On the Distribution of the AreaEnclosed

Let ? 2n be the set of paths with 2n steps of unit length in Z 2 , which begin and end at (0; 0). For 2 ? 2n , let area() 2 Z denote the oriented area enclosed by. We show that for every positive even integer k, there exists a rational function R k with integer coeecients, such that: 1 j? 2n j X 2?2n area()] k = R k (n); n > 2k: We calculate explicitly the degree and leading coeecient of R k. 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 in ? 2n. The formula for the asymptotic distribution can be stated as follows: for < in R lim n!1 f 2 ? 2n j nn < area() < nng j? 2n j = 1 2 (tanh() ? tanh()): Since the appropriately re-normalized random walks with n steps on Z 2 converge, as n ! 1, to the 2-dimensional Brownian motion, this argument can be viewed as a combinatorial approach to the formula of P. L evy for the area enclosed by a random 2-dimensional Brownian path.