Upper tails for intersection local times of random walks in supercritical dimensions. (English) Zbl 1170.60019

Chen, Xia; Mörters, Peter

J. Lond. Math. Soc., II. Ser. 79, No. 1, 186-210 (2009).

Let \((X^{(1)}(n):n\in\mathbb{N}),\ldots,(X^{(p)}(n):n\in\mathbb{N})\) be i.i.d. random walks taking values in \(\mathbb{Z}^d\). The intersection local time \( I= \sum_{i_1=1}^{\infty}\ldots\sum_{i_p=1}^{\infty} \mathbb{I}\{X^{(1)}(i_1)=\ldots = X^{(p)}(i_p)\}. \) The main result states that \[ \lim_{a\uparrow \infty} \frac{1}{a^{1/p}} \log \mathbb{P}\{I>a\} = -p \inf \{\|h\|_q: h\geq 0\;\;\text{with}\;\;\|\mathcal{U}_h\|\geq 1\} \] where \(\mathcal{U}_h\) is a bounded symmetric operator on \(L^2(\mathbb{Z}^d)\) (with an explicit formula), \(q^{-1} = 1- p^{-1}\). To establish this result, the authors propose a method avoiding the use of the Donsker–Varadhan large deviation theory.

Reviewer: Alexander V. Bulinski (Moskva)

MSC:

60G50	Sums of independent random variables; random walks
60F10	Large deviations
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

random walk; intersection local time; supercritical dimensions.