On weak mixing in lattice models

Summary. For lattice models on ℤ ^d , weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region. For a wide class of models on ℤ², including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an appropriate form of connectivity, and a natural coupling property. In particular, on ℤ², the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially, and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range, weak mixing implies ratio weak mixing. Received: 27 August 1996 / In revised form: 15 August 1997     

A new inductive approach to the lace expansion for self-avoiding walks

Summary. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ℤ ^d where loops of length m are penalised by a factor e ^{−β/m p} (0<β≪1) when: (1) d>4, p≥0; (2) d≤4, . In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case d>4, p=0. In addition, we prove a local central limit theorem, with the exception of the case d>4, p=0. Received: 29 October 1997 / In revised form: 15 January 1998     

Transformations of Gibbs measures

We study local transformations of Gibbs measures. We establish sufficient conditions for the quasilocality of the images and obtain results on the existence and continuity properties of their relative energies. General results are illustrated by simple examples. Received: 11 November 1997 / Revised version: 20 February 1998     

Chaotic time dependence in a disordered spin system

Stochastic Ising and voter models on ℤ ^d are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t→∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) – which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature – if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d≥ 2. These results are based on an analysis of the “localization” properties of Random Walks with Random Rates. Received: 10 August 1998     

Hydrodynamical limit for spatially heterogeneous simple exclusion processes

Summary.   We prove hydrodynamical limit for spatially heterogeneous, asymmetric simple exclusion processes on Z ^d . The jump rate of particles depends on the macroscopic position x through some nonnegative, smooth velocity profile α(x). Hydrodynamics are described by the entropy solution to a spatially heterogeneous conservation law of the form

The retrieval phase of the Hopfield model: A rigorous analysis of the overlap distribution*

Summary. Standard large deviation estimates or the use of the Hubbard–Stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function Φ _N,β on ℝ ^M . In this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns. We denote by m ^* (β ) the modulus of the spontaneous magnetization in the Curie–Weiss model and by α the ratio between the number of the stored patterns and the system size. We show that there exist strictly positive numbers 0 < γ _a < γ _c such that (1) If √α≦γ _a (m ^* (β )) ² , then the absolute minima of Φ are located within small balls around the points ± m ^* e ^μ , where e ^μ denotes the μ-th unit vector while (2) if √α≦γ _c (m ^* (β )) ² at least a local minimum surrounded by extensive energy barriers exists near these points. The random location of these minima is given within precise bounds. These are used to prove sharp estimates on the support of the Gibbs measures. Received: 5 August 1995 / In revised form: 22 May 1996     

Scaling identity for crossing Brownian motion in a Poissonian potential

We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|^ξ, whereas the distance fluctuation is of order |y|^χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for this conjecture. Received: 31 December 1997 / Revised version: 14 April 1998     

A central limit theorem for “critical” first-passage percolation in two dimensions

Summary. Consider (independent) first-passage percolation on the edges of ℤ ² . Denote the passage time of the edge e in ℤ ² by t(e), and assume that P{t(e) = 0} = 1/2, P{0<t(e)<C ₀ } = 0 for some constant C ₀ >0 and that E[t ^δ (e)]<∞ for some δ>4. Denote by b _0,n the passage time from 0 to the halfplane {(x,y): x ≧ n}, and by T( 0 ,nu) the passage time from 0 to the nearest lattice point to nu, for u a unit vector. We prove that there exist constants 0<C ₁ , C ₂ <∞ and γ _n such that C ₁ ( log n) ^1/2 ≦γ _n ≦ C ₂ ( log n) ^1/2 and such that γ _n ⁻¹ [b _0,n −Eb _0,n ] and (√ 2γ _n ) ⁻¹ [T( 0 ,nu) − ET( 0 ,nu)] converge in distribution to a standard normal variable (as n →∞, u fixed). A similar result holds for the site version of first-passage percolation on ℤ ² , when the common distribution of the passage times {t(v)} of the vertices satisfies P{t(v) = 0} = 1−P{t(v) ≧ C ₀ } = p _c (ℤ ² , site ) := critical probability of site percolation on ℤ ² , and E[t ^δ (u)]<∞ for some δ>4. Received: 6 February 1996 / In revised form: 17 July 1996     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

Asymptotic behavior of the critical probability for ρ-percolation in high dimensions

We consider oriented bond or site percolation on ℤ ^d ₊. In the case of bond percolation we denote by P _p the probability measure on configurations of open and closed bonds which makes all bonds of ℤ ^d ₊ independent, and for which P _p {e is open} = 1 −P _p e {is closed} = p for each fixed edge e of ℤ ^d ₊. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 < ρ≤ 1, if there exists an oriented infinite path v ₀ = 0, v ₁, v ₂, …, starting at the origin, such that lim inf _{n →∞} (1/n) ∑ _i=1 ⁿ X(e _i ) ≥ρ, where e _i is the edge {v _i−1 , v _i }. [MZ92] showed that there exists a critical probability p _c = p _c (ρ, d) = p _c (ρ, d, bond) such that there is a.s. no ρ-percolation for p < p _c and that P _p {ρ-percolation occurs} > 0 for p > p _c . Here we find lim _{d →∞} d ^1/ρ p _c (ρd, bond) = D ₁ , say. We also find the limit for the analogous quantity for site percolation, that is D ₂ = lim _{d →∞} d ^1/ρ p _c (ρ, d, site). It turns out that for ρ < 1, D ₁ < D ₂ , and neither of these limits equals the analogous limit for the regular d-ary trees. Received: 7 January 1999 / Published online: 14 June 2000     

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     

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000     

Branching random walks and contact processes on homogeneous trees

Summary. Branching random walks and contact processes on the homogeneous tree in which each site has d+1 neighbors have three possible types of behavior (for d≧ 2): local survival, local extinction with global survival, and global extinction. For branching random walks, we show that if there is local extinction, then the probability that an individual ever has a descendent at a site n units away from that individual’s location is at most d ^{− n/2} , while if there is global extinction, this probability is at most d ⁻ⁿ . Next, we consider the structure of the set of invariant measures with finite intensity for the system, and see how this structure depends on whether or not there is local and/or global survival. These results suggest some problems and conjectures for contact processes on trees. We prove some and leave others open. In particular, we prove that for some values of the infection parameter λ, there are nontrivial invariant measures which have a density tending to zero in all directions, and hence are different from those constructed by Durrett and Schinazi in a recent paper. Received: 26 April 1996/In revised form: 20 June 1996     

The true self-repelling motion

We construct and study a continuous real-valued random process, which is of a new type: It is self-interacting (self-repelling) but only in a local sense: it only feels the self-repellance due to its occupation-time measure density in the `immediate neighbourhood' of the point it is just visiting. We focus on the most natural process with these properties that we call `true self-repelling motion'. This is the continuous counterpart to the integer-valued `true' self-avoiding walk, which had been studied among others by the first author. One of the striking properties of true self-repelling motion is that, although the couple (X _t , occupation-time measure of X at time t) is a continuous Markov process, X is not driven by a stochastic differential equation and is not a semi-martingale. It turns out, for instance, that it has a finite variation of order 3/2, which contrasts with the finite quadratic variation of semi-martingales. One of the key-tools in the construction of X is a continuous system of coalescing Brownian motions similar to those that have been constructed by Arratia [A1, A2]. We derive various properties of X (existence and properties of the occupation time densities L _t (x), local variation, etc.) and an identity that shows that the dynamics of X can be very loosely speaking described as follows: −dX _t is equal to the gradient (in space) of L _t (x), in a generalized sense, even though x↦L _t (x) is not differentiable. Received: 15 April 1997 / Revised version: 30 January 1998     

The asymmetric random cluster model and comparison of Ising and Potts models

We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (β, h) for which the 0‘s configuration in the Potts lattice gas is dominated by the “+” configuration of the (β, h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain 0.571 ≤ 1 − exp(−β _c ) ≤ 0.600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as the partial FK model, obtained by deleting a fraction of the nonsingleton clusters from a realization of the Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the Potts model. Received: 27 August 2000 / Revised version: 31 August 2000 / Published online: 8 May 2001     

Convection–diffusion equation with space–time ergodic random flow

We prove the homogenization of convection-diffusion in a time-dependent, ergodic, incompressible random flow which has a bounded stream matrix and a constant mean drift. We also prove two variational formulas for the effective diffusivity. As a consequence, we obtain both upper and lower bounds on the effective diffusivity. Received: 17 December 1996/Revised revision: 9 February 1998     

Stability of infinite clusters in supercritical percolation

. A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p ₁<p ₂≤1 and percolation occurs at level p ₁, then every infinite cluster at level p ₂ contains some infinite cluster at level p ₁. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p ₁ there is a unique infinite cluster then the same holds at level p ₂. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls.