The Singular Values of Lévy's Area Matrix

Abstract

The matrix of Lévy's areas of $d$-dimensional Brownian motion is a fundamental object in stochastic analysis. In this article, we study the singular values of this $d \times d$ skew-symmetric random matrix. First, we derive an explicit formula for the density of the singular values and, en passant, present a new short proof of the characteristic function of Lévy's area when $d \ge 3$. This also allows us to extend the well-known formula for the density of Lévy's area to $d \ge 3$. Next, we use these results to characterise the singular spectrum as a determinantal point process with its kernel in explicit form. Finally, we study the asymptotics as $d \to \infty$: the empirical measure of singular values converges to an absolute Cauchy distribution, the largest singular values are of order $d$ with Gaussian fluctuations, the smallest singular values are of order $1/d$, and the local bulk spacings are of order $1/d$, with sine-kernel statistics after rescaling.