Spanning trees, loop measures and zeta-regularized determinants of Laplacian on half-translation surfaces
Thursday, 13 February, 2020 - 14:00
We will study the asymptotics of several combinatorial invariants of the graph discretizations of a half-translation surface, as the mesh of the discretisation tends to zero, and we will see how the geometry of the surface is reflected in this asymptotics.
In particular, we relate the asymptotic expansion of the number of spanning trees on discretizations with the zeta-regularized determinant of the Laplacian, formally defined as the product of all the eigenvalues.
As a by-product of our study, we obtain some information on the topological properties of the loop measure induced by the cycles of cycle-rooted spanning forests, sampled uniformly on the discretizations approaching a given surface. By using recent work of Kassel-Kenyon, we give an explicit formula for the probability that a multiloop sampled from this loop measure has the isotopy type of the given lamination.
No prior knowledge of probability theory, half-translation surfaces or spectral geometry is assumed for this talk.
Institution de l'orateur :
Thème de recherche :
Théorie spectrale et géométrie