arXiv subjects
Publications and source records attributed to Clément Laurent.
We derive asymptotic estimates for the velocity of random walks in random environments which are perturbations of the simple symmetric random walk but have a small local drift in a given direction. Our estimates complement previous results presented by Sznitman and are in the spirit of expansions obtained by Sabot.
Let $(X_t, t \geq 0)$ be an $α$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t δ_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -α) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \geq r_t)$ for all scales $r_t \gg \E(I_t)$. Our result extends previous works by Chen, Li and Rosen 2005, Becker and König 2010, and Laurent 2012.
Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_t(x)= \int_0^t δ_x(X_s)ds$ be the local time at site $x$ and $ I_t= \sum\limits_{x\in\mathbb{Z}^d} l_t(x)^p $ the p-fold self-intersection local time (SILT). Becker and König have recently proved a large deviations principle for $I_t$ for all $(p,d)\in\mathbb{R}^d\times\mathbb{Z}^d$ such that $p(d-2)<2$. We extend these results to a broader scale of deviations and to the whole subcritical domain $p(d-2)<d$. Moreover we unify the proofs of the large deviations principle using a method introduced by Castell for the critical case $p(d-2)=d$ and developed by Laurent for the critical and supercritical case $p(d-α)\geq d$ of $α$-stable random walk.
Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_T(x)= \int_0^T δ_x(X_s)ds$ the local time at the state $x$ and $ I_T= \sum\limits_{x\in\mathbb{Z}^d} l_T(x)^q $ the q-fold self-intersection local time (SILT). In \cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and Mörters obtain in \cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \cite{Asselah5} a large deviations principle for the SILT with $q=2$. We extend these results to an $α$-stable process (i.e. $α\in]0,2]$) in the case where $q(d-α)\geq d$.