2019/01/18 by Terrence George, George, Terrence · 1 citation
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Quantum chaos and dynamical systems #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1901.06353
A biperiodic planar network is a pair (G,c) where G is a graph embedded on the torus and c is a function from the edges of G to non-zero complex numbers. Associated to the discrete Laplacian on a biperiodic planar network is its spectrum: a triple (C,S,ν), where C is a curve and S is a divisor on it. We give a complete classification of networks (modulo a natural equivalence) in terms of their spectral data. The space of networks has a large group of cluster automorphisms arising from the Y-Δ transformations. We show that the spectrum provides action-angle coordinates for the discrete cluster integrable systems defined by these automorphisms.