Expected maximum of bridge random walks & L\'evy flights

TL;DR

This paper derives an exact formula for the expected maximum of symmetric bridge random walks, including Le9vy flights, revealing how the bridge condition affects the maximum and applying results to polymers, particles, and capture problems.

Contribution

It provides a novel exact analytical expression for the expected maximum of bridge RWs with arbitrary symmetric jump distributions, including Le9vy flights, and analyzes its large n behavior.

Findings

Expected maximum scales as n^{1/e9} for large n.

The amplitude h_1(e9) depends on the bridge condition and differs from free RWs.

Second order term is constant for e9=2 and grows with n for e9<2.

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.