Lower bounds for bulk deviations for the simple random walk on $\mathbb{Z}^d$, $d\geq 3$
| Autor: | Chiarini, Alberto, Nitzschner, Maximilian |
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| Rok vydání: | 2023 |
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| Druh dokumentu: | Working Paper |
| Popis: | This article investigates the behavior of the continuous-time simple random walk on $\mathbb{Z}^d$, $d \geq 3$. We derive an asymptotic lower bound on the principal exponential rate of decay for the probability that the average value over a large box of some non-decreasing local function of the field of occupation times of the walk exceeds a given positive value. This bound matches at leading order the corresponding upper bound derived by Sznitman in arXiv:1906.05809, and is given in terms of a certain constrained minimum of the Dirichlet energy of functions on $\mathbb{R}^d$ decaying at infinity. Our proof utilizes a version of tilted random walks, a model originally constructed by Li in arXiv:1412.3959 to derive lower bounds on the probability of the event that the trace of a simple random walk disconnects a macroscopic set from an enclosing box. Comment: 49 pages, 1 figure |
| Databáze: | arXiv |
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