] . In particular,
μ D { t ℓ ∈ d t } = H D ( t ) d t t . (25) \mu_D\{t_\ell\in\mathrm{d}t\}=H_D(t)\frac{\mathrm{d}t}{t}. \tag*{(25)} μ D { t ℓ ∈ d t } = H D ( t ) t d t . (25) The restriction property identifies loops avoiding K K K D ∖ K D\setminus K D ∖ K
μ D { t ℓ ∈ d t , ℓ ∩ K = ∅ } = H D ∖ K ( t ) d t t , \mu_D\{t_\ell\in\mathrm{d}t,\ \ell\cap K=\varnothing\}=H_{D\setminus K}(t)\frac{\mathrm{d}t}{t}, μ D { t ℓ ∈ d t , ℓ ∩ K = ∅ } = H D ∖ K ( t ) t d t , μ D { t ℓ ∈ d t , ℓ ∩ K ≠ ∅ } = [ H D ( t ) − H D ∖ K ( t ) ] d t t . (26) \mu_D\{t_\ell\in\mathrm{d}t,\ \ell\cap K\neq\varnothing\}=[H_D(t)-H_{D\setminus K}(t)]\frac{\mathrm{d}t}{t}. \tag*{(26)} μ D { t ℓ ∈ d t , ℓ ∩ K = ∅ } = [ H D ( t ) − H D ∖ K ( t )] t d t . (26) These identities express the integrals (1.8)–(1.9) through heat traces; see Section 3.2.
Loop identities and inputs for convergence
In this section, we prove the loop identities behind the construction in § 1.2 and then state the covering and support estimates used to pass to the limit. These estimates are used in § 4 and proved in § 7 and Appendix A.
The loop-erasure identity and the discrete spectral cutoff
We first prove the loop-erasure identity (1.6) and show that Definition 1.1 agrees with the boosted walk whenever the latter can be normalized.
Proposition 3.1. If a δ ρ ( p δ , D δ ) < 1 a_\delta\rho(p_\delta,D_\delta)<1 a δ ρ ( p δ , D δ ) < 1 γ \gamma γ K K K w δ , m ( γ ) w_{\delta,m}(\gamma) w δ , m ( γ )
Proof. For γ = ( v 0 , … , v l ) \gamma=(v_0,\ldots,v_l) γ = ( v 0 , … , v l ) K = { v 0 , … , v l − 1 } K=\{v_0,\ldots,v_{l-1}\} K = { v 0 , … , v l − 1 } U j = D δ ∖ { v 0 , … , v j − 1 } U_j=D_\delta\setminus\{v_0,\ldots,v_{j-1}\} U j = D δ ∖ { v 0 , … , v j − 1 } c ∈ { 1 , a δ } c\in\{1,a_\delta\} c ∈ { 1 , a δ } q ( x , y ) = c ( 2 d ) − 1 1 { ∣ x − y ∣ = δ } q(x,y)=c(2d)^{-1}\mathbf{1}_{\{|x-y|=\delta\}} q ( x , y ) = c ( 2 d ) − 1 1 { ∣ x − y ∣ = δ } Q U Q_U Q U U U U [29 Proposition 9.5.1] , with the exit vertex absorbing, gives
∑ LE ( ω ) = γ q ( ω ) = q ( γ ) ∏ j < l [ ( I − Q U j ) − 1 ] ( v j , v j ) = q ( γ ) det ( I − Q D δ ∖ K ) det ( I − Q D δ ) . (27) \sum_{\operatorname{LE}(\omega)=\gamma}q(\omega)=q(\gamma)\prod_{j<l}\left[(I-Q_{U_j})^{-1}\right](v_j,v_j)=q(\gamma)\frac{\det(I-Q_{D_\delta\setminus K})}{\det(I-Q_{D_\delta})}. \tag*{(27)} LE ( ω ) = γ ∑ q ( ω ) = q ( γ ) j < l ∏ [ ( I − Q U j ) − 1 ] ( v j , v j ) = q ( γ ) det ( I − Q D δ ) det ( I − Q D δ ∖ K ) . (27) The sum is over first-exit walks and converges by the spectral assumption. Dividing the boosted and ordinary identities and expanding the logarithms gives (1.6). The factor depending only on loops in D δ D_\delta D δ W δ , m W_{\delta,m} W δ , m
Beyond this range the extended weight is still defined. The next proposition identifies the paths with positive weight and expresses the weight through determinants.
Proposition 3.2. Write ρ ( Q ) \rho(Q) ρ ( Q ) Q Q Q m > 0 m>0 m > 0 W δ , m ( γ ) > 0 W_{\delta,m}(\gamma)>0 W δ , m ( γ ) > 0 a δ ρ ( p δ , D δ ∖ K ) < 1 a_\delta\rho(p_{\delta,D_\delta\setminus K})<1 a δ ρ ( p δ , D δ ∖ K ) < 1 U U U a δ ρ ( p δ , U ) < 1 a_\delta\rho(p_{\delta,U})<1 a δ ρ ( p δ , U ) < 1
e − M δ , m ( U ) = det ( I − a δ p δ , U ) det ( I − p δ , U ) . (28) e^{-M_{\delta,m}(U)}=\frac{\det(I-a_\delta p_{\delta,U})}{\det(I-p_{\delta,U})}. \tag*{(28)} e − M δ , m ( U ) = det ( I − p δ , U ) det ( I − a δ p δ , U ) . (28) Proof. Use log det ( I − Q ) = − ∑ n ≥ 1 Tr ( Q n ) / n \log\det(I-Q)=-\sum_{n\geq1}\operatorname{Tr}(Q^n)/n log det ( I − Q ) = − ∑ n ≥ 1 Tr ( Q n ) / n ρ ( Q ) < 1 \rho(Q)<1 ρ ( Q ) < 1 Tr ( p δ , U 2 k ) ≥ ρ ( p δ , U ) 2 k \operatorname{Tr}(p_{\delta,U}^{2k})\geq\rho(p_{\delta,U})^{2k} Tr ( p δ , U 2 k ) ≥ ρ ( p δ , U ) 2 k ρ ( p δ , U ) = 0 \rho(p_{\delta,U})=0 ρ ( p δ , U ) = 0
Combining (3.2) with the ordinary LERW path weight gives
P δ ( m ) ( γ ) ∝ ( a δ 2 d ) ∣ γ ∣ det ( I − a δ p δ , D δ ∖ K ) 1 { a δ ρ ( p δ , D δ ∖ K ) < 1 } . \mathbb{P}^{(m)}_\delta(\gamma)\propto\left(\frac{a_\delta}{2d}\right)^{|\gamma|}\det(I-a_\delta p_{\delta,D_\delta\setminus K})\mathbf{1}_{\{a_\delta\rho(p_{\delta,D_\delta\setminus K})<1\}}. P δ ( m ) ( γ ) ∝ ( 2 d a δ ) ∣ γ ∣ det ( I − a δ p δ , D δ ∖ K ) 1 { a δ ρ ( p δ , D δ ∖ K ) < 1 } . Exact compensation identities
We now write the short- and long-loop terms of the second normalization (1.7) explicitly and verify it at each fixed mesh, including for paths of zero weight.
Lemma 3.3. For a simple exit path γ \gamma γ K K K
I δ , m , T ( K ) = ∑ n δ 2 ≤ T a δ n − 1 n [ Tr ( p δ , D δ n ) − Tr ( p δ , D δ ∖ K n ) ] , I_{\delta,m,T}(K)=\sum_{n\delta^2\leq T}\frac{a_\delta^n-1}{n}\left[\operatorname{Tr}(p_{\delta,D_\delta}^n)-\operatorname{Tr}(p_{\delta,D_\delta\setminus K}^n)\right], I δ , m , T ( K ) = n δ 2 ≤ T ∑ n a δ n − 1 [ Tr ( p δ , D δ n ) − Tr ( p δ , D δ ∖ K n ) ] , J δ , m , T ( K ) = ∑ n δ 2 > T a δ n − 1 n Tr ( p δ , D δ ∖ K n ) . J_{\delta,m,T}(K)=\sum_{n\delta^2>T}\frac{a_\delta^n-1}{n}\operatorname{Tr}(p_{\delta,D_\delta\setminus K}^n). J δ , m , T ( K ) = n δ 2 > T ∑ n a δ n − 1 Tr ( p δ , D δ ∖ K n ) . Then Y δ , m , T ( γ ) = a δ ∣ γ ∣ exp { I δ , m , T ( K ) − J δ , m , T ( K ) } Y_{\delta,m,T}(\gamma)=a_\delta^{|\gamma|}\exp\{I_{\delta,m,T}(K)-J_{\delta,m,T}(K)\} Y δ , m , T ( γ ) = a δ ∣ γ ∣ exp { I δ , m , T ( K ) − J δ , m , T ( K )} W δ , m ( γ ) W_{\delta,m}(\gamma) W δ , m ( γ ) e C δ , m , T e^{C_{\delta,m,T}} e C δ , m , T
C δ , m , T = ∑ n δ 2 ≤ T a δ n − 1 n Tr ( p δ , D δ n ) . C_{\delta,m,T}=\sum_{n\delta^2\le T}\frac{a_\delta^n-1}{n}\operatorname{Tr}(p_{\delta,D_\delta}^n). C δ , m , T = n δ 2 ≤ T ∑ n a δ n − 1 Tr ( p δ , D δ n ) . Hence the two weights define the same normalized law whenever the normalizing constant is positive. The identity also holds when the long-loop mass is infinite, in which case both weights vanish.
Proof. Split (1.5) at n δ 2 = T n\delta^2=T n δ 2 = T M δ , m ( D δ ∖ K ) = C δ , m , T − I δ , m , T ( K ) + J δ , m , T ( K ) M_{\delta,m}(D_\delta\setminus K)=C_{\delta,m,T}-I_{\delta,m,T}(K)+J_{\delta,m,T}(K) M δ , m ( D δ ∖ K ) = C δ , m , T − I δ , m , T ( K ) + J δ , m , T ( K )
For the continuum weight defined in (1.8)–(1.10), the restriction identities (2.13) give
I m , T ( K ) = ∫ 0 T ( e m 2 t − 1 ) [ H D ( t ) − H D ∖ K ( t ) ] d t t , I_{m,T}(K)=\int_0^T\left(e^{m^2t}-1\right)[H_D(t)-H_{D\setminus K}(t)]\frac{dt}{t}, I m , T ( K ) = ∫ 0 T ( e m 2 t − 1 ) [ H D ( t ) − H D ∖ K ( t )] t d t , J m , T ( K ) = ∫ T ∞ ( e m 2 t − 1 ) H D ∖ K ( t ) d t t . J_{m,T}(K)=\int_T^\infty\left(e^{m^2t}-1\right)H_{D\setminus K}(t)\frac{dt}{t}. J m , T ( K ) = ∫ T ∞ ( e m 2 t − 1 ) H D ∖ K ( t ) t d t . If m 2 < λ 1 ( D ) m^2<\lambda_1(D) m 2 < λ 1 ( D ) D D D
Φ m ( K ) = I m , T ( K ) − J m , T ( K ) + ∫ T ∞ ( e m 2 t − 1 ) H D ( t ) d t t . (29) \Phi_m(K)=I_{m,T}(K)-J_{m,T}(K)+\int_T^\infty\left(e^{m^2t}-1\right)H_D(t)\frac{dt}{t}. \tag*{(29)} Φ m ( K ) = I m , T ( K ) − J m , T ( K ) + ∫ T ∞ ( e m 2 t − 1 ) H D ( t ) t d t . (29) For the relation to regularized determinants on smooth surfaces see [2 ] ; our definition uses heat traces directly on slit domains.
Inputs for the convergence proof
We now state the covering estimate needed for integrability and the support estimate needed for a positive normalizing constant.
Fix a grid Q r = { r ( z + [ 0 , 1 ] d ) : z ∈ Z d } Q_r=\{r(z+[0,1]^d):z\in\mathbb{Z}^d\} Q r = { r ( z + [ 0 , 1 ] d ) : z ∈ Z d }
N r ( K ) = # { Q ∈ Q r : Q ∩ K ≠ ∅ } . N_r(K)=\#\{Q\in Q_r:Q\cap K\ne\varnothing\}. N r ( K ) = # { Q ∈ Q r : Q ∩ K = ∅ } . Thus N r ( K ) N_r(K) N r ( K ) r r r
b 2 = 5 4 , b 3 = β ∈ [ 3 / 2 , 5 / 3 ] , (30) b_2=\frac{5}{4},\qquad b_3=\beta\in[3/2,5/3], \tag*{(30)} b 2 = 4 5 , b 3 = β ∈ [ 3/2 , 5/3 ] , (30) where β \beta β [35 Corollary 1.3] and [51 ] . For 0 < α < 2 0<\alpha<2 0 < α < 2
M α , δ ( K ) = sup r ∈ { 1 , 2 − 1 , 2 − 2 , … } r ≥ δ r α N r ( K ) , M α ( K ) = sup r ∈ { 1 , 2 − 1 , 2 − 2 , … } r α N r ( K ) . (31) M_{\alpha,\delta}(K)=\sup_{\substack{r\in\{1,2^{-1},2^{-2},\ldots\}\\r\ge\delta}}r^\alpha N_r(K),\qquad M_\alpha(K)=\sup_{r\in\{1,2^{-1},2^{-2},\ldots\}}r^\alpha N_r(K). \tag*{(31)} M α , δ ( K ) = r ∈ { 1 , 2 − 1 , 2 − 2 , … } r ≥ δ sup r α N r ( K ) , M α ( K ) = r ∈ { 1 , 2 − 1 , 2 − 2 , … } sup r α N r ( K ) . (31) The restriction r ≥ δ r\ge\delta r ≥ δ
The geometric input is the following box-counting estimate. The proof is given in § 7 .
Theorem 3.4. In each of the radial settings fixed in § 1.1 , there are constants c , C > 0 c,C>0 c , C > 0 δ 0 > 0 \delta_0>0 δ 0 > 0
P δ { N r ( K δ ) > u r − b d } ≤ C e − c u , δ ≤ r ≤ 1 , u ≥ 1 , 0 < δ < δ 0 . (32) \mathbb{P}_\delta\{N_r(K^\delta)>ur^{-b_d}\}\le Ce^{-cu},\qquad\delta\le r\le1,\quad u\ge1,\quad0<\delta<\delta_0. \tag*{(32)} P δ { N r ( K δ ) > u r − b d } ≤ C e − c u , δ ≤ r ≤ 1 , u ≥ 1 , 0 < δ < δ 0 . (32) For every b d < α < 2 b_d < \alpha< 2 b d < α < 2 λ > 0 \lambda> 0 λ > 0
sup 0 < δ < δ 0 E P δ exp { λ M α , δ ( K δ ) } < ∞ . (33) \sup_{0<\delta<\delta_0} \mathbb{E}_{\mathbb{P}_\delta}\exp\{\lambda M_{\alpha,\delta}(K^\delta)\}<\infty. \tag*{(33)} 0 < δ < δ 0 sup E P δ exp { λ M α , δ ( K δ )} < ∞. (33) Here K δ K^\delta K δ
The second input ensures that the limiting normalizer is positive. As it follows from classical estimates, we will postpone its proof to Appendix § 8 .
Proposition 3.5. For standard radial LERW in either dimension,
P { λ 1 ( D ∖ K ) > M } > 0 for every finite M . (34) \mathbb{P}\{\lambda_1(D\setminus K)>M\}>0 \qquad\text{for every finite }M. \tag*{(34)} P { λ 1 ( D ∖ K ) > M } > 0 for every finite M . (34) The following theorem, together with Theorem 3.4 and Proposition 3.5, implies Theorem 1.2. We will give its proof (as well as that of Theorem 1.2) in § 4 .
Theorem 3.6. In either of the two radial settings, suppose that (3.7) holds for some α < 2 \alpha< 2 α < 2 m ≥ 0 m \ge0 m ≥ 0
P { λ 1 ( D ∖ K ) > m 2 } > 0 , (35) \mathbb{P}\{\lambda_1(D\setminus K)>m^2\}>0, \tag*{(35)} P { λ 1 ( D ∖ K ) > m 2 } > 0 , (35) the extended discrete laws are well defined for all sufficiently small δ \delta δ Y m , T P / E Y m , T Y_{m,T}^{\mathbb{P}}/\mathbb{E}Y_{m,T} Y m , T P / E Y m , T 0 < E Y m , T < ∞ 0<\mathbb{E}Y_{m,T}<\infty 0 < E Y m , T < ∞ T T T
Convergence of the boosted laws
In this section we prove Theorem 3.6 and Theorem 1.2 by showing that the compensated weights of Lemma 3.3 converge to (1.10), using the inputs of § 3.3 . In particular, loops with duration in a fixed window are handled by a loop-soup coupling, short loops by the covering estimate, and long loops by spectral decay.
Loops in a fixed duration interval
On a fixed duration interval, we prove convergence of the loop masses hitting and avoiding the curve by combining loop-soup coupling with intersection stability.
Lemma 4.1. In either radial setting, couple the ordinary naturally parametrized curves so that K δ → K K^\delta\to K K δ → K 0 < u < v < ∞ 0<u<v<\infty 0 < u < v < ∞ f : [ u , v ] → R f:[u,v]\to\mathbb{R} f : [ u , v ] → R
∑ u ≤ n δ 2 ≤ v f ( n δ 2 ) n Tr ( p δ , D δ ∖ K δ n ) ⟶ ∫ u v f ( t ) H D ∖ K ( t ) d t t in L 1 . (36) \sum_{u\le n\delta^2\le v}\frac{f(n\delta^2)}{n}\operatorname{Tr}\left(p_{\delta,D_\delta\setminus K^\delta}^{n}\right)\longrightarrow\int_u^v f(t)H_{D\setminus K}(t)\frac{\mathrm{d}t}{t}\quad\text{in }L^1. \tag*{(36)} u ≤ n δ 2 ≤ v ∑ n f ( n δ 2 ) Tr ( p δ , D δ ∖ K δ n ) ⟶ ∫ u v f ( t ) H D ∖ K ( t ) t d t in L 1 . (36) The same holds for the loops hitting the curve, by subtraction from the corresponding sums for D δ D_\delta D δ D D D
Proof. Couple intensity-one loop soups L δ , L \mathcal{L}_\delta,\mathcal{L} L δ , L [48 Theorem 2.2] , with duration normalized by (1.2); the planar approximation originates in [31 ] . On the fixed window, paired loops converge in duration and parametrized trace, with probability tending to one. Their total intensities are bounded by C D ∫ r 2 2 v t − d / 2 − 1 d t C_D\int_{r^2}^{2v}t^{-d/2-1}\,\mathrm{d}t C D ∫ r 2 2 v t − d /2 − 1 d t r > 0 r>0 r > 0
Pairing alone does not preserve intersection with the curve. Let N δ ( h , r ) N_\delta(h,r) N δ ( h , r ) L δ \mathcal{L}_\delta L δ D δ D_\delta D δ r r r K δ K^\delta K δ h h h [48 Proposition 5.2] gives, for some ζ > 0 \zeta>0 ζ > 0
lim sup δ ↓ 0 P ( N δ ( h , r ) ) ≤ C ( h 1 / 2 + h ζ − 5 ) , 0 < h < r . (37) \limsup_{\delta\downarrow0}\mathbb{P}(\mathcal{N}_{\delta}(h,r))\le C(h^{1/2}+h^{\zeta-5}),\qquad0<h<r. \tag*{(37)} δ ↓ 0 lim sup P ( N δ ( h , r )) ≤ C ( h 1/2 + h ζ − 5 ) , 0 < h < r . (37) Its hypotheses are radial hittability of the curve and independence of the loop soup. In dimension two, the proof of [48 Proposition 5.2] applies with the planar Beurling estimate as its hittability input: the interpolated set K δ ∪ D c K^{\delta}\cup D^{c} K δ ∪ D c lim h ↓ 0 lim sup δ ↓ 0 P ( N δ ( h , r ) ) = 0 \lim_{h\downarrow0}\limsup_{\delta\downarrow0}\mathbb{P}(\mathcal{N}_{\delta}(h,r))=0 lim h ↓ 0 lim sup δ ↓ 0 P ( N δ ( h , r )) = 0 r > 0 r>0 r > 0
A limiting loop avoiding K K K K K K K δ K^{\delta} K δ δ ↓ 0 \delta\downarrow0 δ ↓ 0 h ↓ 0 h\downarrow0 h ↓ 0 D D D [48 Proposition 5.1] . Let X δ , r X_{\delta,r} X δ , r f ( t δ ( ℓ ) ) f(t_{\delta}(\ell)) f ( t δ ( ℓ )) L δ \mathcal{L}_{\delta} L δ [ u , v ] [u,v] [ u , v ] r r r D δ D_{\delta} D δ K δ K^{\delta} K δ X r X_r X r X δ , r → X r X_{\delta,r}\to X_r X δ , r → X r sup δ E ∣ X δ , r ∣ 2 + E ∣ X r ∣ 2 ≤ C D , u , v ∥ f ∥ ∞ 2 \sup_{\delta}\mathbb{E}|X_{\delta,r}|^2+\mathbb{E}|X_r|^2\le CD,u,v\|f\|_{\infty}^2 sup δ E ∣ X δ , r ∣ 2 + E ∣ X r ∣ 2 ≤ C D , u , v ∥ f ∥ ∞ 2 L 1 L^1 L 1 G = σ ( K δ , K ) \mathcal{G}=\sigma(K^{\delta},K) G = σ ( K δ , K )
E ∣ E [ X δ , r − X r ∣ G ] ∣ ≤ E ∣ X δ , r − X r ∣ ⟶ 0. (38) \mathbb{E}\left|\mathbb{E}[X_{\delta,r}-X_r\mid\mathcal{G}]\right|\le\mathbb{E}|X_{\delta,r}-X_r|\longrightarrow0. \tag*{(38)} E ∣ E [ X δ , r − X r ∣ G ] ∣ ≤ E ∣ X δ , r − X r ∣ ⟶ 0. (38) Loops of duration at least u u u r r r C r Cr C r C D ∫ u v t − d / 2 − 1 e − c t / r 2 d t → 0 C_D\int_u^v t^{-d/2-1}e^{-ct/r^2}\,dt\to0 C D ∫ u v t − d /2 − 1 e − c t / r 2 d t → 0 δ ↓ 0 \delta\downarrow0 δ ↓ 0 [ u , v ] [u,v] [ u , v ]
a δ n δ 2 − 1 = e ν δ n δ 2 − 1 , ν δ = δ − 2 log ( 1 + m 2 δ 2 ) ⟶ m 2 . (39) a_{\delta}^{n\delta^2}-1=e^{\nu_{\delta}n\delta^2}-1,\qquad\nu_{\delta}=\delta^{-2}\log(1+m^2\delta^2)\longrightarrow m^2. \tag*{(39)} a δ n δ 2 − 1 = e ν δ n δ 2 − 1 , ν δ = δ − 2 log ( 1 + m 2 δ 2 ) ⟶ m 2 . (39) The factor two in the even-step local central limit theorem cancels the duration spacing 2 δ 2 2\delta^2 2 δ 2
Short loops
For loops of duration t t t t \sqrt{t} t t t t [7 Proposition 6.13] .
Lemma 4.2. For 0 < α < 2 0<\alpha<2 0 < α < 2
H D ( t ) − H D ∖ K ( t ) ≤ C M α ( K ) t − α / 2 , 0 < t ≤ 1 , (40) H_D(t)-H_{D\setminus K}(t)\le CM_{\alpha}(K)t^{-\alpha/2},\qquad0<t\le1, \tag*{(40)} H D ( t ) − H D ∖ K ( t ) ≤ C M α ( K ) t − α /2 , 0 < t ≤ 1 , (40) I m , T ( K ) ≤ C d , α m 2 e m 2 T M α ( K ) T 1 − α / 2 , 0 < T ≤ 1 , (41) I_{m,T}(K)\le C_{d,\alpha}m^2e^{m^2T}M_{\alpha}(K)T^{1-\alpha/2},\qquad0<T\le1, \tag*{(41)} I m , T ( K ) ≤ C d , α m 2 e m 2 T M α ( K ) T 1 − α /2 , 0 < T ≤ 1 , (41) I δ , m , T ( K δ ) ≤ C d , α m 2 e m 2 T M α , δ ( K δ ) T 1 − α / 2 , 0 < T ≤ 1. (42) I_{\delta,m,T}(K^{\delta})\le C_{d,\alpha}m^2e^{m^2T}M_{\alpha,\delta}(K^{\delta})T^{1-\alpha/2},\qquad0<T\le1. \tag*{(42)} I δ , m , T ( K δ ) ≤ C d , α m 2 e m 2 T M α , δ ( K δ ) T 1 − α /2 , 0 < T ≤ 1. (42) Proof. For an r r r Q Q Q r 2 ≤ n δ 2 < 4 r 2 r^2\le n\delta^2<4r^2 r 2 ≤ n δ 2 < 4 r 2 τ Q \tau_Q τ Q
∑ x P x ( S n = x , S [ 0 , n ] ∩ Q ≠ ∅ ) ≤ C n − d / 2 ∑ x P x ( τ Q ≤ n / 2 ) ≤ C . \sum_x\mathbb{P}^x(S_n=x,\ S[0,n]\cap Q\ne\varnothing)\le Cn^{-d/2}\sum_x\mathbb{P}^x(\tau_Q\le n/2)\le C. x ∑ P x ( S n = x , S [ 0 , n ] ∩ Q = ∅ ) ≤ C n − d /2 x ∑ P x ( τ Q ≤ n /2 ) ≤ C . The last sum is O ( n d / 2 ) O(n^{d/2}) O ( n d /2 )
0 ≤ Tr ( p δ , D δ n ) − Tr ( p δ , D δ ∖ K δ n ) ≤ C N r ( K δ ) , r 2 ≤ n δ 2 < 4 r 2 . (43) 0\le\operatorname{Tr}(p_{\delta,D_{\delta}}^n)-\operatorname{Tr}(p_{\delta,D_{\delta}\setminus K^{\delta}}^n)\le CN_r(K^{\delta}),\qquad r^2\le n\delta^2<4r^2. \tag*{(43)} 0 ≤ Tr ( p δ , D δ n ) − Tr ( p δ , D δ ∖ K δ n ) ≤ C N r ( K δ ) , r 2 ≤ n δ 2 < 4 r 2 . (43) The identical bridge argument in physical time gives H D ( t ) − H D ∖ K ( t ) ≤ C N r ( K ) H_D(t)-H_{D\setminus K}(t) \le CN_r(K) H D ( t ) − H D ∖ K ( t ) ≤ C N r ( K ) r ≍ t r \asymp\sqrt{t} r ≍ t (40) . Integrate against ( e m 2 t − 1 ) d t / t ≤ m 2 e m 2 T d t (e^{m^2t}-1)\,\mathrm{d}t/t \le m^2e^{m^2T}\,\mathrm{d}t ( e m 2 t − 1 ) d t / t ≤ m 2 e m 2 T d t (41) . For (42) use ( a δ n − 1 ) / n ≤ m 2 δ 2 e m 2 n δ 2 (a_\delta^n-1)/n \le m^2\delta^2e^{m^2n\delta^2} ( a δ n − 1 ) / n ≤ m 2 δ 2 e m 2 n δ 2
δ 2 ∑ n ≤ T / δ 2 ( n δ 2 ) − α / 2 ≤ C α T 1 − α / 2 . \delta^2\sum_{n\le T/\delta^2}(n\delta^2)^{-\alpha/2}\le C_\alpha T^{1-\alpha/2}. δ 2 n ≤ T / δ 2 ∑ ( n δ 2 ) − α /2 ≤ C α T 1 − α /2 . Loops with a bounded number of steps are covered at the mesh scale δ \delta δ 2 δ 2 2\delta^2 2 δ 2
Since the coefficient of the covering content tends to zero with T T T
Long loops and the spectral cutoff
We first show that changing the cutoff multiplies the weight by a deterministic factor and that positivity of the weight is decided by the slit-domain eigenvalue; we then control long loops in the limit.
Proposition 4.3. For compact K K K M α ( K ) < ∞ M_\alpha(K)<\infty M α ( K ) < ∞ α < 2 \alpha<2 α < 2 I m , T ( K ) < ∞ I_{m,T}(K)<\infty I m , T ( K ) < ∞ m ≥ 0 m\ge0 m ≥ 0 T > 0 T>0 T > 0 0 < T 1 < T 2 0<T_1<T_2 0 < T 1 < T 2 m > 0 m>0 m > 0
Y m , T ( K ) > 0 ⟺ m 2 < λ 1 ( D ∖ K ) . (44) Y_{m,T}(K)>0\quad\Longleftrightarrow\quad m^2<\lambda_1(D\setminus K). \tag*{(44)} Y m , T ( K ) > 0 ⟺ m 2 < λ 1 ( D ∖ K ) . (44) Proof. Short-loop finiteness follows from (41) ; any remaining finite duration interval has finite loop mass in D D D T 1 T_1 T 1 T 2 T_2 T 2 I I I J J J D D D J J J U = D ∖ K ≠ ∅ U=D\setminus K\ne\varnothing U = D ∖ K = ∅
e − t λ 1 ( U ) ≤ H U ( t ) ≤ H U ( 1 ) e − ( t − 1 ) λ 1 ( U ) , t ≥ 1. e^{-t\lambda_1(U)}\le H_U(t)\le H_U(1)e^{-(t-1)\lambda_1(U)},\qquad t\ge1. e − t λ 1 ( U ) ≤ H U ( t ) ≤ H U ( 1 ) e − ( t − 1 ) λ 1 ( U ) , t ≥ 1. Thus J m , T J_{m,T} J m , T m 2 < λ 1 ( U ) m^2<\lambda_1(U) m 2 < λ 1 ( U ) ∫ ∞ ( 1 − e − m 2 t ) d t / t \int^\infty(1-e^{-m^2t})\,\mathrm{d}t/t ∫ ∞ ( 1 − e − m 2 t ) d t / t
The next lemma controls the long-loop tail and proves convergence of the weights, including at the spectral threshold.
Lemma 4.4. Suppose (4.1) holds and Q δ : = ∣ γ δ ∣ log a δ + I δ , m , T ( K δ ) Q_\delta:=|\gamma^\delta|\log a_\delta+I_{\delta,m,T}(K^\delta) Q δ := ∣ γ δ ∣ log a δ + I δ , m , T ( K δ ) I m , T ( K ) I_{m,T}(K) I m , T ( K ) Y δ , m , T → Y m , T Y_{\delta,m,T}\to Y_{m,T} Y δ , m , T → Y m , T { λ 1 ( D ∖ K ) = m 2 } \{\lambda_1(D\setminus K)=m^2\} { λ 1 ( D ∖ K ) = m 2 }
Proof. The case m = 0 m=0 m = 0 U = D ∖ K U=D\setminus K U = D ∖ K U δ = D δ ∖ K δ U_\delta=D_\delta\setminus K^\delta U δ = D δ ∖ K δ p δ = p δ , U δ p_\delta=p_{\delta,U_\delta} p δ = p δ , U δ ρ δ = ρ ( p δ ) \rho_\delta=\rho(p_\delta) ρ δ = ρ ( p δ ) E κ = { λ 1 ( U ) ≥ m 2 + 4 κ } E_\kappa=\{\lambda_1(U)\ge m^2+4\kappa\} E κ = { λ 1 ( U ) ≥ m 2 + 4 κ } κ > 0 \kappa>0 κ > 0
∫ S S + 1 H U ( t ) d t t ≤ C D S e − ( m 2 + 4 κ ) ( S − 1 ) , S ≥ 2. \int_S^{S+1}H_U(t)\,\frac{\mathrm{d}t}{t}\le\frac{C_D}{S}e^{-(m^2+4\kappa)(S-1)},\qquad S\ge2. ∫ S S + 1 H U ( t ) t d t ≤ S C D e − ( m 2 + 4 κ ) ( S − 1 ) , S ≥ 2. If ρ δ ≥ e − ( m 2 + 2 κ ) δ 2 \rho_\delta\ge e^{-(m^2+2\kappa)\delta^2} ρ δ ≥ e − ( m 2 + 2 κ ) δ 2
∑ S ≤ n δ 2 ≤ S + 1 Tr ( p δ n ) n ≥ c S + 1 e − ( m 2 + 2 κ ) ( S + 1 ) . \sum_{S\le n\delta^2\le S+1}\frac{\operatorname{Tr}(p_\delta^n)}{n}\ge\frac{c}{S+1}e^{-(m^2+2\kappa)(S+1)}. S ≤ n δ 2 ≤ S + 1 ∑ n Tr ( p δ n ) ≥ S + 1 c e − ( m 2 + 2 κ ) ( S + 1 ) . Indeed, for even n n n Tr ( p δ n ) ≥ ρ δ n \operatorname{Tr}(p_\delta^n) \geq\rho_\delta^n Tr ( p δ n ) ≥ ρ δ n δ − 2 \delta^{-2} δ − 2 1 / n ≍ δ 2 / ( S + 1 ) 1/n \asymp\delta^2/(S+1) 1/ n ≍ δ 2 / ( S + 1 ) S S S
P ( E κ ∩ { ρ δ ≥ e − ( m 2 + 2 κ ) δ 2 } ) ⟶ 0. (45) \mathbb{P}\left(E_\kappa\cap\left\{\rho_\delta\geq e^{-(m^2+2\kappa)\delta^2}\right\}\right) \longrightarrow0. \tag*{(45)} P ( E κ ∩ { ρ δ ≥ e − ( m 2 + 2 κ ) δ 2 } ) ⟶ 0. (45) On E κ E_\kappa E κ n 0 n_0 n 0 n 0 δ 2 ∈ [ 1 , 1 + 2 δ 2 ] n_0\delta^2 \in[1,1+2\delta^2] n 0 δ 2 ∈ [ 1 , 1 + 2 δ 2 ] Tr ( p δ n 0 ) ≤ C D \operatorname{Tr}(p_\delta^{n_0}) \leq C_D Tr ( p δ n 0 ) ≤ C D n ≥ n 0 n \geq n_0 n ≥ n 0
Tr ( p δ n ) ≤ ρ δ n − n 0 Tr ( p δ n 0 ) ≤ C D , m , κ e − ( m 2 + 2 κ ) n δ 2 . \operatorname{Tr}(p_\delta^n) \leq\rho_\delta^{n-n_0}\operatorname{Tr}(p_\delta^{n_0}) \leq C_{D,m,\kappa}e^{-(m^2+2\kappa)n\delta^2}. Tr ( p δ n ) ≤ ρ δ n − n 0 Tr ( p δ n 0 ) ≤ C D , m , κ e − ( m 2 + 2 κ ) n δ 2 . Since odd traces vanish and a δ n ≤ e m 2 n δ 2 a_\delta^n \leq e^{m^2n\delta^2} a δ n ≤ e m 2 n δ 2 R ≥ 2 R \geq2 R ≥ 2
C ∑ n δ 2 > R e − 2 κ n δ 2 n ≤ C ∫ R / 2 ∞ e − κ t d t t . C\sum_{n\delta^2>R}\frac{e^{-2\kappa n\delta^2}}{n}
\leq C\int_{R/2}^{\infty}e^{-\kappa t}\frac{\mathrm{d}t}{t}. C n δ 2 > R ∑ n e − 2 κn δ 2 ≤ C ∫ R /2 ∞ e − κ t t d t . The continuum tail has the analogous bound.
For R > T R>T R > T
J δ , m , T , R = ∑ T < n δ 2 ≤ R a δ n − 1 n Tr ( p δ n ) , J_{\delta,m,T,R}=\sum_{T<n\delta^2\leq R}\frac{a_\delta^n-1}{n}\operatorname{Tr}(p_\delta^n), J δ , m , T , R = T < n δ 2 ≤ R ∑ n a δ n − 1 Tr ( p δ n ) , J m , T , R = ∫ T R ( e m 2 t − 1 ) H U ( t ) d t t . (46) J_{m,T,R}=\int_T^R(e^{m^2t}-1)H_U(t)\frac{\mathrm{d}t}{t}. \tag*{(46)} J m , T , R = ∫ T R ( e m 2 t − 1 ) H U ( t ) t d t . (46) At fixed R R R δ ↓ 0 \delta\downarrow0 δ ↓ 0 R ↑ ∞ R\uparrow\infty R ↑ ∞ J J J E κ E_\kappa E κ Q δ → I m , T Q_\delta\to I_{m,T} Q δ → I m , T E κ E_\kappa E κ κ ↓ 0 \kappa\downarrow0 κ ↓ 0 { λ 1 ( U ) > m 2 } \{\lambda_1(U)>m^2\} { λ 1 ( U ) > m 2 }
On E 0 = { λ 1 ( U ) ≤ m 2 } E_0=\{\lambda_1(U)\leq m^2\} E 0 = { λ 1 ( U ) ≤ m 2 } J δ , m , T ≥ J δ , m , T , R J_{\delta,m,T}\geq J_{\delta,m,T,R} J δ , m , T ≥ J δ , m , T , R 0 < ϵ < 1 0<\epsilon<1 0 < ϵ < 1 h = log ( 1 / ϵ ) h=\log(1/\epsilon) h = log ( 1/ ϵ )
P ( E 0 ∩ { Y δ , m , T > ϵ } ) ≤ P ( Q δ > M ) + P ( E 0 ∩ { J m , T , R ≤ M + h + 1 } ) + P ( ∣ J δ , m , T , R − J m , T , R ∣ > 1 ) . (47) \mathbb{P}(E_0\cap\{Y_{\delta,m,T}>\epsilon\})\leq\mathbb{P}(Q_\delta>M)+\mathbb{P}(E_0\cap\{J_{m,T,R}\leq M+h+1\})
+\mathbb{P}(\lvert J_{\delta,m,T,R}-J_{m,T,R}\rvert>1). \tag*{(47)} P ( E 0 ∩ { Y δ , m , T > ϵ }) ≤ P ( Q δ > M ) + P ( E 0 ∩ { J m , T , R ≤ M + h + 1 }) + P (∣ J δ , m , T , R − J m , T , R ∣ > 1 ) . (47) After δ ↓ 0 \delta\downarrow0 δ ↓ 0 M M M R R R □ \square □
Uniform integrability and convergence of the densities
We now combine the three duration regimes and prove uniform integrability, which gives convergence of the boosted laws.
Proof of Theorems 3.6 and 1.2. Use the covering and positive-support hypotheses of Theorem 3.6; in the radial models they follow from Theorem 3.4 and Proposition 3.5. Couple η δ → η \eta^\delta\to\eta η δ → η K δ , K K^\delta,K K δ , K M α , δ ( K δ ) M_{\alpha,\delta}(K^\delta) M α , δ ( K δ ) M α , δ M_{\alpha,\delta} M α , δ (33) gives tightness and bounded first moments of these contents. Self-avoidance bounds the number of vertices in each mesh-scale box, giving
∣ γ δ ∣ log a δ ≤ C m 2 δ 2 − α M α , δ ⟶ 0 \lvert\gamma^\delta\rvert\log a_\delta\leq Cm^{2}\delta^{2-\alpha}M_{\alpha,\delta}\longrightarrow0 ∣ γ δ ∣ log a δ ≤ C m 2 δ 2 − α M α , δ ⟶ 0 in probability.
At each fixed scale, Hausdorff convergence covers K K K K δ K^\delta K δ
M α ( K ) ≤ C α lim inf δ ↓ 0 M α , δ (48) M_\alpha(K)\leq C_\alpha\liminf_{\delta\downarrow0}M_{\alpha,\delta} \tag*{(48)} M α ( K ) ≤ C α δ ↓ 0 lim inf M α , δ (48) almost surely. Fatou’s lemma gives E e ( c / C α ) M α ( K ) < ∞ \mathbb{E}e^{(c/C_{\alpha})M_{\alpha}(K)}<\infty E e ( c / C α ) M α ( K ) < ∞
For 0 < u < T 0<u<T 0 < u < T
∣ I δ , m , T − I m , T ∣ ≤ I δ , m , u + I m , u + ∣ ( I δ , m , T − I δ , m , u ) − ( I m , T − I m , u ) ∣ . (49) |I_{\delta,m,T}-I_{m,T}|\le I_{\delta,m,u}+I_{m,u}+|(I_{\delta,m,T}-I_{\delta,m,u})-(I_{m,T}-I_{m,u})|. \tag*{(49)} ∣ I δ , m , T − I m , T ∣ ≤ I δ , m , u + I m , u + ∣ ( I δ , m , T − I δ , m , u ) − ( I m , T − I m , u ) ∣. (49) The first two terms have expectation at most C m u 1 − α / 2 C_m u^{1-\alpha/2} C m u 1 − α /2 δ ↓ 0 \delta\downarrow0 δ ↓ 0 u ↓ 0 u\downarrow0 u ↓ 0 I δ , m , T → I m , T I_{\delta,m,T}\to I_{m,T} I δ , m , T → I m , T { Y m , T = 0 } \{Y_{m,T}=0\} { Y m , T = 0 }
Fix p > 1 p>1 p > 1 0 < T ≤ 1 0<T\le1 0 < T ≤ 1 p C d , α m 2 e m 2 T T 1 − α / 2 < c / 2 pC_{d,\alpha}m^2e^{m^2T}T^{1-\alpha/2}<c/2 p C d , α m 2 e m 2 T T 1 − α /2 < c /2 δ 0 \delta_0 δ 0 p C m 2 δ 0 2 − α < c / 2 pCm^2\delta_0^{2-\alpha}<c/2 pC m 2 δ 0 2 − α < c /2
Y δ , m , T p ≤ exp { p [ C m 2 δ 2 − α + C d , α m 2 e m 2 T T 1 − α / 2 ] M α , δ } ≤ e c M α , δ . Y_{\delta,m,T}^{p}\le\exp\{p[Cm^2\delta^{2-\alpha}+C_{d,\alpha}m^2e^{m^2T}T^{1-\alpha/2}]M_{\alpha,\delta}\}\le e^{cM_{\alpha,\delta}}. Y δ , m , T p ≤ exp { p [ C m 2 δ 2 − α + C d , α m 2 e m 2 T T 1 − α /2 ] M α , δ } ≤ e c M α , δ . Consequently
sup 0 < δ < δ 0 E Y δ , m , T p < ∞ . (50) \sup_{0<\delta<\delta_0}\mathbb{E}Y_{\delta,m,T}^{p}<\infty. \tag*{(50)} 0 < δ < δ 0 sup E Y δ , m , T p < ∞. (50) and convergence is in L 1 L^1 L 1
Z δ , m , T : = E Y δ , m , T ⟶ Z m , T : = E Y m , T ∈ ( 0 , ∞ ) . (51) Z_{\delta,m,T}:=\mathbb{E}Y_{\delta,m,T}\longrightarrow Z_{m,T}:=\mathbb{E}Y_{m,T}\in(0,\infty). \tag*{(51)} Z δ , m , T := E Y δ , m , T ⟶ Z m , T := E Y m , T ∈ ( 0 , ∞ ) . (51) For m = 0 m=0 m = 0 Y δ , m , T Y_{\delta,m,T} Y δ , m , T W δ , m W_{\delta,m} W δ , m
For bounded continuous F F F
∣ E [ F ( η δ ) Y δ , m , T ] − E [ F ( η ) Y m , T ] ∣ ≤ ∥ F ∥ ∞ E ∣ Y δ , m , T − Y m , T ∣ + E [ ∣ F ( η δ ) − F ( η ) ∣ Y m , T ] ⟶ 0. |\mathbb{E}[F(\eta^\delta)Y_{\delta,m,T}]-\mathbb{E}[F(\eta)Y_{m,T}]|
\le\|F\|_{\infty}\mathbb{E}|Y_{\delta,m,T}-Y_{m,T}|+\mathbb{E}[|F(\eta^\delta)-F(\eta)|Y_{m,T}]\longrightarrow0. ∣ E [ F ( η δ ) Y δ , m , T ] − E [ F ( η ) Y m , T ] ∣ ≤ ∥ F ∥ ∞ E ∣ Y δ , m , T − Y m , T ∣ + E [ ∣ F ( η δ ) − F ( η ) ∣ Y m , T ] ⟶ 0. The first term uses L 1 L^1 L 1
E P δ ( m ) F = E [ F ( η δ ) Y δ , m , T ] Z δ , m , T ⟶ E [ F ( η ) Y m , T ] Z m , T = E P ( m ) F . \mathbb{E}_{\mathbb{P}^{(m)}_{\delta}}F
=\frac{\mathbb{E}[F(\eta^\delta)Y_{\delta,m,T}]}{Z_{\delta,m,T}}
\longrightarrow
\frac{\mathbb{E}[F(\eta)Y_{m,T}]}{Z_{m,T}}
=\mathbb{E}_{\mathbb{P}^{(m)}}F. E P δ ( m ) F = Z δ , m , T E [ F ( η δ ) Y δ , m , T ] ⟶ Z m , T E [ F ( η ) Y m , T ] = E P ( m ) F . The cutoff identity (1.11) multiplies the density and its mean by the same deterministic factor. Its discrete counterpart and convergence for loops in D D D L 1 L^1 L 1 T > 0 T>0 T > 0 m 2 < λ 1 ( D ) m^2<\lambda_1(D) m 2 < λ 1 ( D ) E e a δ 2 T δ < ∞ \mathbb{E}e^{a\delta^2T_\delta}<\infty E e a δ 2 T δ < ∞ δ \delta δ
Absolute continuity and geometric consequences
We now identify the subcritical density, determine the regimes of absolute continuity, and bound the size of unvisited regions.
The subcritical density
Below the spectral threshold of D D D D D D
Lemma 5.1. Let T D T_D T D Δ / ( 2 d ) \Delta/(2d) Δ/ ( 2 d ) T δ T_\delta T δ a < λ 1 ( D ) a<\lambda_1(D) a < λ 1 ( D ) δ 0 ( a ) > 0 \delta_0(a)>0 δ 0 ( a ) > 0
sup 0 < δ ≤ δ 0 ( a ) sup x ∈ D δ E x e a δ 2 T δ < ∞ , E 0 e a δ 2 T δ ⟶ E 0 e a T D . \sup_{0<\delta\le\delta_0(a)}\sup_{x\in D_\delta}\mathbb{E}^x e^{a\delta^2T_\delta}<\infty,\qquad
\mathbb{E}^0e^{a\delta^2T_\delta}\longrightarrow\mathbb{E}^0e^{aT_D}. 0 < δ ≤ δ 0 ( a ) sup x ∈ D δ sup E x e a δ 2 T δ < ∞ , E 0 e a δ 2 T δ ⟶ E 0 e a T D . Here δ 2 T δ \delta^2T_\delta δ 2 T δ
Proof. Fix max { a , 0 } < c < c ′ < λ 1 ( D ) \max\{a,0\}<c<c'<\lambda_1(D) max { a , 0 } < c < c ′ < λ 1 ( D ) sup x P x ( T D > T ) < e − c ′ T \sup_x \mathbb{P}^x(T_D>T)<e^{-c'T} sup x P x ( T D > T ) < e − c ′ T T T T sup x P x ( T δ > ⌊ T / δ 2 ⌋ ) ≤ e − c T \sup_x \mathbb{P}^x(T_\delta>\lfloor T/\delta^2\rfloor)\le e^{-cT} sup x P x ( T δ > ⌊ T / δ 2 ⌋) ≤ e − c T δ \delta δ δ n ↓ 0 \delta_n\downarrow0 δ n ↓ 0 x n ∈ D δ n x_n\in D_{\delta_n} x n ∈ D δ n x n x_n x n D D D
sup x ∈ D δ P x ( δ 2 T δ > t ) ≤ C c e − c t , t ≥ 0. (52) \sup_{x\in D_\delta}\mathbb{P}^x(\delta^2T_\delta>t)\le C_c e^{-ct},\qquad t\ge0. \tag*{(52)} x ∈ D δ sup P x ( δ 2 T δ > t ) ≤ C c e − c t , t ≥ 0. (52) Integration gives the moment bound. Using an exponent between max { a , 0 } \max\{a,0\} max { a , 0 } c c c
Proof of Theorem 1.3. Fix m 2 < λ 1 ( D ) m^2<\lambda_1(D) m 2 < λ 1 ( D ) D D D Φ m ( K ) < ∞ \Phi_m(K)<\infty Φ m ( K ) < ∞
log Y m , T ( K ) = Φ m ( K ) − ∫ 0 ∞ ( e m 2 t − 1 ) H D ( t ) d t t . \log Y_{m,T}(K)=\Phi_m(K)-\int_0^\infty\left(e^{m^2t}-1\right)H_D(t)\,\frac{\mathrm{d}t}{t}. log Y m , T ( K ) = Φ m ( K ) − ∫ 0 ∞ ( e m 2 t − 1 ) H D ( t ) t d t . Lemma 5.1 makes the boosted walk normalizable for small δ \delta δ
w δ , m = E 0 [ a δ T δ ∣ γ δ ] = Y δ , m , T exp { ∑ n δ 2 > T a δ n − 1 n Tr ( p δ , D δ n ) } . w_{\delta,m}=\mathbb{E}^0\left[a_\delta^{T_\delta}\mid\gamma^\delta\right]
=Y_{\delta,m,T}\exp\left\{\sum_{n\delta^2>T}\frac{a_\delta^n-1}{n}\operatorname{Tr}(p_{\delta,D_\delta}^n)\right\}. w δ , m = E 0 [ a δ T δ ∣ γ δ ] = Y δ , m , T exp { n δ 2 > T ∑ n a δ n − 1 Tr ( p δ , D δ n ) } . The deterministic sum converges to the Brownian integral. Finite windows follow from Lemma 4.1; for the tail, choose m 2 < c < λ 1 ( D ) m^2<c<\lambda_1(D) m 2 < c < λ 1 ( D ) Tr ( p δ , D δ n ) ≤ C e − c n δ 2 \operatorname{Tr}(p_{\delta,D_\delta}^n)\le Ce^{-cn\delta^2} Tr ( p δ , D δ n ) ≤ C e − c n δ 2 n δ 2 ≥ 2 n\delta^2\ge2 n δ 2 ≥ 2 δ − 2 \delta^{-2} δ − 2 (52) , uniform in the starting vertex, to the other. The Brownian tail follows from the killed semigroup. Hence § 4 gives w δ , m → e Φ m ( K ) w_{\delta,m}\to e^{\Phi_m(K)} w δ , m → e Φ m ( K )
For p > 1 p>1 p > 1 p m 2 < λ 1 ( D ) pm^2<\lambda_1(D) p m 2 < λ 1 ( D )
E P δ w δ , m p ≤ E 0 a δ p T δ ≤ E 0 e p m 2 δ 2 T δ ≤ C p , m . \mathbb{E}_{\mathbb{P}_\delta}w_{\delta,m}^p\le\mathbb{E}^0a_\delta^{pT_\delta}\le\mathbb{E}^0e^{pm^2\delta^2T_\delta}\le C_{p,m}. E P δ w δ , m p ≤ E 0 a δ p T δ ≤ E 0 e p m 2 δ 2 T δ ≤ C p , m . Since ν δ = δ − 2 log a δ → m 2 \nu_\delta=\delta^{-2}\log a_\delta\to m^2 ν δ = δ − 2 log a δ → m 2
E P e Φ m ( K ) = lim δ ↓ 0 E 0 a δ T δ = E 0 e m 2 T D . \mathbb{E}_{\mathbb{P}}e^{\Phi_m(K)}
=\lim_{\delta\downarrow0}\mathbb{E}^0a_\delta^{T_\delta}
=\mathbb{E}^0e^{m^2T_D}. E P e Φ m ( K ) = δ ↓ 0 lim E 0 a δ T δ = E 0 e m 2 T D . This proves (12) . Fatou gives the L p L^p L p
Spectral support and absolute continuity regimes
We identify the essential infimum of the slit-domain eigenvalue and show that the eigenvalue lies strictly above it almost surely in the following proposition, proved in Appendix § 8 .
Proposition 5.2. Let X = λ 1 ( D ∖ K ) X=\lambda_1(D\setminus K) X = λ 1 ( D ∖ K ) ess inf X = λ 1 ( D ) \operatorname{ess\,inf}X=\lambda_1(D) ess inf X = λ 1 ( D ) P ( X > λ 1 ( D ) ) = 1 \mathbb{P}(X>\lambda_1(D))=1 P ( X > λ 1 ( D )) = 1
In the standard planar radial setting, Λ 2 : = ess inf X = π 2 / 4 \Lambda_2:=\operatorname{ess\,inf}X=\pi^2/4 Λ 2 := ess inf X = π 2 /4
Proof of Corollary 1.4. The density formula gives P ( m ) ≪ P \mathbb{P}^{(m)}\ll\mathbb{P} P ( m ) ≪ P
P ≪ P ( m ) ⟺ P { Y m , T > 0 } = 1 ⟺ P ( X > m 2 ) = 1. \mathbb{P}\ll\mathbb{P}^{(m)}
\quad\Longleftrightarrow\quad
\mathbb{P}\{Y_{m,T}>0\}=1
\quad\Longleftrightarrow\quad
\mathbb{P}(X>m^2)=1. P ≪ P ( m ) ⟺ P { Y m , T > 0 } = 1 ⟺ P ( X > m 2 ) = 1. Proposition 5.2 gives both equivalences, including their endpoints.
Unvisited regions at large boost
Proof of Corollary 1.5. Theorem 1.2 gives m 2 < λ 1 ( D ∖ K ) m^{2}<\lambda_{1}(D\setminus K) m 2 < λ 1 ( D ∖ K ) P ( m ) \mathbb{P}^{(m)} P ( m ) Q ⊂ D ∖ K Q\subset D\setminus K Q ⊂ D ∖ K − Δ / ( 2 d ) -\Delta/(2d) − Δ/ ( 2 d )
m 2 < λ 1 ( D ∖ K ) ≤ λ 1 ( Q ) = π 2 2 side ( Q ) 2 . m^{2}<\lambda_{1}(D\setminus K)\leq\lambda_{1}(Q)=\frac{\pi^{2}}{2\operatorname{side}(Q)^{2}}. m 2 < λ 1 ( D ∖ K ) ≤ λ 1 ( Q ) = 2 side ( Q ) 2 π 2 . This proves (1.13), simultaneously for all such cubes. If m > M ( ℓ ) m>M(\ell) m > M ( ℓ ) ℓ \ell ℓ □ \square □
The chordal driving function in two dimensions
In this section, we construct a continuum chordal law using the loop-decoration theorem of Berestycki–Sauzedde [7 ] , and then identify the drift of its driving function throughout the subcritical range q < λ 1 ( D ) q<\lambda_{1}(D) q < λ 1 ( D ) § 1 , the calculation follows the martingale strategy of Makarov–Smirnov [40 ] and Chelkak–Wan [15 ] ; its additional input is a Green-kernel comparison uniform over the evolving slits.
Continuum tilt and the driving statement
Let D D D a , b ∈ ∂ D a,b\in\partial D a , b ∈ ∂ D P D ; a , b \mathbb{P}_{D;a,b} P D ; a , b SLE 2 \operatorname{SLE}_{2} SLE 2 η \eta η K K K L = Δ / 4 L=\Delta/4 L = Δ/4 A U = − Δ U / 4 A_{U}=-\Delta_{U}/4 A U = − Δ U /4 G U = A U − 1 G_{U}=A_{U}^{-1} G U = A U − 1 q = m 2 < λ 1 ( D ) q=m^{2}<\lambda_{1}(D) q = m 2 < λ 1 ( D )
V m ( K ) = ∫ ℓ ∈ L D ( e q t ℓ − 1 ) 1 { ℓ ∩ K ≠ ∅ } μ D ( d ℓ ) . (53) V_{m}(K)=\int_{\ell\in\mathcal{L}_{D}}\left(e^{q t_{\ell}}-1\right)\mathbf{1}_{\{\ell\cap K\neq\varnothing\}}\mu_{D}(\mathrm{d}\ell). \tag*{(53)} V m ( K ) = ∫ ℓ ∈ L D ( e q t ℓ − 1 ) 1 { ℓ ∩ K = ∅ } μ D ( d ℓ ) . (53) Here t ℓ t_{\ell} t ℓ ϕ : D → H \phi:D\to\mathbb{H} ϕ : D → H ϕ ( a ) = 0 \phi(a)=0 ϕ ( a ) = 0 ϕ ( b ) = ∞ \phi(b)=\infty ϕ ( b ) = ∞ Im ( ϕ ( o ) ) = 1 \operatorname{Im}(\phi(o))=1 Im ( ϕ ( o )) = 1 o ∈ D o\in D o ∈ D
g t ( z ) = z + 2 t z + O ( ∣ z ∣ − 2 ) , ∂ t g t ( z ) = 2 g t ( z ) − ξ t . (54) g_{t}(z)=z+\frac{2t}{z}+O(|z|^{-2}),\qquad\partial_{t}g_{t}(z)=\frac{2}{g_{t}(z)-\xi_{t}}. \tag*{(54)} g t ( z ) = z + z 2 t + O ( ∣ z ∣ − 2 ) , ∂ t g t ( z ) = g t ( z ) − ξ t 2 . (54) Thus hcap ( ϕ ( η [ 0 , t ] ) ) = 2 t \operatorname{hcap}(\phi(\eta[0,t]))=2t hcap ( ϕ ( η [ 0 , t ])) = 2 t D t = D ∖ η [ 0 , t ] D_{t}=D\setminus\eta[0,t] D t = D ∖ η [ 0 , t ] a t = η ( t ) a_{t}=\eta(t) a t = η ( t ) f t = ( g t − ξ t ) ∘ ϕ f_{t}=(g_{t}-\xi_{t})\circ\phi f t = ( g t − ξ t ) ∘ ϕ
The normalizer is an exponential moment of the duration of a Brownian excursion.
Proposition 6.1. For q < λ 1 ( D ) q<\lambda_{1}(D) q < λ 1 ( D )
Z m ( D ; a , b ) : = E D ; a , b e V m ( K ) = E D ; a , b e x c e q T < ∞ , (55) Z_{m}(D;a,b):=\mathbb{E}_{D;a,b}e^{V_{m}(K)}=\mathbb{E}^{\mathrm{exc}}_{D;a,b}e^{qT}<\infty, \tag*{(55)} Z m ( D ; a , b ) := E D ; a , b e V m ( K ) = E D ; a , b exc e q T < ∞ , (55) where T T T
d P D ; a , b ( m ) d P D ; a , b ( K ) = e V m ( K ) Z m ( D ; a , b ) (56) \frac{\mathrm{d}\mathbb{P}^{(m)}_{D;a,b}}{\mathrm{d}\mathbb{P}_{D;a,b}}(K)=\frac{e^{V_{m}(K)}}{Z_{m}(D;a,b)} \tag*{(56)} d P D ; a , b d P D ; a , b ( m ) ( K ) = Z m ( D ; a , b ) e V m ( K ) (56) is equivalent to SLE 2 \operatorname{SLE}_{2} SLE 2 L p L^{p} L p p > 1 p>1 p > 1 p q < λ 1 ( D ) pq<\lambda_{1}(D) pq < λ 1 ( D )
Proof. The chordal extension of [7 Theorem 1.1, Section 1.1] attaches an independent intensity-one loop soup to SLE 2 \operatorname{SLE}_{2} SLE 2 ( H ; 0 , ∞ ) (\mathbb{H};0,\infty) ( H ; 0 , ∞ ) [7 Section 1.2] is d x d t / ( 2 π t 2 ) \mathrm{d}x\,\mathrm{d}t/(2\pi t^{2}) d x d t / ( 2 π t 2 ) Δ / 2 \Delta/2 Δ/2 D D D d x d t / ( π t 2 ) \mathrm{d}x\,\mathrm{d}t/(\pi t^{2}) d x d t / ( π t 2 ) Δ / 4 \Delta/4 Δ/4 μ D \mu_{D} μ D T T T E [ e q T ∣ K ] = e V m ( K ) \mathbb{E}[e^{qT}\mid K]=e^{V_m(K)} E [ e q T ∣ K ] = e V m ( K )
The excursion duration density is proportional to the inward normal derivative at both endpoints of the killed heat kernel. At large times it is bounded by C e − λ 1 ( D ) t Ce^{-\lambda_1(D)t} C e − λ 1 ( D ) t L 2 L^2 L 2 L 2 L^2 L 2 (55) is finite. Conditional Jensen gives E e p V m ( K ) ≤ E e x c e p q T < ∞ \mathbb{E}e^{pV_m(K)}\leq\mathbb{E}^{\mathrm{exc}}e^{pqT}<\infty E e p V m ( K ) ≤ E exc e pq T < ∞
Assume that D ⊂ B ( 0 , R ) D\subset B(0,R) D ⊂ B ( 0 , R ) C 0 C_0 C 0
ρ : = q λ 1 ( D ) < 1 , M R = C 0 R 2 . (57) \rho:=\frac{q}{\lambda_1(D)}<1,\qquad M_R=C_0R^2. \tag*{(57)} ρ := λ 1 ( D ) q < 1 , M R = C 0 R 2 . (57) Throughout the subcritical range q < λ 1 ( D ) q<\lambda_1(D) q < λ 1 ( D )
Theorem 6.2. For q < λ 1 ( D ) q<\lambda_1(D) q < λ 1 ( D ) 2 t 2t 2 t
d ξ t = 2 d B t ( m ) + 2 λ t d t , λ t = Q t ( m ) ( b ) P t ( m ) ( b ) , (58) d\xi_t=\sqrt{2}\,dB_t^{(m)}+2\lambda_t\,dt,\qquad\lambda_t=\frac{Q_t^{(m)}(b)}{P_t^{(m)}(b)}, \tag*{(58)} d ξ t = 2 d B t ( m ) + 2 λ t d t , λ t = P t ( m ) ( b ) Q t ( m ) ( b ) , (58) where B ( m ) B^{(m)} B ( m ) (73) .
The semimartingale assertion follows from Proposition 6.1 and Girsanov’s theorem, as in [15 Section 2.6] . Below we identify the drift using [15 Sections 4.1–4.3] . To translate their conventions, replace their operator − Δ + μ 2 -\Delta+\mu^2 − Δ + μ 2 − Δ − 4 q -\Delta-4q − Δ − 4 q
Boosted Green and Poisson kernels
We now introduce the boosted kernels and establish comparisons that are uniform over all slit domains inside D D D
For U ⊂ D U\subset D U ⊂ D q < λ 1 ( U ) q<\lambda_1(U) q < λ 1 ( U )
G U ( m ) = ( A U − q ) − 1 = G U + q G U G U ( m ) . (59) G_U^{(m)}=(A_U-q)^{-1}=G_U+qG_UG_U^{(m)}. \tag*{(59)} G U ( m ) = ( A U − q ) − 1 = G U + q G U G U ( m ) . (59) All kernels are extended by zero outside their domains.
The key estimate compares boosted and ordinary kernels, with a constant that does not depend on the shape of the domain.
Lemma 6.3. For every bounded planar open set U ⊂ B ( 0 , R ) U\subset B(0,R) U ⊂ B ( 0 , R ) x , y x,y x , y
∫ U G U ( x , w ) G U ( w , y ) d w ≤ M R G U ( x , y ) , M R = C 0 R 2 . (60) \int_U G_U(x,w)G_U(w,y)\,dw\leq M_RG_U(x,y),\qquad M_R=C_0R^2. \tag*{(60)} ∫ U G U ( x , w ) G U ( w , y ) d w ≤ M R G U ( x , y ) , M R = C 0 R 2 . (60) If, in addition, U ⊂ D U\subset D U ⊂ D q < λ 1 ( D ) q<\lambda_1(D) q < λ 1 ( D )
G U ≤ G U ( m ) ≤ C m G U , C m = C m ( D , q ) < ∞ , (61) G_U\leq G_U^{(m)}\leq C_mG_U,\qquad C_m=C_m(D,q)<\infty, \tag*{(61)} G U ≤ G U ( m ) ≤ C m G U , C m = C m ( D , q ) < ∞ , (61) where C m C_m C m U U U
Proof. Bass–Burdzy [4 Theorem 1.1] bounds G U ( x , w ) G U ( w , y ) / G U ( x , y ) G_U(x,w)G_U(w,y)/G_U(x,y) G U ( x , w ) G U ( w , y ) / G U ( x , y ) 1 + log + ( 1 / ∣ x − w ∣ ) + log + ( 1 / ∣ w − y ∣ ) 1+\log^+(1/|x-w|)+\log^+(1/|w-y|) 1 + log + ( 1/∣ x − w ∣ ) + log + ( 1/∣ w − y ∣ ) 3 G 3G 3 G (60) ; the change from their generator Δ / 2 \Delta/2 Δ/2 Δ / 4 \Delta/4 Δ/4 C 0 C_0 C 0
For simply connected U ⊂ D U\subset D U ⊂ D G U G_U G U ψ : U → H \psi:U\to\mathbb{H} ψ : U → H h = Im ( ψ ) h=\operatorname{Im}(\psi) h = Im ( ψ ) G U ( x , y ) / ( h ( x ) h ( y ) ) G_U(x,y)/(h(x)h(y)) G U ( x , y ) / ( h ( x ) h ( y ))
d ( z , w ) = Im ( z ) Im ( w ) log ( 1 + 4 Im ( z ) Im ( w ) ∣ w z − w 2 ∣ ) d(z,w)=\frac{\operatorname{Im}(z)\operatorname{Im}(w)}{\log\left(1+4\operatorname{Im}(z)\operatorname{Im}(w)\left|\frac{w}{z-w^2}\right|\right)} d ( z , w ) = log ( 1 + 4 Im ( z ) Im ( w ) z − w 2 w ) Im ( z ) Im ( w ) To check the quasi-triangle inequality, put r = ∣ z − w ∣ r=|z-w| r = ∣ z − w ∣ H = max { Im ( z ) , Im ( w ) } H=\max\{\operatorname{Im}(z),\operatorname{Im}(w)\} H = max { Im ( z ) , Im ( w )} d ( z , w ) ≥ r 2 / 4 d(z,w)\ge r^2/4 d ( z , w ) ≥ r 2 /4 d ( z , w ) ≍ r 2 d(z,w)\asymp r^2 d ( z , w ) ≍ r 2 r ≥ H / 2 r\ge H/2 r ≥ H /2 r < H / 2 r<H/2 r < H /2 d ( z , w ) ≤ C H 2 d(z,w)\le CH^2 d ( z , w ) ≤ C H 2 H / 4 H/4 H /4 r / 2 r/2 r /2 a / log ( 1 + 4 a / s 2 ) a/\log(1+4a/s^2) a / log ( 1 + 4 a / s 2 ) a , s > 0 a,s>0 a , s > 0
For q > 0 q>0 q > 0 [17 Corollary 3.3] , which holds for quasi-metrically modifiable kernels when the associated operator has norm less than one. Apply this with kernel G U G_U G U q d x q\,dx q d x L 2 ( q d x ) L^2(q\,dx) L 2 ( q d x ) q / λ 1 ( U ) ≤ ρ < 1 q/\lambda_1(U)\le\rho<1 q / λ 1 ( U ) ≤ ρ < 1
G U ( m ) ( x , y ) ≤ G U ( x , y ) exp { C ρ q ∫ U G U ( x , w ) G U ( w , y ) d w G U ( x , y ) } ≤ e C ρ q M R G U ( x , y ) . G_U^{(m)}(x,y)\le G_U(x,y)\exp\left\{C_\rho q\frac{\int_U G_U(x,w)G_U(w,y)\,dw}{G_U(x,y)}\right\}\le e^{C_\rho qM_R}G_U(x,y). G U ( m ) ( x , y ) ≤ G U ( x , y ) exp { C ρ q G U ( x , y ) ∫ U G U ( x , w ) G U ( w , y ) d w } ≤ e C ρ q M R G U ( x , y ) . Here C ρ C_\rho C ρ U U U q = 0 q=0 q = 0
For the boundary statement, fix a simply connected U U U a , b a,b a , b ψ : U → H \psi:U\to\mathbb{H} ψ : U → H ψ ( a ) = 0 \psi(a)=0 ψ ( a ) = 0 ψ ( b ) = ∞ \psi(b)=\infty ψ ( b ) = ∞ P ( w ) = − 1 π Im ( ψ ( w ) − 1 ) P(w)=-\frac{1}{\pi}\operatorname{Im}(\psi(w)^{-1}) P ( w ) = − π 1 Im ( ψ ( w ) − 1 ) k ( w ) = 4 Im ( ψ ( w ) ) k(w)=4\operatorname{Im}(\psi(w)) k ( w ) = 4 Im ( ψ ( w )) lim y → b G U ( x , y ) / P ( y ) = k ( x ) \lim_{y\to b}G_U(x,y)/P(y)=k(x) lim y → b G U ( x , y ) / P ( y ) = k ( x ) lim x → a G U ( x , y ) / k ( x ) = P ( y ) \lim_{x\to a}G_U(x,y)/k(x)=P(y) lim x → a G U ( x , y ) / k ( x ) = P ( y ) G U [ j ] G_U^{[j]} G U [ j ] j j j G U [ 0 ] = I G_U^{[0]}=I G U [ 0 ] = I
F j ( x , y ) = G U [ j + 1 ] ( x , y ) G U ( x , y ) , B x , y ( w ) = G U ( x , w ) G U ( w , y ) G U ( x , y ) . F_j(x,y)=\frac{G_U^{[j+1]}(x,y)}{G_U(x,y)},\qquad B_{x,y}(w)=\frac{G_U(x,w)G_U(w,y)}{G_U(x,y)}. F j ( x , y ) = G U ( x , y ) G U [ j + 1 ] ( x , y ) , B x , y ( w ) = G U ( x , y ) G U ( x , w ) G U ( w , y ) . The 3 G 3G 3 G 0 ≤ F j ≤ M R j 0\le F_j\le M_R^j 0 ≤ F j ≤ M R j sup x ≠ y ∈ U ∫ U B x , y ( w ) 2 d w ≤ C R \sup_{x\ne y\in U}\int_U B_{x,y}(w)^2\,dw\le C_R sup x = y ∈ U ∫ U B x , y ( w ) 2 d w ≤ C R L 2 L^2 L 2 B ( 0 , R ) B(0,R) B ( 0 , R ) w w w
B x , y ( w ) ⟶ { G U ( x , w ) k ( w ) / k ( x ) , y → b , x fixed , P ( w ) k ( w ) , x → a , y → b . B_{x,y}(w)\longrightarrow
\begin{cases}
G_U(x,w)k(w)/k(x), & y\to b,\quad x\text{ fixed},\\
P(w)k(w), & x\to a,\quad y\to b.
\end{cases} B x , y ( w ) ⟶ { G U ( x , w ) k ( w ) / k ( x ) , P ( w ) k ( w ) , y → b , x fixed , x → a , y → b . Using the recursion F j ( x , y ) = ∫ U B x , y ( w ) F j − 1 ( w , y ) d w F_j(x,y)=\int_U B_{x,y}(w)F_{j-1}(w,y)\,dw F j ( x , y ) = ∫ U B x , y ( w ) F j − 1 ( w , y ) d w F 0 = 1 F_0=1 F 0 = 1 j ≥ 0 j\ge0 j ≥ 0
lim y → b G U [ j + 1 ] ( x , y ) P ( y ) = ( G U [ j ] k ) ( x ) , \lim_{y\to b}\frac{G_U^{[j+1]}(x,y)}{P(y)}=(G_U^{[j]}k)(x), y → b lim P ( y ) G U [ j + 1 ] ( x , y ) = ( G U [ j ] k ) ( x ) , and, by symmetry, the analogous limit at a a a P P P k k k j ≥ 1 j\ge1 j ≥ 1
lim x → a y → b G U [ j + 1 ] ( x , y ) G U ( x , y ) = ∫ U P ( w ) ( G U [ j − 1 ] k ) ( w ) d w . \lim_{\substack{x\to a\\y\to b}}\frac{G_U^{[j+1]}(x,y)}{G_U(x,y)}
=\int_U P(w)(G_U^{[j-1]}k)(w)\,dw. x → a y → b lim G U ( x , y ) G U [ j + 1 ] ( x , y ) = ∫ U P ( w ) ( G U [ j − 1 ] k ) ( w ) d w . All these integrals are finite by the same 3 G 3G 3 G
It remains to interchange the boundary limits with the Neumann series. Choose q < q ′ < λ 1 ( D ) q<q'<\lambda_1(D) q < q ′ < λ 1 ( D ) q ′ q' q ′ x x x y y y U U U
∑ j ≥ n q j G U [ j + 1 ] ( x , y ) G U ( x , y ) ≤ ( q q ′ ) n G U ( q ′ ) ( x , y ) G U ( x , y ) ≤ C ( D , q ′ ) ( q q ′ ) n , n ≥ 1. \frac{\sum_{j\ge n}q^jG_U^{[j+1]}(x,y)}{G_U(x,y)}
\le\left(\frac{q}{q'}\right)^n\frac{G_U^{(\sqrt{q'})}(x,y)}{G_U(x,y)}
\le C(D,q')\left(\frac{q}{q'}\right)^n,\qquad n\ge1. G U ( x , y ) ∑ j ≥ n q j G U [ j + 1 ] ( x , y ) ≤ ( q ′ q ) n G U ( x , y ) G U ( q ′ ) ( x , y ) ≤ C ( D , q ′ ) ( q ′ q ) n , n ≥ 1. Thus the normalized series converges uniformly, and its boundary limits can be taken term by term. The one-sided limits give the boosted Poisson kernels, while the two-sided limit gives the boosted excursion kernel. Passing to the limits in the interior comparison proves the asserted boundary comparisons with the same constant C m C_m C m
Define the chart-dependent kernels
P t ( z ) = − π − 1 Im ( f t ( z ) − 1 ) , Q t ( z ) = − π − 1 Im ( f t ( z ) − 2 ) , (62) P_t(z)=-\pi^{-1}\operatorname{Im}\left(f_t(z)^{-1}\right),\qquad Q_t(z)=-\pi^{-1}\operatorname{Im}\left(f_t(z)^{-2}\right), \tag*{(62)} P t ( z ) = − π − 1 Im ( f t ( z ) − 1 ) , Q t ( z ) = − π − 1 Im ( f t ( z ) − 2 ) , (62) and, for H = P , Q H=P,Q H = P , Q
H t ( m ) ( z ) = H t ( z ) + q ∫ D t H t ( w ) G D t ( m ) ( w , z ) d w . (63) H_t^{(m)}(z)=H_t(z)+q\int_{D_t}H_t(w)G_{D_t}^{(m)}(w,z)\,dw. \tag*{(63)} H t ( m ) ( z ) = H t ( z ) + q ∫ D t H t ( w ) G D t ( m ) ( w , z ) d w . (63) The estimates below justify the Q Q Q F : U → H F:U\to\mathbb{H} F : U → H q ∣ ( F − 1 ) ′ ∣ 2 q|(F^{-1})'|^2 q ∣ ( F − 1 ) ′ ∣ 2
k t ( m ) ( w ) = lim z → b G D t ( m ) ( w , z ) P t ( z ) , R t = lim z → b P t ( m ) ( z ) P t ( z ) . (64) k_t^{(m)}(w)=\lim_{z\to b}\frac{G_{D_t}^{(m)}(w,z)}{P_t(z)},\qquad R_t=\lim_{z\to b}\frac{P_t^{(m)}(z)}{P_t(z)}. \tag*{(64)} k t ( m ) ( w ) = z → b lim P t ( z ) G D t ( m ) ( w , z ) , R t = z → b lim P t ( z ) P t ( m ) ( z ) . (64) Our Green normalization gives k t ( w ) = 4 Im ( f t ( w ) ) k_t(w)=4\operatorname{Im}(f_t(w)) k t ( w ) = 4 Im ( f t ( w ))
k t ≤ k t ( m ) ≤ C m k t , P t ≤ P t ( m ) ≤ C m P t , 1 ≤ R t ≤ C m , (65) k_t\le k_t^{(m)}\le C_mk_t,\qquad P_t\le P_t^{(m)}\le C_mP_t,\qquad1\le R_t\le C_m, \tag*{(65)} k t ≤ k t ( m ) ≤ C m k t , P t ≤ P t ( m ) ≤ C m P t , 1 ≤ R t ≤ C m , (65) R t = 1 + q ∫ D t P t ( w ) k t ( m ) ( w ) d w . (66) R_t=1+q\int_{D_t}P_t(w)k_t^{(m)}(w)\,dw. \tag*{(66)} R t = 1 + q ∫ D t P t ( w ) k t ( m ) ( w ) d w . (66) We shall repeatedly use the direct half-plane bounds
P t k t ≤ 4 / π , ∣ Q t ∣ k t ≤ 4 P t . (67) P_tk_t\le4/\pi,\qquad|Q_t|k_t\le4P_t. \tag*{(67)} P t k t ≤ 4/ π , ∣ Q t ∣ k t ≤ 4 P t . (67) Conditional density and domain variation
We express the conditional density through Poisson kernels and compute how these kernels change as the slit grows.
Conditioning on an initial segment separates the loop weight accumulated so far from the partition function of the remaining domain.
Lemma 6.4. For q < λ 1 ( D ) q<\lambda_1(D) q < λ 1 ( D ) F t = V m ( η [ 0 , t ] ) F_t=V_m(\eta[0,t]) F t = V m ( η [ 0 , t ]) F t F_t F t
E D ; a , b [ e V m ( K ) / Z m ∣ F t ] = e F t R t / R 0 . (68) \mathbb{E}_{D;a,b}\left[e^{V_m(K)}/Z_m\mid\mathcal{F}_t\right]=e^{F_t}R_t/R_0. \tag*{(68)} E D ; a , b [ e V m ( K ) / Z m ∣ F t ] = e F t R t / R 0 . (68) Proof. Split loops hitting K K K D t D_t D t 2 _2 2 ( D t ; a t , b ) (D_t;a_t,b) ( D t ; a t , b ) R t R_t R t Proposition 6.1 ; Lemma 6.3 justifies the boundary passage. Finite-time slits have locally connected boundary, tip prime end a t a_t a t t = 0 t=0 t = 0 Z m = R 0 Z_m=R_0 Z m = R 0 (68) .
The weighted loop measure hitting K K K s s s η ( s ) \eta(s) η ( s ) s s s F t F_t F t
:::
The variation of the kernels is given by Hadamard’s formula.
Lemma 6.5. For q < λ 1 ( D ) q < \lambda_1(D) q < λ 1 ( D ) w ≠ z w \ne z w = z
G D s ( m ) ( w , z ) − G D t ( m ) ( w , z ) = 8 π ∫ s t P u ( m ) ( w ) P u ( m ) ( z ) d u , (69) G_{D_s}^{(m)}(w,z)-G_{D_t}^{(m)}(w,z)=8\pi\int_s^t P_u^{(m)}(w)P_u^{(m)}(z)\,\mathrm{d}u, \tag*{(69)} G D s ( m ) ( w , z ) − G D t ( m ) ( w , z ) = 8 π ∫ s t P u ( m ) ( w ) P u ( m ) ( z ) d u , (69) d k t ( m ) ( w ) = − 8 π P t ( m ) ( w ) R t d t . (70) \mathrm{d}k_t^{(m)}(w)=-8\pi P_t^{(m)}(w)R_t\mathrm{d}t. \tag*{(70)} d k t ( m ) ( w ) = − 8 π P t ( m ) ( w ) R t d t . (70) Under ordinary S L E 2 \mathrm{SLE}_2 SLE 2
∫ 0 ∞ ( ∫ D t P t ) 2 d t < ∞ , ∫ 0 ∞ ∫ D t P t 2 d w d t < ∞ . (71) \int_0^\infty\left(\int_{D_t}P_t\right)^2\mathrm{d}t<\infty,\qquad\int_0^\infty\int_{D_t}P_t^2\,\mathrm{d}w\,\mathrm{d}t<\infty. \tag*{(71)} ∫ 0 ∞ ( ∫ D t P t ) 2 d t < ∞ , ∫ 0 ∞ ∫ D t P t 2 d w d t < ∞. (71) The first integral has a deterministic bound depending on D D D ( 2 / π ) log ∣ w − z ∣ (2/\pi)\log|w-z| ( 2/ π ) log ∣ w − z ∣ z → w z\to w z → w
Proof. We adapt the argument of [15 Lemma 4.7] , replacing their increment domination by the positive Neumann series. Extend all kernels by zero to D D D G t [ 0 ] = I G_t^{[0]}=I G t [ 0 ] = I G s ( w , z ) − G t ( w , z ) = 8 π ∫ s t P u ( w ) P u ( z ) d u G_s(w,z)-G_t(w,z)=8\pi\int_s^t P_u(w)P_u(z)\,\mathrm{d}u G s ( w , z ) − G t ( w , z ) = 8 π ∫ s t P u ( w ) P u ( z ) d u
G s [ n + 1 ] ( w , z ) − G t [ n + 1 ] ( w , z ) = 8 π ∫ s t ∑ j = 0 n ( G u [ j ] P u ) ( w ) ( G u [ n − j ] P u ) ( z ) d u . G_s^{[n+1]}(w,z)-G_t^{[n+1]}(w,z)=8\pi\int_s^t\sum_{j=0}^{n}(G_u^{[j]}P_u)(w)(G_u^{[n-j]}P_u)(z)\,\mathrm{d}u. G s [ n + 1 ] ( w , z ) − G t [ n + 1 ] ( w , z ) = 8 π ∫ s t j = 0 ∑ n ( G u [ j ] P u ) ( w ) ( G u [ n − j ] P u ) ( z ) d u . All terms are nonnegative, so Tonelli’s theorem permits summation against q n q^n q n (61) ensures convergence, while the resolvent expansion identifies ∑ j ≥ 0 q j G u [ j ] P u = P u ( m ) \sum_{j\ge0}q^jG_u^{[j]}P_u=P_u^{(m)} ∑ j ≥ 0 q j G u [ j ] P u = P u ( m )
For the target variation, divide the integrated identity by P s ( z ) P_s(z) P s ( z ) z → b z\to b z → b P u ( z ) / P s ( z ) → 1 P_u(z)/P_s(z)\to1 P u ( z ) / P s ( z ) → 1 P u ( m ) ≤ C m P u P_u^{(m)}\le C_mP_u P u ( m ) ≤ C m P u
k s ( m ) ( w ) − k t ( m ) ( w ) = 8 π ∫ s t P u ( m ) ( w ) R u d u , k_s^{(m)}(w)-k_t^{(m)}(w)=8\pi\int_s^t P_u^{(m)}(w)R_u\,\mathrm{d}u, k s ( m ) ( w ) − k t ( m ) ( w ) = 8 π ∫ s t P u ( m ) ( w ) R u d u , which gives (6.18). The diagonal identity follows by subtracting the common logarithmic singularity. Finally, the integrability statements are [15 Corollary 4.6] ; the change of Green normalization replaces their coefficient 2 π 2\pi 2 π 8 π 8\pi 8 π
Identification of the drift {#sec:6.4} Proof of the drift formula in Theorem 6.2. Apply [15 Lemmas 4.8–4.9 and equation (4.7)] with the same replacements, using (65) and ∣ Q t ∣ k t ( m ) ≤ 4 C m P t |Q_t|k_t^{(m)}\le4C_mP_t ∣ Q t ∣ k t ( m ) ≤ 4 C m P t I t = ∫ D t P t P t ( m ) d w I_t=\int_{D_t}P_tP_t^{(m)}\,\mathrm{d}w I t = ∫ D t P t P t ( m ) d w
d R t = 2 q ( ∫ D t Q t k t ( m ) d w ) d B t − 8 π q R t I t d t , (72) \mathrm{d}R_t=\sqrt{2}q\left(\int_{D_t}Q_tk_t^{(m)}\,\mathrm{d}w\right)\mathrm{d}B_t-8\pi qR_tI_t\,\mathrm{d}t, \tag*{(72)} d R t = 2 q ( ∫ D t Q t k t ( m ) d w ) d B t − 8 π q R t I t d t , (72) and the boundary identity
Q t ( m ) ( b ) P t ( m ) ( b ) : = lim z → b Q t ( m ) ( z ) P t ( m ) ( z ) = q R t ∫ D t Q t k t ( m ) d w = : λ t . (73) \frac{Q_t^{(m)}(b)}{P_t^{(m)}(b)}:=\lim_{z\to b}\frac{Q_t^{(m)}(z)}{P_t^{(m)}(z)}=\frac{q}{R_t}\int_{D_t}Q_tk_t^{(m)}\,\mathrm{d}w=:\lambda_t. \tag*{(73)} P t ( m ) ( b ) Q t ( m ) ( b ) := z → b lim P t ( m ) ( z ) Q t ( m ) ( z ) = R t q ∫ D t Q t k t ( m ) d w =: λ t . (73) Since e F t R t / R 0 e^{F_t}R_t/R_0 e F t R t / R 0 F t F_t F t 2 λ t d B t \sqrt{2}\lambda_t\,\mathrm{d}B_t 2 λ t d B t (58) .
With the domain and chart fixed, ∂ x [ − π − 1 Im ( [ g t ( ϕ ( z ) ) − x ] − 1 ) ∣ x = ξ t = Q t ( z ) \left.\partial_x[-\pi^{-1}\operatorname{Im}([g_t(\phi(z))-x]^{-1})\right|_{x=\xi_t}=Q_t(z) ∂ x [ − π − 1 Im ([ g t ( ϕ ( z )) − x ] − 1 ) x = ξ t = Q t ( z ) [40 , 15 ] ; (73) is the precise drift used here.
As in [15 Remark 4.10] , the bound ∣ λ t ∣ ≤ 4 q C m q ∫ D t P t |\lambda_t|\leq4qC_m^{q}\int_{D_t}P_t ∣ λ t ∣ ≤ 4 q C m q ∫ D t P t (71) imply Novikov’s condition and weak uniqueness among driving laws locally equivalent to that of ordinary chordal SLE2 _2 2 q < λ 1 ( D ) q<\lambda_1(D) q < λ 1 ( D )
Sharp box-counting estimates
In this section we prove Theorem 3.4. The argument mimics the moment method of Barlow–Masson [3 Section 5] in dimension two and of Shiraishi [50 Section 8] in dimension three, with points replaced by boxes. Two steps need a modification: the decomposition at the ordered entrances into the boxes, where visits that are later erased must be discarded, and the estimate at a single box, where the sharp hitting exponents enter. We state these two steps precisely and only indicate the remaining, standard, arguments.
Throughout this section we work on Z d \mathbb{Z}^d Z d D N = B ( 0 , N ) ∩ Z d D_N=B(0,N)\cap\mathbb{Z}^d D N = B ( 0 , N ) ∩ Z d S S S 0 0 0 γ N = LE ( S [ 0 , T D N ] ) \gamma_N=\operatorname{LE}(S[0,T_{D_N}]) γ N = LE ( S [ 0 , T D N ]) 1 ≤ L ≤ N 1\leq L\leq N 1 ≤ L ≤ N Q L \mathcal{Q}_L Q L L L L
M L = # { Q ∈ Q L : Q ∩ Range ( γ N ) ≠ ∅ } . M_L=\#\{Q\in\mathcal{Q}_L:Q\cap\operatorname{Range}(\gamma_N)\ne\varnothing\}. M L = # { Q ∈ Q L : Q ∩ Range ( γ N ) = ∅ } . Recall (30) for the definition of b d b_d b d θ d = 2 − b d \theta_d=2-b_d θ d = 2 − b d
Theorem 7.1. There are constants c , C > 0 c,C>0 c , C > 0 d d d
P { M L > u ( N / L ) b d } ≤ C e − c u , 1 ≤ L ≤ N , u ≥ 1. (74) \mathbb{P}\{M_L>u(N/L)^{b_d}\}\leq Ce^{-cu},\qquad1\leq L\leq N,\quad u\geq1. \tag*{(74)} P { M L > u ( N / L ) b d } ≤ C e − c u , 1 ≤ L ≤ N , u ≥ 1. (74) Rescaling by N − 1 N^{-1} N − 1 (32) with δ = N − 1 \delta=N^{-1} δ = N − 1 r = L / N r=L/N r = L / N δ ≤ r \delta\leq r δ ≤ r γ δ \gamma^\delta γ δ N r ( K δ ) N_r(K^\delta) N r ( K δ ) 3 d 3^d 3 d γ δ \gamma^\delta γ δ
Reduction to ordered box entrances
Fix a family F ⊂ Q L \mathcal{F}\subset\mathcal{Q}_L F ⊂ Q L F \mathcal{F} F γ N \gamma_N γ N Q 1 , … , Q k ∈ F Q_1,\ldots,Q_k\in\mathcal{F} Q 1 , … , Q k ∈ F { Q 1 → ⋯ → Q k } \{Q_1\to\cdots\to Q_k\} { Q 1 → ⋯ → Q k } γ N \gamma_N γ N X X X F \mathcal{F} F γ N \gamma_N γ N
E ( X k ) = ∑ Q 1 , … , Q k ∈ F distinct P { Q 1 → ⋯ → Q k } . (75) \mathbb{E}\binom{X}{k}=\sum_{\substack{Q_1,\ldots,Q_k\in\mathcal{F}\\\text{distinct}}}\mathbb{P}\{Q_1\to\cdots\to Q_k\}. \tag*{(75)} E ( k X ) = Q 1 , … , Q k ∈ F distinct ∑ P { Q 1 → ⋯ → Q k } . (75) If E ( X ) k ≤ B k \mathbb{E}(X)_k\leq B^k E ( X ) k ≤ B k k ≥ 1 k\geq1 k ≥ 1 B ≥ 1 B\geq1 B ≥ 1 E ( 1 + ( 2 B ) − 1 ) X ≤ 2 \mathbb{E}(1+(2B)^{-1})^X\leq2 E ( 1 + ( 2 B ) − 1 ) X ≤ 2 log ( 1 + t ) ≥ t / 2 \log(1+t)\geq t/2 log ( 1 + t ) ≥ t /2 [ 0 , 1 ] [0,1] [ 0 , 1 ] E e X / ( 4 B ) ≤ 2 \mathbb{E}e^{X/(4B)}\leq2 E e X / ( 4 B ) ≤ 2
E ( X k ) ≤ [ C ( N / L ) b d ] k , k ≥ 1 , (76) \mathbb{E}\binom{X}{k}\leq[C(N/L)^{b_d}]^k,\qquad k\geq1, \tag*{(76)} E ( k X ) ≤ [ C ( N / L ) b d ] k , k ≥ 1 , (76) for each of a bounded number of families F \mathcal{F} F
The families are chosen as follows. Fix a large constant A 0 A_0 A 0 N / L ≤ A 0 N/L\leq A_0 N / L ≤ A 0 A 0 L A_0L A 0 L 0 0 0 O ( 1 ) O(1) O ( 1 ) A 0 L A_0L A 0 L ∂ B ( 0 , N ) \partial B(0,N) ∂ B ( 0 , N ) F \mathcal{F} F A 0 L A_0L A 0 L
Lemma 7.2. Let M L ∂ M^\partial_L M L ∂ Range ( γ N ) \operatorname{Range}(\gamma_N) Range ( γ N ) A 0 L A_0L A 0 L ∂ B ( 0 , N ) \partial B(0,N) ∂ B ( 0 , N )
P ( M L ∂ > u ) ≤ C e − c u , u ≥ 1 , (77) \mathbb{P}(M^\partial_L > u) \le Ce^{-cu}, \qquad u \ge1, \tag*{(77)} P ( M L ∂ > u ) ≤ C e − c u , u ≥ 1 , (77) uniformly in 1 ≤ L ≤ N 1 \le L \le N 1 ≤ L ≤ N
Proof. We count the cubes met by S [ 0 , T D N ] S[0,T_{D_N}] S [ 0 , T D N ] S S S ( A 0 + 2 d ) L (A_0+2\sqrt{d})L ( A 0 + 2 d ) L A L AL A L A A A D N D_N D N p > 0 p>0 p > 0 c L cL c L B ( 0 , N ) B(0,N) B ( 0 , N ) A L / 2 AL/2 A L /2 C A C_A C A
The decomposition at surviving entrances
The first visit of the walk to a cube may be erased by a later loop. We therefore mark the last visit to the first vertex of the cube retained by the final loop erasure, that is, the index t Q t_Q t Q y Q y_Q y Q (17) . For a finite set U ⊂ Z d U \subset\mathbb{Z}^d U ⊂ Z d x , y ∈ U x,y \in U x , y ∈ U Green bridge measure is
B U x , y ( ω ) = p ( ω ) G U ( x , y ) (78) B_U^{x,y}(\omega) = \frac{p(\omega)}{G_U(x,y)} \tag*{(78)} B U x , y ( ω ) = G U ( x , y ) p ( ω ) (78) on finite nearest-neighbor paths ω \omega ω x x x y y y U U U (19) it is a probability measure.
Lemma 7.3. Let Q ⊂ U Q \subset U Q ⊂ U x ∈ U ∖ Q x \in U \setminus Q x ∈ U ∖ Q y ∈ Q y \in Q y ∈ Q S [ 0 , T U ] S[0,T_U] S [ 0 , T U ] x x x y y y t t t y y y α = LE ( S [ 0 , t ] ) \alpha=\operatorname{LE}(S[0,t]) α = LE ( S [ 0 , t ]) y y y Q Q Q LE ( S [ 0 , T U ] ) \operatorname{LE}(S[0,T_U]) LE ( S [ 0 , T U ])
α ∩ Q = { y } , α ∩ S [ t + 1 , T U ] = ∅ . (79) \alpha\cap Q = \{y\}, \qquad\alpha\cap S[t+1,T_U] = \varnothing. \tag*{(79)} α ∩ Q = { y } , α ∩ S [ t + 1 , T U ] = ∅ . (79) Consequently, if Z ∼ B U x , y Z \sim B_U^{x,y} Z ∼ B U x , y S ~ \widetilde{S} S y y y U U U
P x { y Q = y } = G U ( x , y ) P { LE ( Z ) ∩ Q = { y } , LE ( Z ) ∩ S ~ [ 1 , T ~ U ] = ∅ } . (80) \mathbb{P}^x\{y_Q=y\}=G_U(x,y)\mathbb{P}\{\operatorname{LE}(Z)\cap Q=\{y\},\ \operatorname{LE}(Z)\cap\widetilde{S}[1,\widetilde{T}_U]=\varnothing\}. \tag*{(80)} P x { y Q = y } = G U ( x , y ) P { LE ( Z ) ∩ Q = { y } , LE ( Z ) ∩ S [ 1 , T U ] = ∅ } . (80) Proof. Write α = ( a 0 , … , a l ) \alpha=(a_0,\ldots,a_l) α = ( a 0 , … , a l ) a l = y a_l=y a l = y S [ t + 1 , T U ] S[t+1,T_U] S [ t + 1 , T U ] α \alpha α a j a_j a j j < l j<l j < l t t t y y y y y y LE ( S [ 0 , T U ] ) = α ⊕ LE ( S [ t , T U ] ) \operatorname{LE}(S[0,T_U])=\alpha\oplus\operatorname{LE}(S[t,T_U]) LE ( S [ 0 , T U ]) = α ⊕ LE ( S [ t , T U ]) (79) . Splitting at the last visit to y y y y Q = y y_Q=y y Q = y (80) . Indeed, the second condition ensures that the concatenation point is the last visit to y y y (80) follows from the definition of the Green bridge.
Iterating Lemma 7.3 at the marked vertices gives an exact expansion for several boxes. This is the multiple-mark decomposition of [3 Proposition 5.2] , in which we retain the first-entrance condition. Fix pairwise disjoint cubes Q 1 , … , Q k Q_1,\ldots,Q_k Q 1 , … , Q k 0 0 0 y 0 = 0 y_0=0 y 0 = 0 y i ∈ ∂ i n Q i y_i\in\partial_{\mathrm{in}}Q_i y i ∈ ∂ in Q i Z j ∼ B D N y j , y j + 1 Z^j\sim B_{D_N}^{y_j,y_{j+1}} Z j ∼ B D N y j , y j + 1 0 ≤ j < k 0\le j<k 0 ≤ j < k Z k Z^k Z k y k y_k y k D N D_N D N n j = ∣ Z j ∣ n_j=|Z^j| n j = ∣ Z j ∣ j ≤ k j\le k j ≤ k γ j = LE ( Z j ) \gamma^j=\operatorname{LE}(Z^j) γ j = LE ( Z j ) j < k j<k j < k
H = ⋂ i = 1 k { γ i − 1 ∩ ⋃ j = i k Z j [ 1 , n j ] = ∅ } , (81) \mathcal{H}=\bigcap_{i=1}^{k}\left\{\gamma^{i-1}\cap\bigcup_{j=i}^{k}Z^j[1,n_j]=\varnothing\right\}, \tag*{(81)} H = i = 1 ⋂ k { γ i − 1 ∩ j = i ⋃ k Z j [ 1 , n j ] = ∅ } , (81) V = ⋂ i = 1 k { ( γ 0 ⊕ ⋯ ⊕ γ i − 1 ) ∩ Q i = { y i } } . (82) \mathcal{V}=\bigcap_{i=1}^{k}\left\{(\gamma^0\oplus\cdots\oplus\gamma^{i-1})\cap Q_i=\{y_i\}\right\}. \tag*{(82)} V = i = 1 ⋂ k { ( γ 0 ⊕ ⋯ ⊕ γ i − 1 ) ∩ Q i = { y i } } . (82) Lemma 7.4. With this notation,
P { Q 1 → ⋯ → Q k } = ∑ y i ∈ ∂ i n Q i 1 ≤ i ≤ k [ ∏ i = 1 k G D N ( y i − 1 , y i ) ] P ( H ∩ V ) . (83) \mathbb{P}\{Q_1 \to\cdots\to Q_k\} = \sum_{\substack{y_i \in\partial_{\mathrm{in}} Q_i\\1 \le i \le k}} \left[\prod_{i=1}^{k} G_{D_N}(y_{i-1},y_i)\right]\mathbb{P}(\mathcal{H} \cap\mathcal{V}). \tag*{(83)} P { Q 1 → ⋯ → Q k } = y i ∈ ∂ in Q i 1 ≤ i ≤ k ∑ [ i = 1 ∏ k G D N ( y i − 1 , y i ) ] P ( H ∩ V ) . (83) On H ∩ V \mathcal{H} \cap\mathcal{V} H ∩ V 1 ≤ i ≤ k 1 \le i \le k 1 ≤ i ≤ k
γ i − 1 ∩ Q i = { y i } , γ i − 1 ∩ Z i [ 1 , n i ] = ∅ . (84) \gamma^{i-1} \cap Q_i = \{y_i\}, \qquad\gamma^{i-1} \cap Z^{i}[1,n_i] = \varnothing. \tag*{(84)} γ i − 1 ∩ Q i = { y i } , γ i − 1 ∩ Z i [ 1 , n i ] = ∅ . (84) Proof. Since 0 ∉ Q i 0 \notin Q_i 0 ∈ / Q i y Q i y_{Q_i} y Q i Q i Q_i Q i γ N \gamma_N γ N ∂ i n Q i \partial_{\mathrm{in}}Q_i ∂ in Q i { Q 1 → ⋯ → Q k } \{Q_1 \to\cdots\to Q_k\} { Q 1 → ⋯ → Q k } t 1 , … , t k t_1,\ldots,t_k t 1 , … , t k y Q 1 , … , y Q k y_{Q_1},\ldots,y_{Q_k} y Q 1 , … , y Q k γ N \gamma_N γ N t 1 < ⋯ < t k t_1 < \cdots< t_k t 1 < ⋯ < t k t i t_i t i t i t_i t i LE ( S [ 0 , t i ] ) \operatorname{LE}(S[0,t_i]) LE ( S [ 0 , t i ]) LE ( S [ 0 , t i ] ) = γ 0 ⊕ ⋯ ⊕ γ i − 1 \operatorname{LE}(S[0,t_i]) = \gamma^0 \oplus\cdots\oplus\gamma^{i-1} LE ( S [ 0 , t i ]) = γ 0 ⊕ ⋯ ⊕ γ i − 1 H ∩ V \mathcal{H} \cap\mathcal{V} H ∩ V H ∩ V \mathcal{H} \cap\mathcal{V} H ∩ V
The first condition in (7.11) is the reason for marking surviving entrances. Without it, the two-arm probability at a single vertex at scale r r r r − θ d r^{-\theta_d} r − θ d L d − 1 L^{d-1} L d − 1 L b d − 1 L^{b_d-1} L b d − 1
The estimate at a single box
For a cube Q Q Q y ∈ ∂ i n Q y \in\partial_{\mathrm{in}}Q y ∈ ∂ in Q r ≥ 1 r \ge1 r ≥ 1 γ ∞ \gamma^\infty γ ∞ S S S y y y j r j_r j r τ r \tau_r τ r B ( y , r ) B(y,r) B ( y , r )
q Q ( y , r ) = P ( A r Q ) , A r Q = { γ ∞ [ 1 , j r ] ∩ Q = ∅ , S [ 1 , τ r ] ∩ γ ∞ [ 0 , j r ] = ∅ } . (85) q_Q(y,r) = \mathbb{P}(A_r^Q), \qquad A_r^Q = \left\{\gamma^\infty[1,j_r] \cap Q = \varnothing,\ S[1,\tau_r] \cap\gamma^\infty[0,j_r] = \varnothing\right\}. \tag*{(85)} q Q ( y , r ) = P ( A r Q ) , A r Q = { γ ∞ [ 1 , j r ] ∩ Q = ∅ , S [ 1 , τ r ] ∩ γ ∞ [ 0 , j r ] = ∅ } . (85) For Q = { y } Q = \{y\} Q = { y } Es ( r ) \operatorname{Es}(r) Es ( r ) [3 , 50 ] ; in dimension three, Es ( n ) = c n β − 2 [ 1 + O ( n − κ ) ] \operatorname{Es}(n) = cn^{\beta-2}[1+O(n^{-\kappa})] Es ( n ) = c n β − 2 [ 1 + O ( n − κ )] n ≥ 1 n \ge1 n ≥ 1
Lemma 7.5. There are constants C 0 , C < ∞ C_0,C < \infty C 0 , C < ∞ d d d Q Q Q L ≥ 1 L \ge1 L ≥ 1 r ≥ C 0 L r \ge C_0L r ≥ C 0 L
∑ y ∈ ∂ i n Q q Q ( y , r ) ≤ C L d − 2 ( L r ) θ d . (86) \sum_{y \in\partial_{\mathrm{in}}Q} q_Q(y,r) \le CL^{d-2}\left(\frac{L}{r}\right)^{\theta_d}. \tag*{(86)} y ∈ ∂ in Q ∑ q Q ( y , r ) ≤ C L d − 2 ( r L ) θ d . (86) Proof. Let η \eta η r r r Q Q Q r r r H y H_y H y y ∈ ∂ i n Q y \in\partial_{\mathrm{in}}Q y ∈ ∂ in Q Q Q Q η \eta η
The last-visit decomposition of Lemma 7.3 (and the loop-erased path-weight formula [29 Propositions 9.3.3 and 9.5.1] ) gives
P ( H y ) = κ y p y , κ y ≍ r 2 − d , \mathbb{P}(H_y) = \kappa_y p_y, \qquad\kappa_y \asymp r^{2-d}, P ( H y ) = κ y p y , κ y ≍ r 2 − d , where p y p_y p y Q Q Q y y y p y p_y p y q Q ( y , r ) q_Q(y,r) q Q ( y , r ) Q Q Q Sep r \operatorname{Sep}_r Sep r r r r ϵ r \epsilon r ϵr ϵ > 0 \epsilon> 0 ϵ > 0
P ( A r Q ∩ Sep r ) ≥ c P ( A r Q ) , \mathbb{P}(A_r^Q \cap\operatorname{Sep}_r) \ge c\mathbb{P}(A_r^Q), P ( A r Q ∩ Sep r ) ≥ c P ( A r Q ) , e n d s m i d = 1 {ends_mid=1} e n d s m i d = 1 with constants independent of Q Q Q y y y r r r
In dimension three, this follows from the history-uniform separation estimate [50 equation (6.15), in the proof of Theorem 6.1.5] . Indeed, at scale r / 2 r/2 r /2 y y y Q ∩ Z 3 Q \cap\mathbb{Z}^3 Q ∩ Z 3 Q Q Q
In dimension two, the iteration of [41 Theorem 4.7 and Lemmas 4.8–4.9] applies conditionally on both exposed prefixes, from radius r / 2 r/2 r /2 r r r Q Q Q y y y Q Q Q B ( y , r / 4 ) B(y,r/4) B ( y , r /4 ) Q Q Q C 0 ≥ 4 d C_0 \ge4\sqrt{d} C 0 ≥ 4 d
The laws of the local prefixes of the two arms in p y p_y p y [41 Propositions 4.2, 4.4 and Corollary 4.5] . On Sep r \operatorname{Sep}_r Sep r η \eta η η \eta η [50 equations (6.21)–(6.22)] and [41 Proposition 3.5] . Hence p y ≥ c q Q ( y , r ) p_y \ge c q_Q(y,r) p y ≥ c q Q ( y , r )
q Q ( y , r ) ≤ C r d − 2 P ( H y ) , y ∈ ∂ i n Q . (87) q_Q(y,r) \le C r^{d-2}\mathbb{P}(\mathcal{H}_y), \qquad y \in\partial_{\mathrm{in}} Q. \tag*{(87)} q Q ( y , r ) ≤ C r d − 2 P ( H y ) , y ∈ ∂ in Q . (87) The events H y \mathcal{H}_y H y y y y [21 Lemma 4.4] for d = 3 d=3 d = 3 [32 Theorem 6.1 and Corollary 6.3] for d = 2 d=2 d = 2
∑ y ∈ ∂ i n Q q Q ( y , r ) ≤ C r d − 2 P ( η ∩ Q ≠ ∅ ) ≤ C r d − 2 ( L r ) d − b d = C L d − 2 ( L r ) θ d , \sum_{y\in\partial_{\mathrm{in}}Q} q_Q(y,r) \le C r^{d-2}\mathbb{P}(\eta\cap Q \ne\varnothing) \le C r^{d-2}\left(\frac{L}{r}\right)^{d-b_d} = C L^{d-2}\left(\frac{L}{r}\right)^{\theta_d}, y ∈ ∂ in Q ∑ q Q ( y , r ) ≤ C r d − 2 P ( η ∩ Q = ∅ ) ≤ C r d − 2 ( r L ) d − b d = C L d − 2 ( r L ) θ d , as required.
Multiplying the local estimates
To bound P ( H ∩ V ) \mathbb{P}(\mathcal{H}\cap\mathcal{V}) P ( H ∩ V ) y i y_i y i γ i − 1 \gamma^{i-1} γ i − 1 Z i Z^{i} Z i R \mathcal{R} R y i − 1 y_{i-1} y i − 1 y i y_i y i LE ( R ω ) = ( LE ( ω R ) ) R \operatorname{LE}(\mathcal{R}\omega) = (\operatorname{LE}(\omega^{\mathcal{R}}))^{\mathcal{R}} LE ( R ω ) = ( LE ( ω R ) ) R Z i − 1 Z^{i-1} Z i − 1 R Z i − 1 \mathcal{R}Z^{i-1} R Z i − 1 γ i − 1 \gamma^{i-1} γ i − 1 LE ( ( Z i − 1 ) R ) \operatorname{LE}((Z^{i-1})^{\mathcal{R}}) LE (( Z i − 1 ) R ) y i y_i y i
Lemma 7.6. Let U ⊂ Z d U \subset\mathbb{Z}^d U ⊂ Z d Z ∼ B U a , b Z \sim B_U^{a,b} Z ∼ B U a , b r a , r b ≥ 1 r_a,r_b \ge1 r a , r b ≥ 1 B ( a , 8 r a ) B(a,8r_a) B ( a , 8 r a ) B ( b , 8 r b ) B(b,8r_b) B ( b , 8 r b ) U U U Z a Z_a Z a Z Z Z B ( a , r a ) B(a,r_a) B ( a , r a ) γ ^ \widehat{\gamma} γ LE ( Z R ) \operatorname{LE}(Z^{\mathcal{R}}) LE ( Z R ) B ( b , r b ) B(b,r_b) B ( b , r b ) ω , η \omega,\eta ω , η
P ( Z a = ω , γ ^ = η ) ≤ C P a ( S [ 0 , T B ( a , r a ) ] = ω ) P ( γ ∞ [ 0 , τ ‾ b ] = η ) , (88) \mathbb{P}(Z_a=\omega,\widehat{\gamma}=\eta) \le C\mathbb{P}^{a}(S[0,T_{B(a,r_a)}]=\omega)\mathbb{P}(\gamma^\infty[0,\underline{\tau}_b]=\eta), \tag*{(88)} P ( Z a = ω , γ = η ) ≤ C P a ( S [ 0 , T B ( a , r a ) ] = ω ) P ( γ ∞ [ 0 , τ b ] = η ) , (88) where γ ∞ \gamma^\infty γ ∞ b b b C C C d d d P ( γ ^ = η ) ≤ C P ( γ ∞ [ 0 , τ ‾ b ] = η ) \mathbb{P}(\widehat{\gamma}=\eta) \le C\mathbb{P}(\gamma^\infty[0,\underline{\tau}_b]=\eta) P ( γ = η ) ≤ C P ( γ ∞ [ 0 , τ b ] = η ) B ( b , 8 r b ) ⊂ U B(b,8r_b)\subset U B ( b , 8 r b ) ⊂ U a ∉ B ( b , 8 r b ) a\notin B(b,8r_b) a ∈ / B ( b , 8 r b )
Sketch of proof. Given γ ^ = η \widehat{\gamma}=\eta γ = η v v v Z Z Z Range ( η ) \operatorname{Range}(\eta) Range ( η ) v v v Z Z Z a a a Range ( η ) \operatorname{Range}(\eta) Range ( η ) v v v U U U Z a Z_a Z a x ↦ P x { S x\mapsto\mathbb{P}^x\{S x ↦ P x { S Range ( η ) \operatorname{Range}(\eta) Range ( η ) v v v τ U } \tau_U\} τ U } B ( a , 8 r a ) B(a,8r_a) B ( a , 8 r a ) C C C γ ^ \widehat{\gamma} γ [41 Propositions 4.2, 4.4 and Corollary 4.5] , since LE ( Z R ) \operatorname{LE}(Z^{\mathcal{R}}) LE ( Z R ) b b b a a a
For the cubes of one family, with centers q i q_i q i q 0 = 0 q_0=0 q 0 = 0
ρ i = { min { dist ( q i , ∂ B ( 0 , N ) ) , ∣ q i − q i − 1 ∣ , ∣ q i − q i + 1 ∣ } , i < k , min { dist ( q k , ∂ B ( 0 , N ) ) , ∣ q k − q k − 1 ∣ } , i = k . (89) \rho_i =
\begin{cases}
\min\{\operatorname{dist}(q_i,\partial B(0,N)),\lvert q_i-q_{i-1}\rvert,\lvert q_i-q_{i+1}\rvert\}, & i<k,\\
\min\{\operatorname{dist}(q_k,\partial B(0,N)),\lvert q_k-q_{k-1}\rvert\}, & i=k.
\end{cases} \tag*{(89)} ρ i = { min { dist ( q i , ∂ B ( 0 , N )) , ∣ q i − q i − 1 ∣ , ∣ q i − q i + 1 ∣} , min { dist ( q k , ∂ B ( 0 , N )) , ∣ q k − q k − 1 ∣} , i < k , i = k . (89) By the choice of the families, ρ i ≥ A 0 L \rho_i\ge A_0L ρ i ≥ A 0 L
Proposition 7.7. For distinct cubes Q 1 , … , Q k Q_1,\ldots,Q_k Q 1 , … , Q k
P { Q 1 → ⋯ → Q k } ≤ C k ∏ i = 1 k g d ( q i − 1 , q i ) L d − 2 ( L ρ i ) θ d , (90) \mathbb{P}\{Q_1\to\cdots\to Q_k\}\le C^k\prod_{i=1}^{k}g_d(q_{i-1},q_i)L^{d-2}\left(\frac{L}{\rho_i}\right)^{\theta_d}, \tag*{(90)} P { Q 1 → ⋯ → Q k } ≤ C k i = 1 ∏ k g d ( q i − 1 , q i ) L d − 2 ( ρ i L ) θ d , (90) where g 2 ( x , y ) = log C N L + ∣ x − y ∣ g_2(x,y)=\log\frac{CN}{L+\lvert x-y\rvert} g 2 ( x , y ) = log L + ∣ x − y ∣ C N g 3 ( x , y ) = ( L + ∣ x − y ∣ ) − 1 g_3(x,y)=(L+\lvert x-y\rvert)^{-1} g 3 ( x , y ) = ( L + ∣ x − y ∣ ) − 1 N N N L L L k k k
Proof. Put R i = c 1 ρ i R_i=c_1\rho_i R i = c 1 ρ i c 1 > 0 c_1>0 c 1 > 0 A 0 ≥ C 0 / c 1 A_0\ge C_0/c_1 A 0 ≥ C 0 / c 1 B ( y i , 8 R i ) B(y_i,8R_i) B ( y i , 8 R i ) D N D_N D N 0 ∉ B ( y 1 , 8 R 1 ) 0\notin B(y_1,8R_1) 0 ∈ / B ( y 1 , 8 R 1 ) R i ≥ C 0 L R_i\ge C_0L R i ≥ C 0 L P ( H ∩ V ) \mathbb{P}(H\cap V) P ( H ∩ V ) B ( y i , R i ) B(y_i,R_i) B ( y i , R i ) i i i y i y_i y i Z k Z^k Z k
P { Q 1 → ⋯ → Q k } ≤ C k ∑ y i ∈ ∂ i n Q i 1 ≤ i ≤ k ∏ i = 1 k G D N ( y i − 1 , y i ) q Q i ( y i , R i ) . (91) \mathbb{P}\{Q_1\to\cdots\to Q_k\}\le C^k\sum_{\substack{y_i\in\partial_{\mathrm{in}}Q_i\\1\le i\le k}}\prod_{i=1}^{k}G_{D_N}(y_{i-1},y_i)q_{Q_i}(y_i,R_i). \tag*{(91)} P { Q 1 → ⋯ → Q k } ≤ C k y i ∈ ∂ in Q i 1 ≤ i ≤ k ∑ i = 1 ∏ k G D N ( y i − 1 , y i ) q Q i ( y i , R i ) . (91) Since ∣ q i − q i − 1 ∣ ≥ A 0 L \lvert q_i-q_{i-1}\rvert\ge A_0L ∣ q i − q i − 1 ∣ ≥ A 0 L G D N ( y i − 1 , y i ) ≤ C g d ( q i − 1 , q i ) G_{D_N}(y_{i-1},y_i)\le Cg_d(q_{i-1},q_i) G D N ( y i − 1 , y i ) ≤ C g d ( q i − 1 , q i ) y i y_i y i R i ≍ ρ i R_i\asymp\rho_i R i ≍ ρ i
Spatial summation
Proposition 7.7 is the box analogue of the multi-point estimates used in [3 Section 5] and [50 Section 8.1] , with a point replaced by a box of side L L L L L L R ≥ 1 R\ge1 R ≥ 1 V R = B ( 0 , R ) ∩ Z d V_R=B(0,R)\cap\mathbb{Z}^d V R = B ( 0 , R ) ∩ Z d h R ( x ) = 1 + dist ( x , ∂ B ( 0 , R ) ) h_R(x)=1+\operatorname{dist}(x,\partial B(0,R)) h R ( x ) = 1 + dist ( x , ∂ B ( 0 , R ))
g 2 , R ( x , y ) = log A R 1 + ∣ x − y ∣ , g 3 , R ( x , y ) = 1 1 + ∣ x − y ∣ , g_{2,R}(x,y)=\log\frac{AR}{1+\lvert x-y\rvert},\qquad g_{3,R}(x,y)=\frac{1}{1+\lvert x-y\rvert}, g 2 , R ( x , y ) = log 1 + ∣ x − y ∣ A R , g 3 , R ( x , y ) = 1 + ∣ x − y ∣ 1 , with A A A x 0 ∈ B ( 0 , R ) x_0\in B(0,R) x 0 ∈ B ( 0 , R ) s i = 1 + ∣ x i − x i − 1 ∣ s_i=1+\lvert x_i-x_{i-1}\rvert s i = 1 + ∣ x i − x i − 1 ∣ d i = min { s i , s i + 1 , h R ( x i ) } d_i=\min\{s_i,s_{i+1},h_R(x_i)\} d i = min { s i , s i + 1 , h R ( x i )} i < k i<k i < k d k = min { s k , h R ( x k ) } d_k=\min\{s_k,h_R(x_k)\} d k = min { s k , h R ( x k )}
Lemma 7.8. For d ∈ { 2 , 3 } d\in\{2,3\} d ∈ { 2 , 3 } 0 ≤ θ < 1 0\le\theta<1 0 ≤ θ < 1
∑ x 1 , … , x k ∈ V R ∏ i = 1 k g d , R ( x i − 1 , x i ) d i − θ ≤ [ C θ R 2 − θ ] k . (92) \sum_{x_1,\ldots,x_k\in V_R}\prod_{i=1}^{k}g_{d,R}(x_{i-1},x_i)d_i^{-\theta}\le[C_\theta R^{2-\theta}]^k. \tag*{(92)} x 1 , … , x k ∈ V R ∑ i = 1 ∏ k g d , R ( x i − 1 , x i ) d i − θ ≤ [ C θ R 2 − θ ] k . (92) The same holds for any set of centers with bounded density in unit cubes in place of V R V_R V R
Sketch of proof. Expand d i − θ ≤ s i − θ + s i + 1 − θ + h R ( x i ) − θ d_i^{-\theta} \le s_i^{-\theta} + s_{i+1}^{-\theta} + h_R(x_i)^{-\theta} d i − θ ≤ s i − θ + s i + 1 − θ + h R ( x i ) − θ i = k i=k i = k 3 k 3^k 3 k x i x_i x i s i − a i h R ( x i ) − z i s_i^{-a_i}h_R(x_i)^{-z_i} s i − a i h R ( x i ) − z i z i ≤ θ < 1 z_i \le\theta< 1 z i ≤ θ < 1 a i + z i ≤ 2 θ < 2 a_i+z_i \le2\theta< 2 a i + z i ≤ 2 θ < 2 ∑ i ( a i + z i ) = k θ \sum_i(a_i+z_i)=k\theta ∑ i ( a i + z i ) = k θ x k , x k − 1 , … , x 1 x_k,x_{k-1},\ldots,x_1 x k , x k − 1 , … , x 1
sup y ∈ B ( 0 , R ) ∑ x ∈ V R g d , R ( y , x ) ( 1 + ∣ x − y ∣ ) − a h R ( x ) − z ≤ C a , z R 2 − a − z , z < 1 , a + z < 2. \sup_{y\in B(0,R)}\sum_{x\in V_R}g_{d,R}(y,x)(1+|x-y|)^{-a}h_R(x)^{-z}\le C_{a,z}R^{2-a-z},\qquad z<1,\ a+z<2. y ∈ B ( 0 , R ) sup x ∈ V R ∑ g d , R ( y , x ) ( 1 + ∣ x − y ∣ ) − a h R ( x ) − z ≤ C a , z R 2 − a − z , z < 1 , a + z < 2. which follows by decomposing into dyadic annuli around y y y z < 1 z<1 z < 1
Proof of Theorem 7.1. Let X X X γ N \gamma_N γ N R = N / L R=N/L R = N / L x i = q i / L x_i=q_i/L x i = q i / L L g 3 ( q i − 1 , q i ) ≤ C g 3 , R ( x i − 1 , x i ) L\,g_3(q_{i-1},q_i)\le Cg_{3,R}(x_{i-1},x_i) L g 3 ( q i − 1 , q i ) ≤ C g 3 , R ( x i − 1 , x i ) g 2 ( q i − 1 , q i ) ≤ C g 2 , R ( x i − 1 , x i ) g_2(q_{i-1},q_i)\le Cg_{2,R}(x_{i-1},x_i) g 2 ( q i − 1 , q i ) ≤ C g 2 , R ( x i − 1 , x i ) ρ i / L \rho_i/L ρ i / L d i d_i d i θ = θ d \theta=\theta_d θ = θ d
E ( X k ) ≤ [ C R 2 − θ d ] k = [ C ( N / L ) b d ] k . (93) \mathbb{E}\binom{X}{k}\le\left[CR^{2-\theta_d}\right]^k=\left[C(N/L)^{b_d}\right]^k. \tag*{(93)} E ( k X ) ≤ [ C R 2 − θ d ] k = [ C ( N / L ) b d ] k . (93) This is (7.3). Together with the bounded number of cubes near 0 and Lemma 7.2, it proves (7.1).
Covering content over all scales
Recall M α , δ M_{\alpha,\delta} M α , δ α > b d \alpha>b_d α > b d
Lemma 7.9. For every b d < α < d b_d<\alpha<d b d < α < d
sup 0 < δ ≤ 1 P δ { M α , δ ( K δ ) > u } ≤ C α exp { − c α u ( d − b d ) / ( d − α ) } , u ≥ 1. (94) \sup_{0<\delta\le1}\mathbb{P}_\delta\{M_{\alpha,\delta}(K^\delta)>u\}\le C_\alpha\exp\left\{-c_\alpha u^{(d-b_d)/(d-\alpha)}\right\},\qquad u\ge1. \tag*{(94)} 0 < δ ≤ 1 sup P δ { M α , δ ( K δ ) > u } ≤ C α exp { − c α u ( d − b d ) / ( d − α ) } , u ≥ 1. (94) Consequently, for every α > b d \alpha>b_d α > b d t > 0 t>0 t > 0 sup 0 < δ ≤ 1 E P δ exp { t M α , δ ( K δ ) } < ∞ \sup_{0<\delta\le1}\mathbb{E}_{\mathbb{P}_\delta}\exp\{tM_{\alpha,\delta}(K^\delta)\}<\infty sup 0 < δ ≤ 1 E P δ exp { t M α , δ ( K δ )} < ∞ r ∈ [ δ , 1 ] r\in[\delta,1] r ∈ [ δ , 1 ]
Proof. Since N r ( K δ ) ≤ C D r − d N_r(K^\delta)\le C_Dr^{-d} N r ( K δ ) ≤ C D r − d { r α N r ( K δ ) > u } \{r^\alpha N_r(K^\delta)>u\} { r α N r ( K δ ) > u } r ≤ r ∗ ( u ) = ( C D / u ) 1 / ( d − α ) r\le r_*(u)=(C_D/u)^{1/(d-\alpha)} r ≤ r ∗ ( u ) = ( C D / u ) 1/ ( d − α ) j j j r ∗ ( u ) r_*(u) r ∗ ( u ) u r b d − α ≥ c u ( d − b d ) / ( d − α ) 2 j ( α − b d ) ur^{b_d-\alpha}\ge cu^{(d-b_d)/(d-\alpha)}2^{j(\alpha-b_d)} u r b d − α ≥ c u ( d − b d ) / ( d − α ) 2 j ( α − b d ) j j j ( d − b d ) / ( d − α ) > 1 (d-b_d)/(d-\alpha)>1 ( d − b d ) / ( d − α ) > 1 α ≥ d \alpha\ge d α ≥ d r α N r ≤ C s α N s r^\alpha N_r\le Cs^\alpha N_s r α N r ≤ C s α N s s ∈ [ r , 2 r ] s\in[r,2r] s ∈ [ r , 2 r ]
Proof of Theorem 3.4. The fixed-scale bound (3.6) is Theorem 7.1 after rescaling, and (3.7) follows from Lemma 7.9.
Spectral support and capacitary constructions
In this appendix, we prove Propositions 3.5 and 5.2. We use the capacity theory of [42 Chapters 4 and 8] and the Sobolev formulation in [9 Section 2] ; see also [20 ] for related spectral results.
Brownian containment and capacitary obstacles
Throughout the appendix, Brownian motion has generator Δ / ( 2 d ) \Delta/(2d) Δ/ ( 2 d )
Lemma A.1. There is a coupling of the trace of the ordinary LERW scaling limit K K K B = B [ 0 , T D ] B = B[0,T_D] B = B [ 0 , T D ] K ⊂ B K \subset B K ⊂ B z ∈ B ∖ { 0 , B T D } z \in B \setminus\{0,B_{T_D}\} z ∈ B ∖ { 0 , B T D } 0 0 0 B T D B_{T_D} B T D B B B K K K
Proof. The trace/loop-soup identity [48 Theorem 1.1] supplies the inclusion; disintegration over the trace supplies a parametrized Brownian motion. By [48 Theorem 1.2] , K K K
We use the capacity associated with the Dirichlet form of A U A_U A U cap \operatorname{cap} cap cap ( z + r F ) = r cap ( F ) \operatorname{cap}(z+rF)=r\operatorname{cap}(F) cap ( z + r F ) = r cap ( F )
We next record two properties of the principal eigenvalue of slit domains.
Lemma A.2. If K ∩ D K \cap D K ∩ D λ 1 ( D ∖ K ) > λ 1 ( D ) \lambda_1(D \setminus K) > \lambda_1(D) λ 1 ( D ∖ K ) > λ 1 ( D ) K ⊂ D ‾ K \subset\overline{D} K ⊂ D K ε = { x : dist ( x , K ) ≤ ε } K^\varepsilon=\{x:\operatorname{dist}(x,K)\le\varepsilon\} K ε = { x : dist ( x , K ) ≤ ε }
λ 1 ( D ∖ K ε ) ↓ λ 1 ( D ∖ K ) ( ε ↓ 0 ) . (95) \lambda_1(D \setminus K^\varepsilon) \downarrow\lambda_1(D \setminus K) \qquad(\varepsilon\downarrow0). \tag*{(95)} λ 1 ( D ∖ K ε ) ↓ λ 1 ( D ∖ K ) ( ε ↓ 0 ) . (95) For classical LERW the strict inequality holds almost surely in both dimensions.
Proof. Equality of the first eigenvalues would make a minimizer on the slit domain a multiple of the positive ground state on D D D D ∖ K D \setminus K D ∖ K K ε K^\varepsilon K ε ε \varepsilon ε dim H K > 1 \dim_{\mathrm{H}} K > 1 dim H K > 1 [48 Theorem 1.4] , so an interior compact part has positive capacity by Frostman’s criterion. This proves the last assertion.
If K K K K K K
Lemma A.3. In dimension three, let K ⊂ D ‾ K \subset\overline{D} K ⊂ D A , L , κ > 0 A,L,\kappa>0 A , L , κ > 0 K K K F i ⊂ B ( z i , A r ) F_i \subset B(z_i,Ar) F i ⊂ B ( z i , A r ) cap ( F i ) ≥ κ r \operatorname{cap}(F_i)\ge\kappa r cap ( F i ) ≥ κ r D D D L r Lr L r L r Lr L r z i z_i z i c , r 0 > 0 c,r_0>0 c , r 0 > 0 λ 1 ( D ∖ K ) ≥ c r − 2 \lambda_1(D \setminus K)\ge cr^{-2} λ 1 ( D ∖ K ) ≥ c r − 2 r < r 0 r<r_0 r < r 0
Proof. Within L r Lr L r D D D T r 2 Tr^2 T r 2 F i F_i F i M r Mr M r M M M c / r c/r c / r ( c / r ) cap ( F i ) ≥ c κ (c/r)\operatorname{cap}(F_i)\ge c\kappa ( c / r ) cap ( F i ) ≥ c κ T T T K K K D D D T r 2 Tr^2 T r 2 p > 0 p>0 p > 0 L ∞ L^\infty L ∞ 1 − p 1-p 1 − p L 2 L^2 L 2 λ 1 ( D ∖ K ) ≥ c r − 2 \lambda_1(D \setminus K)\ge cr^{-2} λ 1 ( D ∖ K ) ≥ c r − 2
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Positive support at arbitrarily large eigenvalues
This subsection is dedicated to Proposition 3.5 . We obtain traces with arbitrarily large slit-domain eigenvalue by confining the coupled Brownian motion to suitable regions. In dimension three, the obstacles come from the following construction, which joins Brownian pieces in small cubes while keeping enough global cutpoints.
Lemma A.4. There are a bounded cube V ⊂ R 3 V \subset\mathbb{R}^{3} V ⊂ R 3 κ > 0 \kappa> 0 κ > 0 0 < u − < u + < 1 0 < u_{-} < u_{+} < 1 0 < u − < u + < 1 V V V z i + r V z_i + rV z i + r V D D D r r r 0 0 0 F i F_i F i
F i ⊂ z i + r V , cap ( F i ) ≥ κ r for every room i . F_i \subset z_i + rV, \qquad\operatorname{cap}(F_i) \ge\kappa r \qquad\text{for every room } i. F i ⊂ z i + r V , cap ( F i ) ≥ κ r for every room i . The template constants are independent of r r r
Proof. Consider a unit-time Brownian piece confined to a sufficiently large cube V V V [27 Theorem 1.1 and the discussion before Corollary 1.2] and Frostman’s criterion give a positive-measure event on which the closure of the cutpoints in a fixed interval [ u − , u + ] ⊂ ( 0 , 1 ) [u_{-}, u_{+}] \subset(0,1) [ u − , u + ] ⊂ ( 0 , 1 ) κ > 0 \kappa> 0 κ > 0 u − u_{-} u − u + u_{+} u + V V V T − , T + T^{-}, T^{+} T − , T +
T − ∩ ω [ u − , 1 ] = ∅ , T + ∩ ω [ 0 , u + ] = ∅ . T^{-} \cap\omega[u_{-},1] = \varnothing, \qquad T^{+} \cap\omega[0,u_{+}] = \varnothing. T − ∩ ω [ u − , 1 ] = ∅ , T + ∩ ω [ 0 , u + ] = ∅ . The gate balls are contained in the corresponding tubes.
Place scaled copies of this construction in the prescribed rooms. Join their outer stubs in order by disjoint narrow tubes, avoiding all nonincident rooms; such polygonal routes exist in three dimensions. Include an initial connector from 0 0 0 κ r \kappa r κ r
Proof of Proposition 3.5. In dimension three put room centers on a lattice of spacing a fixed large multiple of r r r 0 0 0 K K K L L L λ 1 ( D ∖ K ) ≥ c r − 2 \lambda_1(D \setminus K) \ge cr^{-2} λ 1 ( D ∖ K ) ≥ c r − 2
In the plane, confine Brownian motion to a narrow corridor around a simple spiral from 0 0 0 r r r K K K K ∪ D c K \cup D^{c} K ∪ D c C r Cr C r [8 ] then gives a uniformly positive probability to hit this set before leaving a ball of radius L r Lr L r T r 2 Tr^{2} T r 2 λ 1 ( D ∖ K ) ≥ c r − 2 \lambda_1(D \setminus K) \ge cr^{-2} λ 1 ( D ∖ K ) ≥ c r − 2 r ↓ 0 r \downarrow0 r ↓ 0
The lower edges of the spectral support
We now turn to Proposition 5.2. The lower edge of the support is found by comparison with a radial segment: the segment is polar in dimension three, and it is the extremal slit in dimension two.
Proof of Proposition 5.2. In dimension three, strictness above λ 1 ( D ) \lambda_1(D) λ 1 ( D ) Lemma A.2 . A straight radial segment γ ∗ \gamma_\ast γ ∗ H 0 1 ( D ) H_0^1(D) H 0 1 ( D ) λ 1 ( D ∖ γ ∗ ) ↓ λ 1 ( D ) \lambda_1(D \setminus\gamma_\ast) \downarrow\lambda_1(D) λ 1 ( D ∖ γ ∗ ) ↓ λ 1 ( D ) K ⊂ γ ∗ K \subset\gamma_\ast K ⊂ γ ∗ X ≤ λ 1 ( D ∖ γ ∗ ) X \le\lambda_1(D \setminus\gamma_\ast) X ≤ λ 1 ( D ∖ γ ∗ )
In the plane, a radial arc K K K 0 0 0 f ∈ C ∞ ( D ∖ K ) f \in C^\infty(D \setminus K) f ∈ C ∞ ( D ∖ K ) v ( r , θ ) = r f ( r e i θ ) v(r,\theta) = \sqrt{r}\,f(re^{i\theta}) v ( r , θ ) = r f ( r e i θ ) 2 π 2\pi 2 π ∫ ∣ ∂ θ v ∣ 2 ≥ 1 4 ∫ ∣ v ∣ 2 \int|\partial_\theta v|^2 \ge\frac{1}{4}\int|v|^2 ∫ ∣ ∂ θ v ∣ 2 ≥ 4 1 ∫ ∣ v ∣ 2 r r r ( 0 , 1 ) (0,1) ( 0 , 1 )
∫ D ∣ ∇ f ∣ 2 = ∫ 0 1 ∫ 0 2 π [ ∣ ∂ r v ∣ 2 + ∣ ∂ θ v ∣ 2 − ∣ v ∣ 2 / 4 r 2 ] d θ d r ≥ π 2 ∫ D ∣ f ∣ 2 . (96) \int_D |\nabla f|^2 = \int_0^1 \int_0^{2\pi} \left[|\partial_r v|^2 + \frac{|\partial_\theta v|^2 - |v|^2/4}{r^2}\right]\,d\theta\,dr \ge\pi^2 \int_D |f|^2. \tag*{(96)} ∫ D ∣∇ f ∣ 2 = ∫ 0 1 ∫ 0 2 π [ ∣ ∂ r v ∣ 2 + r 2 ∣ ∂ θ v ∣ 2 − ∣ v ∣ 2 /4 ] d θ d r ≥ π 2 ∫ D ∣ f ∣ 2 . (96) Thus X ≥ π 2 / 4 X \ge\pi^2/4 X ≥ π 2 /4 γ ∗ \gamma_\ast γ ∗ r − 1 / 2 sin ( π r ) sin ( θ / 2 ) r^{-1/2}\sin(\pi r)\sin(\theta/2) r − 1/2 sin ( π r ) sin ( θ /2 ) 0 < θ < 2 π 0 < \theta< 2\pi 0 < θ < 2 π ess inf X = π 2 / 4 \operatorname{ess\,inf} X = \pi^2/4 ess inf X = π 2 /4
Moreover, equality in (A.2) determines the arc. The identity and both nonnegative remainders extend to H 0 1 ( D ∖ K ) H_0^1(D \setminus K) H 0 1 ( D ∖ K ) v / r v/r v / r v ( r , θ ) = sin ( π r ) a ( θ ) v(r,\theta) = \sin(\pi r)a(\theta) v ( r , θ ) = sin ( π r ) a ( θ ) a ( θ ) = c ∣ sin ( ( θ − θ 0 ) / 2 ) ∣ a(\theta) = c|\sin((\theta-\theta_0)/2)| a ( θ ) = c ∣ sin (( θ − θ 0 ) /2 ) ∣ K K K K K K SLE 2 \operatorname{SLE}_2 SLE 2 P ( X > π 2 / 4 ) = 1 \mathbb{P}(X > \pi^2/4) = 1 P ( X > π 2 /4 ) = 1
