Abstract
Determinants of Laplacians on discretizations of flat surfaces and analytic torsion
Highlights
Over the discretizations of a given surface, we study the asymptotic behaviour of the number of spanning trees and the partition function of cycle-rooted spanning forests, weighted by the monodromy of the unitary connection on a vector bundle, as the mesh of the discretization of the surface goes to zero
By a cycle-rooted spanning forest (CRSF in what follows) on a graph we mean a subset of edges, spanning all vertices and with the property that each connected component of the subset has as many vertices as edges, cf
The number of spanning trees on a finite graph G is often called the complexity of the graph, denoted here by t (G)
Summary
Over the discretizations of a given surface, we study the asymptotic behaviour of the number of spanning trees and the partition function of cycle-rooted spanning forests, weighted by the monodromy of the unitary connection on a vector bundle, as the mesh of the discretization of the surface goes to zero. Our main result shows that up to some universal contribution, depending only on the angles of conical points and interior angles of the corners on the boundary of Σ, the normalized logarithm of the determinant of the discrete Laplacian converges to the logarithm of the analytic torsion of the surface, which is an invariant introduced by Ray–Singer in [21]. This gives a complete answer to Open problem 2 and a partial answer to Open problem 4 in Kenyon [14, §8]. Details of the results announced here are developed in [9] and [10]
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