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Journal Articles Journal of Statistical Mechanics: Theory and Experiment Year : 2021

Expected maximum of bridge random walks & Levy flights

Abstract

We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions $f(\eta)$, including the case of L\'evy flights. We study the expected maximum ${\mathbb E}[M_n]$ of bridge RWs, i.e., RWs starting and ending at the origin after $n$ steps. We obtain an exact analytical expression for ${\mathbb E}[M_n]$ valid for any $n$ and jump distribution $f(\eta)$, which we then analyze in the large $n$ limit up to second leading order term. For jump distributions whose Fourier transform behaves, for small $k$, as $\hat f(k) \sim 1 - |a\, k|^\mu$ with a L\'evy index $0<\mu \leq 2$ and an arbitrary length scale $a>0$, we find that, at leading order for large $n$, ${\mathbb E}[M_n]\sim a\, h_1(\mu)\, n^{1/\mu}$. We obtain an explicit expression for the amplitude $h_1(\mu)$ and find that it carries the signature of the bridge condition, being different from its counterpart for the free random walk. For $\mu=2$, we find that the second leading order term is a constant, which, quite remarkably, is the same as its counterpart for the free RW. For generic $0< \mu < 2$, this second leading order term is a growing function of $n$, which depends non-trivially on further details of $\hat f (k)$, beyond the L\'evy index $\mu$. Finally, we apply our results to compute the mean perimeter of the convex hull of the $2d$ Rouse polymer chain and of the $2d$ run-and-tumble particle, as well as to the computation of the survival probability in a bridge version of the well-known "lamb-lion" capture problem.

Dates and versions

hal-03388319 , version 1 (20-10-2021)

Identifiers

Cite

Benjamin de Bruyne, Satya N. Majumdar, Gregory Schehr. Expected maximum of bridge random walks & Levy flights. Journal of Statistical Mechanics: Theory and Experiment, 2021, 8, pp.083215. ⟨10.1088/1742-5468/ac150c⟩. ⟨hal-03388319⟩
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