Densities of short uniform random walks in higher dimensions

Abstract

We study arithmetic properties of short uniform random walks in arbitrary dimensions, with a focus on explicit (hypergeometric) evaluations of the moment functions and probability densities in the case of up to five steps. Somewhat to our surprise, we are able to provide complete extensions to arbitrary dimensions for most of the central results known in the twodimensional case.

Wick rotations, Eichler integrals, and multi-loop Feynman diagrams

“…Statement of results and plan of proof. In this article, we supplement our previous work with complex analysis and modular forms, which are two powerful devices that not only produce new algebraic relations among different IKM moments, but also connect Feynman diagrams to special L-values and Kluyver's "random walk integrals" JYM(n, 0, 1), n ∈ Z ≥5 [24,8,7]. The layout of this paper is described in the next four paragraphs.…”

Some Algebraic and Arithmetic Properties of Feynman Diagrams

“…characterizes the probability density of the distance x traveled by a rambler, who walks in the Euclidean plane, taking n consecutive steps of unit lengths, aiming at uniformly distributed random directions. The analytic properties of such probability densities have been extensively studied [4,8,7,6,51]. Recently, we have shown [51, Theorem 5.1] that p n (x) is expressible through Feynman diagrams when n is odd, as stated in the theorem below.…”

On Borwein’s Conjectures for Planar Uniform Random Walks

“…The analytic and arithmetic properties of the 3-step density p 3 (x) and the 4-step density p 4 (x) for planar uniform random walks have been thoroughly explored by Borwein and coworkers [3,6,5]. In [6, §5], Borwein et al investigated the Maclaurin expansion p 5 (x) = ∑ ∞ k=0 r 5,k x 2k+1 for small and positive x, arriving at a closed-form evaluation [ 1 In this work, we write f ′ (0 + ) for the one-sided limit of the derivative, namely, lim where J 1 (t) = − d J 0 (t)/ d t. The integral on the left-hand side of (1.5) can be evaluated in closed form [4,Example 4.15], so the original conjecture in (1.4) has been verified by a connection between 2dimensional and 4-dimensional random walks [4,Theorem 4.17]. Let be Pearson's n-step ramble integral for complex-valued s. For n ∈ Z >1 and Re s > 0, the convergent ramble integral is related to Kluyver's probability density by a moment formula W n (s) = ∞ 0 x s p n (x) d x [6, (2.3)].…”

“…p 3 (x), p 4 (x) or p 5 (x). As p ′ 6 (0 + ) is finite, the function p 6 (x) differs qualitatively from p4 (x) = − 3x 2π 2 log x+ O(x), x → 0 + [6,(4.4)]. Even though (4.8) holds, numerical computations reveal that one cannot equatep 6 (x) with 30 π 0 (xt)[I 0 (t)] 2 [K 0 (t)] 4 xt d t(4.14) for x > 0, contrary to the situations in p 3 (x) and p 5 (x).…”