Three-Step and Four-Step Random Walk Integrals

Abstract

We investigate the moments of 3-step and 4-step uniform random walk in the plane. In particular, we further analyse a formula conjectured in [BNSW09] expressing 4-step moments in terms of 3-step moments. Diverse related results including hypergeometric and elliptic closed forms for W 4 (±1) are given and two new conjectures are recorded.

High-Precision Arithmetic in Mathematical Physics

“…Integrals such as (4) are the s-th moment of the distance to the origin after n steps. In [77], it is shown that when s = 0, the first derivatives of these integrals, which arise independently in the study of Mahler measures, are…”

Densities of short uniform random walks in higher dimensions

“…We note that, as in [BSW13], this Meijer G-function expression can be expressed as a sum of hypergeometric functions by Slater's Theorem [Mar83,p. 57].…”

“…As much as possible, we keep our notation consistent with that in [BNSW11,BSW13], and especially [BSWZ12], to which we refer for details of how to exploit the Mellin transform and similar matters. Random walks in higher dimensions are also briefly discussed in [Wan13,Chapter 4].…”

“…This paper is a companion to [BNSW11,BSW13] and [BSWZ12], which studied the analytic and number theoretic behaviour of short uniform random walks in the plane (five steps or less). In this work we revisit the same issues in higher dimensions.…”

Closed Forms: What They Are and Why We Care

“…These Meijer-G functions, first introduced in 1936, also occur in quantum field theory and many other places [8]. For example, the moments of an n-step random walk in the plane are given for s > 0 by It transpires [24], [36] that for all complex s (6) Example 9 (Trefethen's problem #10 [16], [19]). A particle at the center of a 10 × 1 rectangle undergoes Brownian motion (i.e., 2-D random walk with infinitesimal step lengths) till it hits the boundary.…”