Large deviations for self-intersection local times in subcritical dimensions

Clément Laurent (Université de Provence)


Let $(X_t,t\geq 0)$ be a simple symmetric random walk on $\mathbb{Z}^d$ and for any $x\in\mathbb{Z}^d$, let $ l_t(x)$ be its local time at site $x$. For any $p>1$, we denote by$ I_t= \sum\limits_{x\in\mathbb{Z}^d} l_t(x)^p $ the p-fold self-intersection local times (SILT). Becker and König recently proved a large deviations principle for $I_t$ for all $p>1$ such that $p(d-2/p)<2$. 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 $p(d-2)=d$.

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Pages: 1-20

Publication Date: March 14, 2012

DOI: 10.1214/EJP.v17-1874


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