A Tropical Approach to Ising Problems

Abstract

Tropical geometry offers a piecewise-linear framework that translates algebraic and combinatorial structures into polyhedral geometry. In this dissertation, we explore applications of tropical and polyhedral methods to the analysis of spin systems, particularly the Ising model on graphs. Each spin configuration of a graph with n vertices corresponds to a vertex of the n-cube, and the energy associated to these configurations defines a Laurent polynomial whose Newton polytope encodes all interaction and external field parameters. We show that the tropicalization of this polynomial captures the loci in parameter space where multiple spin configurations minimize the system's energy—so-called degeneracy loci. These loci are described by tropical hypersurfaces, whose combinatorial types are determined by faces of the secondary polytope of the n-cube. Through this connection, vertex-state interactions naturally parameterize regular subdivisions of the cube, and ground-state degeneracies are encoded by the dual secondary fan. We further construct new polytope, whose vertices reflect both spin states and interaction parities, and provide a facet classification for graphs built from trees and cycles. This polyhedral perspective reveals a natural moduli space for studying phase transitions, optimization, and combinatorial symmetries in discrete physical systems.