The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

We study the stationary distribution of the standard Abelian sandpile model in the box Λ_n = [-n, n] ^d ∩ ℤ ^d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤ ^d . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤ ^d . In the case d > 4, we also make use of Wilson’s method. An erratum to this article is available at .     

Invasion Percolation and the Incipient Infinite Cluster in 2D

 We establish two links between two-dimensional invasion percolation and Kesten's incipient infinite cluster (IIC). We first prove that the k ^th moment of the number of invaded sites within the box [−n, n]×[−n, n] is of order (n ²π _n ) ^k , for k≥1, where π _n is the probability that the origin in critical percolation is connected to the boundary of a box of radius n. This improves a result of Y. Zhang. We show that the size of the invaded region, when scaled by n ²π _n , is tight. Secondly, we prove that the invasion cluster looks asymptotically like the IIC, when viewed from an invaded site v, in the limit |v|→∞. We also establish this when an invaded site v is chosen at random from a box of radius n, and n→∞. Received: 3 December 2000 / Accepted: 3 December 2002 Published online: 18 February 2003 RID="⋆" ID="⋆" Present address: CWI, PNA 3, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands. E-mail:jarai@cwi.nl Communicated by M. Aizenman     

Pattern Theorems,Ratio Limit Theorems and Gumbel Maximal Clusters for Random Fields

We study occurrences of patterns on clusters of size n in random fields on ℤ ^d . We prove that for a given pattern, there is a constant a>0 such that the probability that this pattern occurs at most na times on a cluster of size n is exponentially small. Moreover, for random fields obeying a certain Markov property, we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value. This leads to an elegant and simple proof of the ratio limit theorem for these random fields, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges as n→∞. Implications for the maximal cluster in a finite box are discussed.     

On the Ornstein-Zernike Behaviour for the Bernoulli Bond Percolation on $\pmb{\mathbb{Z}}^{d},d\geq3$, in the Supercritical Regime

We prove Ornstein-Zernike behaviour in every direction for finite connection functions of bond percolation on ℤ ^d for d≥3 when p, the probability of occupation of a bond, is sufficiently close to 1. Moreover, we prove that equi-decay surfaces are locally analytic, strictly convex, with positive Gaussian curvature.     

Continuous-Time Quantum Walk on Integer Lattices and Homogeneous Trees

This paper is concerned with the continuous-time quantum walk on ℤ, ℤ ^d , and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on ℤ, and for nearest-neighbor walks on ℤ ^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.     

Asymptotics of Determinants of Bessel Operators

 For aL ^∞ (ℝ₊)∩L ¹ (ℝ₊) the truncated Bessel operator B _τ (a) is the integral operator acting on L ² [0,τ] with the kernel

A Note on the Abelian Sandpile in $\pmb{\mathbb{Z}}^{d}$

We analyze the abelian sandpile model on ℤ ^d for the starting configuration of n particles in the origin and 2d−2 particles otherwise. We give a new short proof of the theorem of Fey, Levine and Peres (J. Stat. Phys. 198:143–159, 2010) that the radius of the toppled cluster of this configuration is O(n ^1/d ).     

Analyticity and Mixing Properties for Random Cluster Model with q >0 on ℤ d

We study the Random Cluster Model on ℤ ^d for p near either 0 or 1 and for all q > 0 and we prove by mean of cluster expansion methods the analyticity of the pressure and finite connectivities in both regimes. These results are valid also in the regime q < 1 and they imply that percolation probability is strictly less than 1.     

Decay Properties of the Connectivity for Mixed Long Range Percolation Models on ℤ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

In this short note we consider mixed short-long range independent bond percolation models on ℤ ^k+d . Let p _uv be the probability that the edge (u,v) will be open. Allowing a x,y-dependent length scale and using a multi-scale analysis due to Aizenman and Newman, we show that the long distance behavior of the connectivity τ _xy is governed by the probability p _xy . The result holds up to the critical point.     

Surface States and Spectra

Let ℤ₊ ^{d +1}= ℤ⁺×ℤ₊, let H ₀ be the discrete Laplacian on the Hilbert space l ²(ℤ₊ ^{d +1}) with a Dirichlet boundary condition, and let V be a potential supported on the boundary ∂ℤ₊ ^{d +1}. We introduce the notions of surface states and surface spectrum of the operator H=H ₀+V and explore their properties. Our main result is that if the potential V is random and if the disorder is either large or small enough, then in dimension two H has no surface spectrum on σ(H ₀) with probability one. To prove this result we combine Aizenman–Molchanov theory with techniques of scattering theory. Received: 18 September 2000 / Accepted: 21 November 2000     

The Current Distribution of the Multiparticle Hopping Asymmetric Diffusion Model

In this paper we treat the multiparticle hopping asymmetric diffusion model (MADM) on ℤ introduced by Sasamoto and Wadati in 1998. The transition probability of the MADM with N particles is provided by using the Bethe ansatz. The transition probability is expressed as the sum of N-dimensional contour integrals of which contours are circles centered at the origin with restrictions on their radii. By using the transition probability we find ℙ(x _m (t)=x), the probability that the mth particle from the left is at x at time t. The probability ℙ(x _m (t)=x) is expressed as the sum of |S|-dimensional contour integrals over all S⊂{1,…,N} with |S|≥m, and is used to give the current distribution of the system. The mapping between the MADM and the pushing asymmetric simple exclusion process (PushASEP) is discussed.     

Entropic Repulsion for the High Dimensional Gaussian Lattice Field Between Two Walls

The main aim of this paper is to discuss the entropic repulsion of random interfaces between two hard walls. We consider the d (≥ 3)-dimensional Gaussian lattice field on ℝ^λ _N , λ _N = [−N, N] ^d ∩ ℤ ^d and identify the repulsion of the field as N → ∞ under the condition that the field lies between two hard walls at the height level 0 and L in Λ _N where L is large enough but finite. We also study the same problem for two layered interfaces case.     

Spectral Properties of Hypoelliptic Operators

 We study hypoelliptic operators with polynomially bounded coefficients that are of the form K=∑ _i=1 ^m X _i ^T X _i +X ₀+f, where the X _j denote first order differential operators, f is a function with at most polynomial growth, and X _i ^T denotes the formal adjoint of X _i in L ². For any ɛ>0 we show that an inequality of the form ||u||_δ,δ≤C(||u||_0,ɛ+||(K+iy)u||_0,0) holds for suitable δ and C which are independent of yR, in weighted Sobolev spaces (the first index is the derivative, and the second the growth). We apply this result to the Fokker-Planck operator for an anharmonic chain of oscillators coupled to two heat baths. Using a method of Hérau and Nier [HN02], we conclude that its spectrum lies in a cusp {x+iy|x≥|y|^τ−c,τ(0,1],cR}. Received: 30 July 2002 / Accepted: 18 October 2002 Published online: 25 February 2003 RID="*" ID="*" Present address: Mathematics Research Centre of the University of Warwick Communicated by A. Kupiainen     

A Small-Scale Density of States Formula

 Let (M,g) be a C ^∞ compact Riemann manifold with classical Hamiltonian, HC ^∞ (T ^* M). Assume that the corresponding -quantization P ₁ :=Op _ (H) is quantum completely integrable. We establish an -microlocal Weyl law on short spectral intervals of size ^2−ε;∀ε>0 for various families of operators P ₁ ^u ;uI containing P ₁ , both in the mean and pointwise a.e. for uI. The -microlocalization refers to a small tubular neighbourhood of a non-degenerate, stable periodic bicharacteristic γ⊂T ^* M−0. Received: 10 December 2001 / Accepted: 23 January 2003 Published online: 2 April 2003 RID="⋆" ID="⋆" Supported in part by an Alfred P. Sloan Research Fellowship and NSERC grant OGP01720280 Communicated by P. Sarnak     

On the Distribution of Free Path Lengths for the Periodic Lorentz Gas III

 For r(0,1), let Z _r ={xR ² |dist(x,Z ²)>r/2} and define τ _r (x,v)=inf{t>0|x+tv∂Z _r }. Let Φ _r (t) be the probability that τ _r (x,v)≥t for x and v uniformly distributed in Z _r and §¹ respectively. We prove in this paper that

Adiabatic Times for Markov Chains and Applications

We state and prove a generalized adiabatic theorem for Markov chains and provide examples and applications related to Glauber dynamics of the Ising model over ℤ ^d /nℤ ^d . The theorems derived in this paper describe a type of adiabatic dynamics for l¹(\mathbbRⁿ₊)\ell^{1}(\mathbb{R}^{n}_{+}) norm preserving, time inhomogeneous Markov transformations, while quantum adiabatic theorems deal with ℓ ²(ℂ ⁿ ) norm preserving ones, i.e. gradually changing unitary dynamics in ℂ ⁿ .     

Limiting Shapes for Deterministic Centrally Seeded Growth Models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in ℤ ^d , Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter h (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension d≥1. For the rotor router model, the limiting shape is a sphere for all values of h. For one of the sandpile models, and h=2d−2 (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit h→−∞. Finally, we prove that the rotor router shape contains a diamond.     

Noncommutative electromagnetism as a large <Emphasis Type="Italic">N</Emphasis> gauge theory

We map noncommutative (NC) U(1) gauge theory on ℝ _C ^d ×ℝ _NC ²ⁿ to U(N→∞) Yang–Mills theory on ℝ _C ^d , where ℝ _C ^d is a d-dimensional commutative spacetime while ℝ _NC ²ⁿ is a 2n-dimensional NC space. The resulting U(N) Yang–Mills theory on ℝ _C ^d is equivalent to that obtained by the dimensional reduction of (d+2n)-dimensional U(N) Yang–Mills theory onto ℝ _C ^d . We show that the gauge-Higgs system (A _μ ,Φ ^a ) in the U(N→∞) Yang–Mills theory on ℝ _C ^d leads to an emergent geometry in the (d+2n)-dimensional spacetime whose metric was determined by Ward a long time ago. In particular, the 10-dimensional gravity for d=4 and n=3 corresponds to the emergent geometry arising from the 4-dimensional N=4{\mathcal{N}}=4 vector multiplet in the AdS/CFT duality. We further elucidate the emergent gravity by showing that the gauge-Higgs system (A _μ ,Φ ^a ) in half-BPS configurations describes self-dual Einstein gravity.     

Lower bounds on the cluster size distribution

We rigorously prove that the probabilityP _n that the origin of ad-dimensional lattice belongs to a cluster of exactlyn sites satisfiesP _n > exp(–n ^(d–1)/d ) whenever percolation occurs. This holds for the usual (noninteracting) percolation models for any concentrationp > p _c , as well as for the equilibrium states of lattice spin systems with quite general interactions. Such a lower bound applies also if no percolation occurs, but if it appears in some other phase of the system.