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

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:1|Γ2n| ∑γ∈Gamma;2n [area(γ)]k=Rk(n, n>2k.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<βinRlimn→∞|{γ∈Γ2n | nα<area(γ)<nβ}||Γ2n|=12 (tanh(πβ)−tanh(πα)).Since the appropriately re-normalized random walks withnsteps onZ2converge, asn→∞, to the 2-dimensional Brownian motion, this argument can be viewed as a combinatorial approach to the formula of Lévy for the area enclosed by a random 2-dimensional Brownian path.