Convex hulls of random walks, hyperplane arrangements, and Weyl chambers

TL;DR

This paper derives a distribution-free formula for the probability that the convex hull of an n-step symmetric random walk in R^d excludes the origin, linking it to hyperplane arrangements and Weyl chambers, with asymptotic analysis.

Contribution

It extends classical results by providing explicit, distribution-free probabilities for convex hulls of random walks and connects these to geometric structures like Weyl chambers.

Findings

Distribution-free probability formulas for convex hulls excluding the origin.

Connection between random walk convex hulls and hyperplane arrangements.

Asymptotic behavior of absorption probabilities as n and d grow.

Abstract

We give an explicit formula for the probability that the convex hull of an $n$-step random walk in $R^d$ does not contain the origin, under the assumption that the distribution of increments of the walk is centrally symmetric and puts no mass on affine hyperplanes. This extends the formula by Sparre Andersen (1949) for the probability that such random walk in dimension one stays positive. Our result is distribution-free, that is, the probability does not depend on the distribution of increments. This probabilistic problem is shown to be equivalent to either of the two geometric ones: 1) Find the number of Weyl chambers of type $B_n$ intersected by a generic linear subspace of $R^n$ of codimension $d$; 2) Find the conic intrinsic volumes of a Weyl chamber of type $B_n$. We solve the first geometric problem using the theory of hyperplane arrangements. A by-product of our method is a new simple proof of the general formula by Klivans and Swartz (2011) relating the coefficients of the characteristic polynomial of a linear hyperplane arrangement to the conic intrinsic volumes of the chambers constituting its complement. We obtain analogous distribution-free results for Weyl chambers of type $A_{n-1}$ (yielding the probability of absorption of the origin by the convex hull of a generic random walk bridge), type $D_n$, and direct products of Weyl chambers (yielding the absorption probability for the joint convex hull of several random walks or bridges). The simplest case of products of the form $B_1\times \dots \times B_1$ recovers the Wendel formula (1962) for the probability that the convex hull of an i.i.d. multidimensional sample chosen from a centrally symmetric distribution does not contain the origin. We also give an asymptotic analysis of the obtained absorption probabilities as $n \to \infty$, in both cases of fixed and increasing dimension $d$.