arXiv subjects

Renato Soares Dos Santos

Publications and source records attributed to Renato Soares Dos Santos.

3 recordsLinked to original sources

arXiv2015

The quenched limiting distributions of a charged-polymer model

The limit distributions of the charged-polymer Hamiltonian of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths and Higgs [Gaussian case] are considered. Two sources of randomness enter in the definition: a random field $q= (q_i)_{i\geq 1}$ of i.i.d. random variables (called random charges) and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the charges. The energy or Hamiltonian $K = (K_n)_{n \geq 2}$ is then defined as $$K_n := \sum_{1\leq i < j\leq n} q_i q_j {\bf 1}_{\{S_i=S_j\}}.$$ The law of $K$ under the joint law of $q$ and $S$ is called "annealed", and the conditional law given $q$ is called "quenched". Recently, strong approximations under the annealed law were proved for $K$. In this paper we consider the limit distributions of $K$ under the quenched law.

math.PR↗

arXiv2014

A Quenched Functional Central Limit Theorem for Random Walks in Random Environments under $(T)_γ$

We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppäläinen in [10] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_γ$ condition of Sznitman we reduce the moment condition to ${\Bbb E}(τ^2(\ln τ)^{1+m})<+\infty$ for $m>1+1/γ$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.

math.PR↗

arXiv2013

A Quenched Functional Central Limit Theorem for Planar Random Walks in Random Sceneries

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $ξ= (ξ_x)_{x \in \Z^d}$ of i.i.d.\ random variables, which is called the \emph{random scenery}, and a random walk $S = (S_n)_{n \in \N}$ evolving in $\Z^d$, independent from the scenery. The RWRS $Z = (Z_n)_{n \in \N}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n ξ_{S_k}$. The law of $Z$ under the joint law of $ξ$ and $S$ is called "annealed", and the conditional law given $ξ$ is called "quenched". Recently, central limit theorems under the quenched law were proved for $Z$ by the first two authors for a class of transient random walks including walks with finite variance in dimension $d \ge 3$. In this paper we extend their results to dimension $d=2$.

math.PR↗