Determinants of Laplacians of converging surfaces

The “tree entropy” of a converging sequence of graphs roughly counts how many spanning trees per vertex each graph has, and can be calculated using the Laplacian of the graph.

In this talk, we will discuss a similar quantity for compact hyperbolic surfaces. We show that, under some assumptions on the eigenvalues and short geodesics, if a sequence of surfaces converges to a random rooted surface, then the logarithm of the determinant of its Laplacian converges to a constant. The proof involves analyzing the return density of Brownian motion to the origin, averaged over the entire surface.

• Speaker: Renan Gross (Tel Aviv)
• Tuesday 24 October 2023, 14:00–15:00
• Venue: MR12.
• Series: Probability; organiser: Jason Miller.