Area versus Length Distribution for Closed Random Walks

TL;DR

This paper derives an exact formula for counting closed random walks of given length and area on a hypercubic lattice in high dimensions, revealing asymptotic behaviors and area distribution for long loops, along with new mathematical identities.

Contribution

It introduces a novel connection between the $q$-oscillator algebra and high temperature expansion coefficients to exactly count closed random walks by length and area.

Findings

Exact formula for walk counts in high dimensions

Asymptotic behavior of area distribution for long loops

Derivation of new mathematical identities

Abstract

Using a connection between the $q$-oscillator algebra and the coefficients of the high temperature expansion of the frustrated Gaussian spin model, we derive an exact formula for the number of closed random walks of given length and area, on a hypercubic lattice, in the limit of infinite number of dimensions. The formula is investigated in detail, and asymptotic behaviours are evaluated. The area distribution in the limit of long loops is computed. As a byproduct, we obtain also an infinite set of new, nontrivial identities.