Filters

2007 - 200922020 - 20211

More filters

2019 - 201922020 - 20221
Reset filters

Settings

Author: Bianchi, Alessandra × Version: publishedVersion ×

Search Results

Now showing 1 - 3 of 3
  • Thumbnail Image
    Item
    Limit theorems for Lévy flights on a 1D Lévy random medium
    ([Madralin] : EMIS ELibEMS, 2021) Stivanello, Samuele; Bet, Gianmarco; Bianchi, Alessandra; Lenci, Marco; Magnanini, Elena
    We study a random walk on a point process given by an ordered array of points (ωk,k∈Z) on the real line. The distances ωk+1−ωk are i.i.d. random variables in the domain of attraction of a β-stable law, with β∈(0,1)∪(1,2). The random walk has i.i.d. jumps such that the transition probabilities between ωk and ωℓ depend on ℓ−k and are given by the distribution of a Z-valued random variable in the domain of attraction of an α-stable law, with α∈(0,1)∪(1,2). Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.
  • Thumbnail Image
    Item
    Glauber dynamics on hyperbolic graphs : boundary conditions and mixing time
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Bianchi, Alessandra
    We study a continuous time Glauber dynamics reversible with respect to the Ising model on hyperbolic graphs and analyze the effect of boundary conditions on the mixing time. Specifically, we consider the dynamics on an $n$-vertex ball of the hyperbolic graph $H(v,s)$, where $v$ is the number of neighbors of each vertex and $s$ is the number of sides of each face, conditioned on having $(+)$-boundary. If $v>4$, $s>3$ and for all low enough temperatures (phase coexistence region) we prove that the spectral gap of this dynamics is bounded below by a constant independent of $n$. This implies that the mixing time grows at most linearly in $n$, in contrast to the free boundary case where it is polynomial with exponent growing with the inverse temperature $b$. Such a result extends to hyperbolic graphs the work done by Martinelli, Sinclair and Weitz for the analogous system on regular tree graphs, and provides a further example of influence of the boundary condition on the mixing time.
  • Thumbnail Image
    Item
    Sharp asymptotics for metastability in the random field Curie-Weiss model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Bianchi, Alessandra; Bovier, Anton; Ioffe, Dmitry
    In this paper we study the metastable behavior of one of the simplest disordered spin system, the random field Curie-Weiss model. We will show how the potential theoretic approach can be used to prove sharp estimates on capacities and metastable exit times also in the case when the distribution of the random field is continuous. Previous work was restricted to the case when the random field takes only finitely many values, which allowed the reduction to a finite dimensional problem using lumping techniques. Here we produce the first genuine sharp estimates in a context where entropy is important.