Fisica Matematica The asymptotic determinant of the discrete Laplacian in non-rectilinear and non-simply-connected polygons Rafael Greenblatt
08-10-2020 - 14:30
Largo San Leonardo Murialdo,1 - Pal.C - Aula 211 Consider the discrete Laplacian Delta with Dirichlet boundary conditions on a domain of the form L Pi, for Pi a bounded region whose boundary is a disjoint union of polygons with vertices in Z^2. I will show that for
large L
- log det Delta = a_0(Pi) L^2 + a_1(Pi) L + a_2(Pi) log L + a_3(Pi) + o(1),
generalizing a previous result by Kenyon (2000). An expansion of the same form holds for the modified Laplacians appearing in the partition functions of dimer models on the associated Temperleyan polyominoes in the
non-simply-connected case, with a_3 changed.
The method of the proof is new, and is based on analyzing an expression for the determinant in terms of the return probabilities of the continuous time random walk generated by Delta. This has the additional feature of relating a_2 and a_3 to the zeta-regularized determinant of the continuum Laplacian on Pi via convergence of the aforementioned random walk to a Brownian motion.
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