Spectral invariants and playing hide-and-seek on surfaces

TL;DR

This paper demonstrates that the expected duration of a hide-and-seek game on a Riemannian surface, played under Brownian motion, is a spectral invariant related to the Laplacian's regularized trace, linking it to Kemeny's constant in Markov chains.

Contribution

It establishes a novel connection between spectral invariants of Riemannian manifolds and classical Markov chain quantities, unifying geometric analysis and stochastic processes.

Findings

Expected game duration is a spectral invariant on surfaces.

The invariant relates to the zeta-regularized trace of the Laplacian.

Analogies with Kemeny's constant in Markov chains are developed.

Abstract

We prove the expected duration of a game of hide-and-seek played on a Riemannian manifold under the laws of Brownian Motion is a spectral invariant: it is a zeta-regularized version of the `trace' of the Laplacian. An analogous hide-and-seek game may be played on Markov chains, where the spectral invariant that emerges is a classical quantity known as Kemeny's constant. We develop the analogies between the two settings in order to highlight the connections between the regularized trace and Kemeny's constant. Our proof relies on the connections between Green's functions and expected hitting times, and the fact that the regularized trace may be approached via the Green's function.