Probability and analysis informal seminar
On a finite connected graph, the product of the non-zero eigenvalues of the Laplacian counts the rooted spanning trees, according to a theorem often attributed to Kirchhoff (1847), or sometimes to Sylvester (1857). Among many generalisations of this classical result, those of Zaslavsky (1982), Forman (1993) and Kenyon (2011) state that when we twist the Laplacian by putting a sign or a phase, complex or quaternionic, on each edge, its determinant counts covering forests of unicycles, with appropriate weights. A common feature of all these results is that the random subgraphs naturally associated to each of these situations (uniform spanning trees and random covering forests of unicycles), seen as random subsets of the (finite) set of edges of the ambient graph, are determinantal point processes.
I will present some results of an ongoing joint work with With Adrien Kassel (CNRS, ENS Lyon) in which we investigate further extensions of these results to the covariant Laplacian associated with an arbitrary unitary connection, that is, to the Laplacian twisted by a unitary matrix on each edge.
In a first part, I will describe the classical results of Kirchhoff and Forman, then (from a perhaps slightly unorthodox point of view) determinantal point processes on finite sets, and explain what the ones have to do with the others. In a second part, I will describe the measures on Grassmannians that we introduced with Adrien Kassel, explain why they are relevant to the understanding of the twistes Laplacian, and finally describe, to the extent that we understand them, the new random objects that appear over a graph endowed with a unitary connection.
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Thierry Bodineau, Pieter Lammers, Yilin Wang
