Adrien Kassel (ENS)

Laplacian determinant and integer coefficients of laminations
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The determinant of a 2-dimensional vector bundle Laplacian on a graph embedded on a surface has a rich combinatorial meaning. It can be seen as an integer-valued function on the space of laminations. The coefficients of these laminations depend on potential theoretic properties of the graph, linked to the combinatorial Laplacian (these can be expressed in several equivalent ways as enumerations of spanning trees, densities of recurrent sandpile configurations, loop soups of random walk, etc.).

One motivation to study these integer-measures on laminations is to define natural probability measures on the space of multicurves (finite collections of disjoint simple closed curves) of a Riemannian surface (independent of any graph embedded on it).

To address this problem, two main steps are needed: (1) extract the integer coefficients of laminations from the Laplacian determinant; (2) show convergence properties of these when we take a sequence of graphs approximating the surface in a certain way.
I will present some ways of going through these two steps and illustrate them with examples, hopefully managing to show why these methods should be quite general and applicable in diverse settings.