Gaussian integrals on symmetric spaces (the complex case and beyond)

TL;DR

This paper investigates Gaussian integrals on symmetric spaces, especially their high-rank limits, by developing a variational approach that generalizes results from the complex case to broader classes of symmetric spaces.

Contribution

It introduces a variational characterization of high-rank limits of Gaussian integrals on symmetric spaces, enabling explicit closed-form solutions beyond the complex case.

Findings

Derived a variational formulation for high-rank limits

Recovered closed-form expressions for limits in symmetric cones and domains

Provided a general approach applicable to various symmetric spaces

Abstract

The present work is concerned with Gaussian integrals on simply connected non-positively curved Riemannian symmetric spaces. It is motivated by the aim of explicitly finding the high-rank limit of these integrals for each of the eleven families of classical Riemannian symmetric spaces. To begin, it deals with the easier complex case (where the isometry group admits a complex Lie group structure). To go beyond this case, it introduces a variational characterisation of the high-rank limit, as the minimum of a certain energy functional over the space of probability distributions on the real line. Using this new variational formulation, it is possible to recover the high-rank limit in closed form, from the expression originally found in the complex case. This two-step approach is illustrated through the examples of two kinds of symmetric spaces : symmetric cones and classical symmetric domains. Gaussian integrals on symmetric spaces, and particularly their high-rank limits, have proved important in both statistics and theoretical physics. The present work proposes an approach for dealing with these limits, which has the merit of yielding general, concrete, closed-form results.