The Maximum of a Random Walk and Its Application to Rectangle Packing

Abstract

Let S0,…,Sn be a symmetric random walk that starts at the origin (S0 = 0) and takes steps uniformly distributed on [— 1,+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate,where c = 0.297952.... This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, O(n½), when the rectangle sides are 2n independent uniform random draws from [0,1].

Unified Solution of the Expected Maximum of a Discrete Time Random Walk and the Discrete Flux to a Spherical Trap

“…(10a) is equal to six times the flux at time n for precisely this one-dimensional problem. Summing the data for a (2) (n)/6 up to time n (some of which data is presented in Table I of that paper), we indeed find that Eq. (3) (including c) is satisfied, with the next correction approximately equal to 0.0921n −1/2 .…”

“…√ n behavior is easy to understand and can be derived from the corresponding behavior of a continuoustime Brownian motion after a suitable rescaling [2]. However, the leading finite-size correction term turns out to be a nontrivial constant −c with c = 0.29795219028 .…”

“…that was computed in Ref. [2] by enumerating a somewhat intricate double series obtained after a lengthy calculation. Recently, it was shown [3] that for arbitrary continuous and symmetric jump distribution f (x) with a finite second moment σ 2 = ∞ 0 x 2 f (x)dx, the expected maximum has a similar asymptotic behavior as in the uniform case, namely,…”

“…The similarity of these constants suggests that these two problems may be intimately related. For the first problem, the constant was first computed numerically by evaluating a rather complicated double series expansion [1,2] and very recently, an exact closed form expression of the constant has been found [3] that is valid not just for the uniform jump distribution, but for any arbitrary continuous and symmetric jump distribution. For the second problem of flux to a spherical trap, the corresponding constant was computed only numerically [4].…”

“…It was shown in Ref. [2] that for the special case of the uniform jump distribution, f (x) = 1/2 for −1 ≤ x ≤ 1 and f (x) = 0 outside, for large n, A typical configuration of a random walker in one dimension up to n steps, starting at 0 at n = 0 with Mn denoting the maximum up to n steps.…”