On the Distribution of the Area Enclosed by a Random Walk onZ2

Abstract

LetΓ2nbe the set of paths with 2nsteps of unit length inZ2, which begin and end at (0,0). Forγ∈Γ2n, letarea(γ)∈Zdenote the oriented area enclosed byγ. We show that for every positive even integerk, there exists a rational functionRkwith integer coefficients, such that:1Γ2n∑γ∈Gamma; 2n[area(γ)]k=Rk(n,n2k.We calculate explicitly the degree and leading coefficient ofRk. We show how as a consequence of this (and by also using the enumeration of up-down permutations, and the exponential formula for cycles of permutations) one can derive the asymptotic distribution of the area enclosed by a random path inΓ2n. The formula for the asymptotic distribution can be stated as follows: forα