Expected maximum of bridge random walks & Lévy flights

Abstract

We consider one-dimensional discrete-time random walks (RWs) with arbitrary symmetric and continuous jump distributions f(η), including the case of Lévy flights. We study the expected maximum of bridge RWs, i.e. RWs starting and ending at the origin after n steps. We obtain an exact analytical expression for valid for any n and jump distribution f(η), 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 with a Lévy index 0 < μ ⩽ 2 and an arbitrary length scale a > 0, we find that, at leading order for large n, . We obtain an explicit expression for the amplitude h 1(μ) and find that it carries the signature of the bridge condition, being different from its counterpart for the free RW. For μ = 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 < μ < 2, this second leading order term is a growing function of n, which depends non-trivially on further details of , beyond the Lévy index μ. 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.