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We consider the Potts model with q colors on a sequence of weighted graphs with adjacency matrices $$A_n$$ , allowing for both positive and negative weights. Under a mild regularity condition on $$A_n$$ we show that the mean-field prediction for the log partition function is asymptotically correct, whenever $${{\mathrm{tr}}}(A_n^2)=o(n)$$ . In particular, our results are applicable for the Ising and the Potts models on any sequence of graphs with average degree going to $$+\infty $$ . Using this, we establish the universality of the limiting log partition function of the ferromagnetic Potts model for a sequence of asymptotically regular graphs, and that of the Ising model for bi-regular bipartite graphs in both ferromagnetic and anti-ferromagnetic domain. We also derive a large deviation principle for the empirical measure of the colors for the Potts model on asymptotically regular graphs.

Let $${\mathcal {T}}$$ be a supercritical Galton–Watson tree with a bounded offspring distribution that has mean $$\mu >1$$, conditioned to survive. Let $$\varphi _{\mathcal {T}}$$ be a random embedding of $${\mathcal {T}}$$ into $${\mathbb {Z}}^d$$ according to a simple random walk step distribution. Let $${\mathcal {T}}_p$$ be percolation on $${\mathcal {T}}$$ with parameter p, and let $$p_c = \mu ^{-1}$$ be the critical percolation parameter. We consider a random walk $$(X_n)_{n \ge 1}$$ on $${\mathcal {T}}_p$$ and investigate the behavior of the embedded process $$\varphi _{{\mathcal {T}}_p}(X_n)$$ as $$n\rightarrow \infty $$ and simultaneously, $${\mathcal {T}}_p$$ becomes critical, that is, $$p=p_n\searrow p_c$$. We show that when we scale time by $$n/(p_n-p_c)^3$$ and space by $$\sqrt{(p_n-p_c)/n}$$, the process $$(\varphi _{{\mathcal {T}}_p}(X_n))_{n \ge 1}$$ converges to a d-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.

We consider simple generalizations of the potlatch and smoothing processes which were introduced in [8] and studied in [5]. These generalizations provide relatively simple examples of infinite interacting systems which exhibit phase transition. The original potlatch and smoothing processes do not exhibit phase transition. Our results show that for the generalized processes, phase transition does not usually occur in one or two dimensions, but usually does occur in higher dimensions. Upper and lower bounds for the relevant critical values are obtained. As one application of our results, we obtain the limiting behavior of the critical values for the linear contact process in d dimensions as d→∞, thus answering a question we raised in [3]. This is done by comparing the contact process with an appropriate generalized smoothing process.

LetX 1,X 2, ...,X n be a sequence of positive, independent, identically distributed random variables with the same distribution function (d.f.)F and denote byX 1:n ≦X 2:n ≦...≦X n:n the order statistics of the sample. We characterize the class of d.f.F for which $$P(X_{1:n} + X_{2:n} + \ldots + X_{n - i:n} > x) \sim P(X_{n - i:n} > x) as x \to \infty $$ for fixedn andi (i≦n-1), and we show that it is independent ofn. This leads to the genesis of a new class of d.f.L i ; we show that the sequence (L i ) ∞ =0 is strictly decreasing and we illustrate how the classesL i determine the probabilistic structure of the classL of subexponential distributions.

We introduce a new percolation model on planar lattices. First, impurities (“holes”) are removed independently from the lattice. On the remaining part, we then consider site percolation with some parameter p close to the critical value $$p_c$$ . The mentioned impurities are not only microscopic, but allowed to be mesoscopic (“heavy-tailed”, in some sense). For technical reasons (the proofs of our results use quite precise bounds on critical exponents in Bernoulli percolation), our study focuses on the triangular lattice. We determine explicitly the range of parameters in the distribution of impurities for which the connectivity properties of percolation remain of the same order as without impurities, for distances below a certain characteristic length. This generalizes a celebrated result by Kesten for classical near-critical percolation (which can be viewed as critical percolation with single-site impurities). New challenges arise from the potentially large impurities. This generalization, which is also of independent interest, turns out to be crucial to study models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate 1. If an occupied vertex is hit by lightning, which occurs at a very small rate $$\zeta $$ , its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities are instrumental in analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of “exceptional scales” (functions of $$\zeta $$ ). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $$\zeta \searrow 0$$ . This surprising behavior, related to the non-monotonicity of these processes, was not predicted in the physics literature.

We construct solutions to vector valued Burgers type equations perturbed by a multiplicative space–time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise.

When k(x, y) is a quasi-monotone function and the random variables X and Y have fixed distributions, it is shown under some further mild conditions that ℰ k(X, Y) is a monotone functional of the joint distribution function of X and Y. Its infimum and supremum are both attained and correspond to explicitly described joint distribution functions.