## Convex hulls of random walks and Weyl chambers

#### Vladislav Vysotsky (Arizona State University)

Frank Adams 2,

What is the probability that the convex hull of a random walk in \(\mathbb{R}^d\) has not absorbed the origin by the time \(n\)? In dimension one this is simply the probability that there was no sign change. The remarkable formula of Sparre Andersen (1949) states that a random walk with symmetric density of increments stays positive with probability \((2n-1)!!/(2n)!!\) We prove a multidimensional distribution-free counterpart of this fact and provide a tractable formula for the absorption probability. Our idea is to show that the absorption problem is equivalent to a geometric problem on counting the number of Weyl chambers intersected by a generic linear subspace. This method is then applied to convex hulls of random walk bridges, and to the joint convex hulls of several symmetric random walks. In particular, we recover a well-known formula by Wendel on the absorption probability for the convex hull of i.i.d. random vectors in \(\mathbb{R}^d\) with a centrally symmetric distribution.

This is a joint work with Zakhar Kabluchko (Munster) and Dmitry Zaporozhets (St. Petersburg).