A layered graph G× is the Cartesian product of a graph G=(V,E) with the linear graph Z, e.g. Z× is the 2D square lattice Z2. For Bernoulli percolation with parameter p∈[0,1] on G× one intuitively would expect that Pp((o,0)↔(v,n))≥Pp((o,0)↔(v,n+1)) for all o,v∈V and n≥0. This is reminiscent of the better known bunkbed conjecture. Here we introduce an approach to the above monotonicity conjecture that makes use of a Markov chain building the percolation pattern layer by layer. In case of finite G we thus can show that for some N≥0 the above holds for all n≥N o,v∈V and p∈[0,1]. One might hope that this Markov chain approach could be useful for other problems concerning Bernoulli percolation on layered graphs.

While 2D Gibbsian particle systems might exhibit orientational order resulting in a lattice-like structure, these particle systems do not exhibit positional order if the interaction between particles satisfies some weak assumptions. Here we investigate to which extent particles within a box of size $2n \times 2n$ may fluctuate from their ideal lattice position. We show that particles near the center of the box typically show a displacement at least of order $\sqrt{log n}$. Thus we extend recent results on the hard disk model to particle systems with fairly arbitrary particle spins and interaction. Our result applies to models such as rather general continuum Potts type models, e.g. with Widom-Rowlinson or Lenard-Jones-type interaction.

The hard disk model is a 2D Gibbsian process of particles interacting via pure hard core repulsion. At high particle density the model is believed to show orientational order, however, it is known not to exhibit positional order. Here we investigate to what extent particle positions may fluctuate. We consider a finite volume version of the model in a box of dimensions 2 n × 2 n with arbitrary boundary configuration, and we show that the mean square displacement of particles near the center of the box is bounded from below by c log n . The result generalizes to a large class of models with fairly arbitrary interaction.

We consider Gibbs distributions on permutations of a locally finite infinite set X subset of R, where a permutation sigma of X is assigned (formal) energy Sigma(x is an element of X) V(sigma(x) - x). This is motivated by Feynman's path representation of the quantum Bose gas; the choice X := Z and V(x) := alpha x(2) is of principal interest. Under suitable regularity conditions on the set X and the potential V, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.

We consider the model of Deijfen, Häggström and Bagley (2004) for competing growth of two infection types in R d , based on the Richardson model on Z d . Stochastic ball-shaped infection outbursts transmit the infection type of the center to all points of the ball that are not yet infected. Relevant parameters of the model are the initial infection configuration, the (type-dependent) growth rates, and the radius distribution of the infection outbursts. The main question is that of coexistence: Which values of the parameters allow the unbounded growth of both types with positive probability? Deijfen, Häggström and Bagley (2004) conjectured that the initial configuration is basically irrelevant for this question, and gave a proof for this under strong assumptions on the radius distribution, which, e.g. do not include the case of a deterministic radius. Here we give a proof that does not rely on these assumptions. One of the tools to be used is a slight generalization of the model with immune regions and delayed initial infection configurations.

Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations.

We consider a type of long-range percolation problem on the positive integers, motivated by earlier work of others on the appearance of (in)finite words within a site percolation model. The main issue is whether a given infinite binary word appears within an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and we provide partial results in others. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

One of the main objectives of equilibrium state statistical physics is to analyze which symmetries of an interacting particle system in equilibrium are broken or conserved. Here we present a general result on the conservation of translational symmetry for two-dimensional Gibbsian particle systems. The result applies to particles with internal degrees of freedom and fairly arbitrary interaction, including the interesting cases of discontinuous, singular, and hard core interaction. In particular we thus show the conservation of translational symmetry for the continuum Widom–Rowlinson model and a class of continuum Potts type models.

Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations.

The conservation of translation as a symmetry in two-dimensional systems with interaction is a classical subject of statistical mechanics. Here we establish such a result for Gibbsian particle systems with two-body interaction, where the interesting cases of singular, hard-core and discontinuous interaction are included. We start with the special case of pure hard core repulsion in order to show how to treat hard cores in general.

The conservation of continuous symmetries in two-dimensional systems with interaction is a classical subject of statistical mechanics. So far, all results of this sort required some smoothness properties of the interaction. Only recently Ioffe et al. (Comm. Math. Phys. 226 (2002) 433) succeeded to treat the case of lattice systems with continuous, rather than smooth, interaction. Here we establish a similar result for Gibbsian systems of point particles with internal degrees of freedom.

Aldous' spectral gap conjecture asserts that on any graph the random walk process and the random transposition (or interchange) process have the same spectral gap. We prove the conjecture using a recursive strategy. The approach is a natural extension of the method already used to prove the validity of the conjecture on trees. The novelty is an idea based on electric network reduction, which reduces the problem to the proof of an explicit inequality for a random transposition operator involving both positive and negative rates. The proof of the latter inequality uses suitable coset decompositions of the associated matrices on permutations. Aldous' conjecture concerns the spectral gap, a quantity that plays an important role in the analysis of the convergence to equilibrium of reversible Markov chains. We begin by reviewing some well known facts about Markov chains and their spectral gaps. For details we refer to [2]. 1.1. Finite state, continuous time Markov chains. Let us consider a continuous time Markov chain Z = (Z t) t 0 with finite state space S and transition rates (q i,j : i = j ∈ S) such that q i,j 0. We will always assume that the Markov chain is irreducible and satisfies q i,j = q j,i for all i = j. Such a Markov chain is reversible with respect to the uniform distribution ν on S, which is the unique stationary distribution of the chain. The infin-itesimal generator L of the Markov chain is defined by Lg(i) = j∈S q i,j (g(j) − g(i)) ,