We prove that the q-state Potts model and the random-cluster model with cluster weight q > 4 undergo a discontinuous phase transition on the square lattice. More precisely, we show 1. Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, 2. Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and 3. Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models.The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical randomcluster and Potts models, and show that they behave as exp(π 2 √ q − 4) as q tends to 4.Section 4: Fourier computations. The study will require certain computations using Fourier decompositions. While these computations are elementary, they may be lengthy, and would break the pace of the proofs. We therefore defer all of them to Section 4.Notation. Most functions hereafter depend on the parameter ∆ = 2−c 2 2 < −1. For ease of notation, we will generally drop the dependency in ∆, and recall it only when it is relevant. We write ∂ i for the partial derivative in the i th coordinate.

In this work we principally study random walk on the supercritical infinite cluster for bond percolation on Z d . We prove a quenched functional central limit theorem for the walk when d ≥ 4. We also prove a similar result for random walk among i.i.d. random conductances along nearest neighbor edges of Z d , when d ≥ 1. IntroductionConsider supercritical bond-percolation on Z d , d ≥ 2, and the simple random walk on the infinite cluster, which at each jump picks with equal probability one of the neighboring sites in the infinite cluster. Is it true that for a.e. configuration such that the origin belongs to the infinite cluster, the random walk starting at the origin exits a symmetric slab through either side with probability tending to 1 2 , as the width of the slab tends to infinity?We give in this work a positive answer to this question when d ≥ 4. This answer comes as a consequence of a more general quenched invariance principle for the walk on the supercritical infinite cluster, when d ≥ 4. We also discuss the related case of a walk evolving in a network of i.i.d. random conductances placed along nearest-neighbor edges of Z d . We are able to prove a quenched invariance principle in this situation, for general d ≥ 1. For partial results in this direction, we refer to Anshelevich-Khanin-Sinai [2], Boivin [5], Boivin-Depauw [6], Kozlov [19].Before discussing our results any further, we describe the models more precisely. We begin with random walk on the supercritical infinite cluster. We let B d stand for the set of nearest-neighbor bonds (or edges) on Z d , d ≥ 2, and = {0, 1} B d for the set of configurations. We denote by P the product measure on endowed with its canonical σ -algebra A, under which the canonical coordinates, ω(b), b ∈ B d , are Bernoulli variables with success probability p ∈ (0, 1). V. Sidoravicius: IMPA, Estrada Dona Castorina 110, Jardim Botanico, CEP 22460-320, Rio de Janeiro, RJ, Brasil. e-mail: vladas@impa.br V. Sidoravicius would like to thank the FIM for financial support and hospitality during his multiple visits to ETH. His research was also partially supported by FAPERJ and CNPq. A.-S. Sznitman:

The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on Z d in d = 3 dimensions. The analysis applies also to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. The proof employs in an essential way an extension of Ising model's random current representation to the model's infinite volume limit. Using it, we relate the continuity of the magnetization to the vanishing of the free boundary condition Gibbs state's Long Range Order parameter. For reflection positive models the resulting criterion for continuity may be established through the infrared bound for all but the borderline lower dimensional cases. The exclusion applies to the one dimensional model with 1 r 2 interaction for which the spontaneous magnetization is known to be discontinuous at T c . x∈Λ hσ x − {x,y}⊂Λ∶x≠y

We study the long-time behavior of conservative interacting particle systems in Z: the activated random walk model for reaction-diffusion systems and the stochastic sandpile. We prove that both systems undergo an absorbing-state phase transition.This preprint has the same numbering for sections, theorems, equations and figures as the published article "Invent. Math. 188 (2012): 127-150."

We consider the following interacting particle system: There is a ``gas'' of particles, each of which performs a continuous-time simple random walk on $\mathbb{Z}^d$, with jump rate $D_A$. These particles are called $A$-particles and move independently of each other. They are regarded as individuals who are ignorant of a rumor or are healthy. We assume that we start the system with $N_A(x,0-)$ $A$-particles at $x$, and that the $N_A(x,0-),x\in\mathbb{Z}^d$, are i.i.d., mean-$\mu_A$ Poisson random variables. In addition, there are $B$-particles which perform continuous-time simple random walks with jump rate $D_B$. We start with a finite number of $B$-particles in the system at time 0. $B$-particles are interpreted as individuals who have heard a certain rumor or who are infected. The $B$-particles move independently of each other. The only interaction is that when a $B$-particle and an $A$-particle coincide, the latter instantaneously turns into a $B$-particle. We investigate how fast the rumor, or infection, spreads. Specifically, if $\widetilde{B}(t):=\{x\in\mathbb{Z}^d:$ a $B$-particle visits $x$ during $[0,t]\}$ and $B(t)=\widetilde{B}(t)+[-1/2,1/2]^d$, then we investigate the asymptotic behavior of $B(t)$. Our principal result states that if $D_A=D_B$ (so that the $A$- and $B$-particles perform the same random walk), then there exist constants $0

No abstract

We investigate random interlacements on Z d , d 3. This model, recently introduced in [8], corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times. A nonnegative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d 3 the vacant set at level u percolates when u is small. This solves an open problem of [8], where this fact has only been established when d 7. It also completes the proof of the nondegeneracy in all dimensions d 3 of the critical parameter u of [8].

We address the question of how a localized microscopic defect, especially if it is small with respect to certain dynamic parameters, affects the macroscopic behavior of a system. In particular we consider two classical exactly solvable models: Ulam's problem of the maximal increasing sequence and the totally asymmetric simple exclusion process. For the first model, using its representation as a Poissonian version of directed last passage percolation on R 2 , we introduce the defect by placing a positive density of extra points along the diagonal line. For the latter, the defect is produced by decreasing the jump rate of each particle when it crosses the origin.The powerful algebraic tools for studying these processes break down in the perturbed versions of the models. Taking a more geometric approach we show that in both cases the presence of an arbitrarily small defect affects the macroscopic behavior of the system: in Ulam's problem the time constant increases, and for the exclusion process the flux of particles decreases. This, in particular, settles the longstanding "Slow Bond Problem".