Edge-reinforced random walk on a ladder

We prove that the edge-reinforced random walk on the ladder ${\mathbb{Z}\times\{1,2\}}$ with initial weights $a>3/4$ is recurrent. The proof uses a known representation of the edge-reinforced random walk on a finite piece of the ladder as a random walk in a random environment. This environment is given by a marginal of a multicomponent Gibbsian process. A transfer operator technique and entropy estimates from statistical mechanics are used to analyze this Gibbsian process. Furthermore, we prove spatially exponentially fast decreasing bounds for normalized local times of the edge-reinforced random walk on a finite piece of the ladder, uniformly in the size of the finite piece.Comment: Published at http://dx.doi.org/10.1214/009117905000000396 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic behavior of edge-reinforced random walks

In this article, we study linearly edge-reinforced random walk on general multi-level ladders for large initial edge weights. For infinite ladders, we show that the process can be represented as a random walk in a random environment, given by random weights on the edges. The edge weights decay exponentially in space. The process converges to a stationary process. We provide asymptotic bounds for the range of the random walker up to a given time, showing that it localizes much more than an ordinary random walker. The random environment is described in terms of an infinite-volume Gibbs measure.Comment: Published at http://dx.doi.org/10.1214/009117906000000674 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Spontaneous breaking of rotational symmetry in the presence of defects

We prove a strong form of spontaneous breaking of rotational symmetry for a simple model of two-dimensional crystals with random defects in thermal equilibrium at low temperature. The defects consist of isolated missing atoms.Comment: 18 page

Infinite volume asymptotics of the ground state energy in a scaled poissonian potential

ABSTRACT. – We investigate the ground state energy of the random Schrödinger operator − 12+ β(log t)−2/dV on the box (−t, t)d with Dirichlet boundary conditions. V denotes the Poissonian potential which is obtained by translating a fixed non-negative compactly supported shape function to all the particles of a d-dimensional Poissonian point process. The scaling (log t)−2/d is chosen to be of critical order, i.e. it is determined by the typical size of the largest hole of the Poissonian cloud in the box (−t, t)d. We prove that the ground state energy (properly rescaled) converges to a deterministic limit I (β) with probability 1 as t →∞. I (β) can be expressed by a (deterministic) variational principle. This approach leads to a completely different method to prove the phase transition picture developed in [4]. Further we derive critical exponents in dimensions d 4 and we investigate the large-β-behavior, which asymptotically approaches a similar picture as for the unscaled Poissonian potential considered by Sznitman [9]. 2002 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. – Nous examinons l’énergie à l’état de base de l’opérateur de Schrödinger aléatoire − 12 + β(log t)−2/dV sur le cube (−t, t)d avec des conditions marginales de Dirichlet. V désigne le potentiel Poissonien obtenu par translation d’une fonction modèle fixe, non-négative e