Spectral invariants and playing hide-and-seek on surfaces

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.