From lattice Quantum Electrodynamics to the distribution of the algebraic areas enclosed by random walks on $Z^2$

TL;DR

This paper explores the relationship between lattice quantum electrodynamics and the distribution of algebraic areas enclosed by random walks, providing a combinatorial framework and algorithms for calculating moments of these areas.

Contribution

It introduces a novel combinatorial approach to analyze moments of algebraic areas in lattice loops, connecting quantum field theory with random walk geometry.

Findings

Moments are products of combinatorial factors and polynomials in step counts.

An explicit algorithm for low-order moment formulas is developed.

The structure of moments is characterized for fixed edge counts in lattice loops.

Abstract

In the worldline formalism, scalar Quantum Electrodynamics on a 2-dimensional lattice is related to the areas of closed loops on this lattice. We exploit this relationship in order to determine the general structure of the moments of the algebraic areas over the set of loops that have fixed number of edges in the two directions. We show that these moments are the product of a combinatorial factor that counts the number of such loops, by a polynomial in the numbers of steps in each direction. Our approach leads to an algorithm for obtaining explicit formulas for the moments of low order.