On the Asymptotics and the Non-Holonomic Character of First Returns in the Standard Euclidean Lattice

TL;DR

This paper provides precise asymptotics for first return walks in the standard Euclidean lattice, proves their generating functions are non-holonomic, and explores their algebraic and differential properties using classical and new methods.

Contribution

It introduces new asymptotic formulas for first return walks, demonstrates their generating functions are not holonomic, and derives associated differential equations in multiple dimensions.

Findings

Asymptotics for first return walks are precisely characterized.

Generated functions are G-functions but not holonomic.

Derived new differential equations for dimensions up to 5.

Abstract

We give precise asymptotics to the number of first time returning random walks in the standard orthogonal lattice in $\mathbb{R}$ and we prove that these numbers do not form a $P$-recursive sequence. In the process, the known asymptotics of the number of closed walks are obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials. By showing that the relevant generating functions are $G$-functions, we use a form of the Hadamard convolution to find their singularities in all dimensions and give the ODEs that they satisfy for $d\leq 5$, some of which seem to be new. We use the Lucas property of the number of closed walks to prove that the corresponding generating function is not invertible as a $G$-function, which immediately implies that the generating function of the first time returning walks is not holonomic. We propose a few conjectures on the form of the asymptotic coefficients and of the ODEs.