The Conley-Zehnder Index of Brownian Paths on Sp(2, R)

TL;DR

This paper studies the distribution of Conley-Zehnder indices for Brownian paths on symplectic groups, revealing asymptotic behaviors similar to random walks, with numerical evidence extending to higher dimensions.

Contribution

It provides the first analysis of Conley-Zehnder index distributions for Brownian paths on Sp(2n, R), including asymptotic results and numerical evidence for general n.

Findings

Distribution has same moment asymptotics as standard random walk for n=1

Numerical evidence suggests similar asymptotics for general n

First analysis of Conley-Zehnder indices for Brownian paths on symplectic groups

Abstract

We investigate the probability distribution of Conley-Zehnder indices associated with Brownian random paths on Sp(2n, R) that start at the identity. In the case of n = 1, we prove that the distribution has the same moment asymptotics as the standard random walk on the real line. We also present numerical evidence suggesting that the same asymptotics should hold for general n.