# From lattice Quantum Electrodynamics to the distribution of the algebraic areas enclosed by random walks on $Z_{2}$

### Thomas Epelbaum

McGill University, Montreal, Canada### François Gelis

CEA/DSM/Saclay , Gif-Sur-Yvette, France### Bin Wu

The Ohio State University, Columbus, USA

## Abstract

In theworldline 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 xed 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.

## Cite this article

Thomas Epelbaum, François Gelis, Bin Wu, From lattice Quantum Electrodynamics to the distribution of the algebraic areas enclosed by random walks on $Z_{2}$. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 3 (2016), no. 4, pp. 381–404

DOI 10.4171/AIHPD/33