Universal Order Statistics for Random Walks & Levy Flights

We consider one-dimensional discrete-time random walks (RWs) of n steps, starting from x(0) = 0, with arbitrary symmetric and continuous jump distributions f (eta), including the important case of Levy flights. We study the statistics of the gaps Delta(k,n) between the kth and (k + 1)th maximum of the set of positions (x(1), ..., x(n)). We obtain an exact analytical expression for the probability distribution P-k,P-n(Delta) valid for any k and n, and jump distribution f (eta), which we then analyse in the large n limit. For jump distributions whose Fourier transform behaves, for small q, as (f) over cap (q) similar to 1 - vertical bar q vertical bar(mu) with a Levy index 0 < mu <= 2, we find that the distribution becomes stationary in the limit of n -> infinity, i.e. lim(n ->infinity) P-k,P-n (Delta) = P-k(Delta). We obtain an explicit expression for its first moment E[Delta(k)], valid for any k and jump distribution f (eta) with mu > 1, and show that it exhibits a universal algebraic decay E[Delta(k)] similar to k(1/mu-1) Gamma (1 - 1/mu)/pi for large k. Furthermore, for mu > 1, we show that in the limit of k -> infinity the stationary distribution exhibits a universal scaling form P-k(Delta) similar to k(1-1/mu )P(mu)(k(1-1/mu) Delta) which depends only on the Levy index mu, but not on the details of the jump distribution. We compute explicitly the limiting scaling function P-mu(x) in terms of Mittag-Leffler functions. For 1 < mu < 2, we show that, while this scaling function captures the distribution of the typical gaps on the scale k(1/mu-1), the atypical large gaps are not described by this scaling function since they occur at a larger scale of order k(1/mu). This atypical part of the distribution is reminiscent of a "condensation bump" that one often encounters in several mass transport models.