Lower bounds for bulk deviations for the simple random walk on $\mathbb{Z}^d$, $d\geq 3$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911070922211328 |
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| author | Chiarini, Alberto Nitzschner, Maximilian |
| author_facet | Chiarini, Alberto Nitzschner, Maximilian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17074 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lower bounds for bulk deviations for the simple random walk on $\mathbb{Z}^d$, $d\geq 3$ Chiarini, Alberto Nitzschner, Maximilian Probability Mathematical Physics 60F10, 60K35, 60J27, 60J55, 82B43 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. |
| title | Lower bounds for bulk deviations for the simple random walk on $\mathbb{Z}^d$, $d\geq 3$ |
| topic | Probability Mathematical Physics 60F10, 60K35, 60J27, 60J55, 82B43 |
| url | https://arxiv.org/abs/2312.17074 |