Gap probabilities and applications to geometry and random topology

TL;DR

This paper derives exact formulas and asymptotics for gap probabilities in Gaussian ensembles, with applications to geometric measures and topological invariants of random algebraic sets, revealing new precise asymptotic behaviors.

Contribution

It provides the first exact formula for the derivative at zero of gap probabilities in finite Gaussian ensembles and applies it to compute geometric and topological properties of random matrices and algebraic sets.

Findings

Exact formula for the derivative at zero of gap probability in Gaussian ensembles.

Asymptotic behavior of order n^(1/2) for large n in various applications.

Precise asymptotics for Betti numbers of random Kostlan quadrics.

Abstract

We give an exact formula for the value of the derivative at zero of the gap probability in finite n x n Gaussian ensembles. As n goes to infinity our computation provides an asymptotic (with an explicit constant) of the order n^(1/2). As a first application, we consider the set of n x n (Real, Complex or Quaternionic) Hermitian matrices with Frobenius norm one and determinant zero. We give an exact formula for the intrinsic volume of this set and as n goes to infinity its asymptotic (with an explicit constant) is of the order n^(1/2). As a second application we consider the problem of computing Betti numbers of an intersection of k random Kostlan quadrics in RP^n. We show that the i-th Betti number is asymptotically expected to be one (for i sufficiently away from n/2). In the case k=2 the the sum of all Betti numbers was recently shown by the first author to equal n+o(n). Here we sharpen this asymptotic proving an asymptotic with two orders of precision and explicit constants.