On Borwein's conjectures for planar uniform random walks

TL;DR

This paper proves Borwein's conjectures related to planar uniform random walks by evaluating specific derivatives of probability densities and establishing uniform convergence of their Maclaurin expansions, connecting probability theory with quantum field theory.

Contribution

It provides a closed-form evaluation of a third-order derivative of the probability density and proves the uniform convergence of Maclaurin expansions for all odd n ≥ 5, confirming Borwein's conjectures.

Findings

Closed-form expression for p_5'''(0+).

Uniform convergence of Maclaurin series for p_n(x), n odd and ≥ 5.

Connection established between random walks and quantum field theory.

Abstract

Let $ p_n(x)=\int_0^\infty J_0(xt)[J_0(t)]^n xt\,\mathrm{d}\, t$ be Kluyver's probability density for $n$-step uniform random walks in the Euclidean plane. Through connection to a similar problem in 2-dimensional quantum field theory, we evaluate the third-order derivative $ p_5'''(0^{+})$ in closed form, thereby giving a new proof for a conjecture of J. M. Borwein. By further analogies to Feynman diagrams in quantum field theory, we demonstrate that $ p_n(x),0\leq x\leq 1$ admits a uniformly convergent Maclaurin expansion for all odd integers $ n\geq5$, thus settling another conjecture of Borwein.