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# Annales de l’Institut Henri Poincaré D

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**Volume 3, Issue 4, 2016, pp. 381–404**

**DOI: 10.4171/AIHPD/33**

Published online: 2016-12-22

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

Thomas Epelbaum^{[1]}, François Gelis

^{[2]}and Bin Wu

^{[3]}(1) McGill University, Montreal, Canada

(2) CEA/DSM/Saclay , Gif-Sur-Yvette, France

(3) The Ohio State University, Columbus, USA

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.

*Keywords: *Random walks on $\mathbf Z^2$, algebraic areas, lattice quantum electrodynamics

Epelbaum Thomas, Gelis François, Wu Bin: From lattice Quantum Electrodynamics to the distribution of the algebraic areas enclosed by random walks on $\mathbf Z^2$. *Ann. Inst. Henri Poincaré Comb. Phys. Interact.* 3 (2016), 381-404. doi: 10.4171/AIHPD/33