Planar lattice gases with nearest-neighbor exclusion

We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists, being the problem of counting binary matrices with no two adjacent 1's. For this case, we use the powerful corner transfer matrix method to numerically evaluate the partition function per site, density and some near-neighbor correlations to high accuracy. In particular, for the square lattice, we obtain the partition function per site to 43 decimal places.     

The random-cluster model on the complete graph

Summary The random-cluster model of Fortuin and Kasteleyn contains as special cases the percolation, Ising, and Potts models of statistical physics. When the underlying graph is the complete graph onn vertices, then the associated processes are called mean-field. In this study of the mean-field random-cluster model with parametersp=/n andq, we show that its properties for any value ofq(0, ) may be derived from those of an Erds-Rényi random graph. In this way we calculate the critical point _c (q) of the model, and show that the associated phase transition is continuous if and only ifq2. Exact formulae are given for _C (q), the density of the largest component, the density of edges of the model, and the free energy. This work generalizes earlier results valid for the Potts model, whereq is an integer satisfyingq2. Equivalent results are obtained for a fixed edge-number random-cluster model. As a consequence of the results of this paper, one obtains large-deviation theorems for the number of components in the classical random-graph models (whereq=1).     

The random-cluster model on a homogeneous tree

Summary The random-cluster model on a homogeneous tree is defined and studied. It is shown that for 1q2, the percolation probability in the maximal random-cluster measure is continuous inp, while forq>2 it has a discontinuity at the critical valuep=p _c (q). It is also shown that forq>2, there is nonuniqueness of random-cluster measures for an entire interval of values ofp. The latter result is in sharp contrast to what happens on the integer lattice Z ^d .Research partially supported by a grant from the Royal Swedish Academy of Sciences     

Localization for branching random walks in random environment

We consider branching random walks in d$d$-dimensional integer lattice with time–space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If d≥3$d\ge 3$ and the environment is “not too random”, then, the total population grows as fast as its expectation with strictly positive probability. If, on the other hand, d≤2$d\le 2$, or the environment is “random enough”, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of “replica overlap”. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.     

Surface order large deviations for Ising,Potts and percolation models

Summary We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq1 andd3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution.     

Random walk on the infinite cluster of the percolation model

Summary We consider random walk on the infinite cluster of bond percolation on ^d . We show that, in the supercritical regime whend3, this random walk is a.s. transient. This conclusion is achieved by considering the infinite percolation cluster as a random electrical network in which each open edge has unit resistance. It is proved that the effective resistance of this network between a nominated point and the points at infinity is almost surely finite.G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell UniversityH.K. was supported in part by the N.S.F. through a grant to Cornell University     

Surface order large deviations for high-density percolation

Summary We derive surface order large deviation estimates for the volume of the largest cluster and for the volume of the largest region surrounded by a cluster of a Bernoulli percolation process restricted to a big finite box, with sufficiently large parameter. We also establish a useful version of the isoperimetric inequality, which is the main tool of our proofs.     

Exact large deviation bounds up toT c for the Ising model in two dimensions

Summary We prove an upper large deviation bound for the block spin magnetization in the 2D Ising model in the phase coexistence region. The precise rate (given by the Wulff construction) is shown to hold true for all > _c. Combined with the lower bounds derived in [I] those results yield an exact second order large deviation theory up to the critical temperature.     

Brownian confinement and pinning in a Poissonian potential. I

Summary We study the behavior of ad dimensional Brownian motion in a soft repulsive Poissonian potential over long time intervals [0,t]. We introduce certaint and configuration dependent scales, which grow almost linearly witht. For typical configurations with probability tending to 1 ast goes to , the size of displacements of the process is bounded above by these scales, (confinement effect). The proof involves calculations beyond leading order. To this end we use a coarse grained picture of the environment (method of enlargement of obstacles) and of the path (a backbone of excursions between clearings and forest parts in the environment). These coarse grained pictures are also used in the sequel [11] to the present article, when proving the pinning effect.This article was processed by the author using the LATEX style filepljour1m from Springer-Verlag.     

Gauge symmetries and percolation in ±J Ising spin glasses

We consider the ±J spin glass on a finite graph G=(V,E), with i.i.d. couplings. Our approach considers the Z ₂ local gauge invariance of the system. We see the gauge group as a graph theoretic linear code ? over GF(2). The gauge is fixed by choosing a convenient linear supplement of ?. Assuming some relation between the disorder parameter p and the inverse temperature of the thermal bath β _pb , we study percolation in the random interaction random cluster model, and extend the results to dilute spin glasses. Received: 5 May 1997 / Revised version: 9 April 1998     

Brownian confinement and pinning in a Poissonian potential. II

Summary We continue our study ofd-dimensional Brownian motion in a soft repulsive Poissonian potential over a long time interval [0,t]. We prove here a pinning effect: for typical configuratons, with probability tending to 1 ast tends to , the particle gets trapped close to locations of near minima of certain variational problems. These locations lie at distances growing almost linearly witht from the origin, and the particle gets pinned within distance smaller than any positive power oft of one such location. In dimension 1, we can push further our estimates and show that in a suitable sense, the particle gets trapped with high probability, within time t and within distance (logt)²⁺ from a suitable location at distance of ordert/(logt)³ from the origin.This article was processed by the author using the LATEX style filepljour1m from Springer-Verlag     

Brownian asymptotics in a Poissonian environment

Summary We consider a Brownian motion moving in a random potential obtained by translating a given fixed non negative shape function at the points of a Poisson cloud. We derive the almost sure principal long time behavior of the expectation of the natural Feynman Kac functional, which is insensitive to the detail of the shape function. We also study the situation of hard obstacles where Brownian motion is killed once it comes within distancea of a point of the cloud. The nature of the results then changes between the case whena is small or large in connection with the presence, or absence of an infinite component in the complement of the obstacles.     

Brownian motion in a Poissonian potential

We study here ad-dimensional Brownian motion in a random potentialV(·, ) obtained as the sum of translations of a given fixed non negative shape function at the points of a Poisson cloud of constant intensityv. We are interested in the larget behavior for typical cloud configurations, of the Brownian path in timet under the influence of the natural Feynman-Kac weight associated toV(·, ). In particular, we show that the location at timet of the process tends to be concentrated near points of suitably low local eigenvalue of –1/2+V(·,), which lie almost at distancet from the origin. Near these points one can find in the cloud a big hole or clearing of size const(logt)^1/d with volume like a ball of radiusR ₀(d, v)(logt)^1/d .     

Sharp asymptotics for Kawasaki dynamics on a finite box with open boundary

In this paper we study the metastable behavior of the lattice gas in two and three dimensions subject to Kawasaki dynamics in the limit of low temperature and low density. We consider the local version of the model, where particles live on a finite box and are created, respectively, annihilated at the boundary of the box in a way that reflects an infinite gas reservoir. We are interested in how the system nucleates, i.e., how it reaches a full box when it starts from an empty box. Our approach combines geometric and potential theoretic arguments. In two dimensions, we identify the full geometry of the set of critical droplets for the nucleation, compute the average nucleation time up to a multiplicative factor that tends to one, show that the nucleation time divided by its average converges to an exponential random variable, express the proportionality constant for the average nucleation time in terms of certain capacities associated with simple random walk, and compute the asymptotic behavior of this constant as the system size tends to infinity. In three dimensions, we obtain similar results but with less control over the geometry and the constant. A special feature of Kawasaki dynamics is that in the metastable regime particles move along the border of a droplet more rapidly than they arrive from the boundary of the box. The geometry of the critical droplet and the sharp asymptotics for the average nucleation time are highly sensitive to this motion.     

Coexistence of infinite (*)-clusters II. Ising percolation in two dimensions

Summary We show a strong type of conditionally mixing property for the Gibbs states ofd-dimensional Ising model when the temperature is above the critical one. By using this property, we show that there is always coexistence of infinite (+ *)-and (–*)-clusters when is smaller than _c andh=0 in two dimensions. It is also possible to show that this coexistence region extends to some non-zero external field case, i.e., for every < _c, there exists someh _c()>0 such that |h|<h _c() implies the coexistence of infinite (*)-clusters with respect to the Gibbs state for (,h).work supported in part by Grant in Aid for Cooperative research no. 03302010, Grant in Aid for Scientific Research no. 03640056 and ISM Cooperative research program (91-ISM,CRP-3)To the memory of Professor Haruo Totoki     

Singularity of full scaling limits of planar nearcritical percolation

We consider full scaling limits of planar nearcritical percolation in the Quad-Crossing-Topology introduced by Schramm and Smirnov. We show that two nearcritical scaling limits with different parameters are singular with respect to each other. The results hold for percolation models on rather general lattices, including bond percolation on the square lattice and site percolation on the triangular lattice.     

Higher correlation inequalities

We prove a correlation inequality for n increasing functions on a distributive lattice, which for n = 2 reduces to a special case of the FKG inequality. The key new idea is to reformulate the inequalities for all n into a single positivity statement in the ring of formal power series. We also conjecture that our results hold in greater generality.     

High temperature regime for a multidimensional Sherrington–Kirkpatrick model of spin glass

Comets and Neveu have initiated in [5] a method to prove convergence of the partition function of disordered systems to a log-normal random variable in the high temperature regime by means of stochastic calculus. We generalize their approach to a multidimensional Sherrington-Kirkpatrick model with an application to the Heisenberg model of uniform spins on a sphere of ℝ ^d , see [9]. The main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical energy path. Received: 30 October 1996 / In revised form: 13 October 1997     

Linear response theory for magnetic Schrödinger operators in disordered media

We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the well-known Kubo-Str?eda formula for the quantum Hall conductivity at zero temperature.