On the first hitting time of a high-dimensional orthant

We consider a collection of independent standard Brownian particles (or random walks), starting from a configuration where at least one particle is negative, and study the first time they all become positive. This is clearly equivalent to studying the first hitting time of the positive orthant or the first exit time from the complement of the positive orthant. While it turns out to be possible to compute the distribution of these hitting times for one and two particles, the distribution (and even its tail asymptotics) is not known in closed form for three or more particles. In this paper we study the tail asymptotics of the distribution as the number of particles tends to infinity. Our main techniques come from spectral geometry: we prove new asymptotic estimates for the principal eigenvalue of the complement of a high-dimensional orthant, which we believe are of independent interest.