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 $2n times 2n$ 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 $Xsubsetmathbb{R}$, where a permutation $sigma$ of $X$ is assigned (formal) energy $sum_{xin X}V(sigma(x)-x)$. This is motivated by Feynmans path representation of the q
uantum Bose gas; the choice $X:=mathbb{Z}$ 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 et al. 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 et al. conjectured that the initial configuration basically is 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 doesnt 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 w
ithin an iid Bernoulli sequence at locations that satisfy certain constraints. We settle the issue in some cases, and provide partial results in others.
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.
