The 3D Dimer and Ising Problems Revisited
Martin Loebl, Lenka Zdeborova
TL;DR
This paper presents a novel mathematical approach to compute the partition functions of 3D Dimer and Ising models using determinants and walk products, leveraging embeddings on high-genus surfaces.
Contribution
It introduces a new method to express 3D Dimer and Ising partition functions via determinants and walk products using lattice embeddings on complex surfaces.
Findings
Partition functions expressed as linear combinations of determinants and walk products
Method applicable to finite 3D lattice models
Utilizes embedding of cubic lattices on high-genus surfaces
Abstract
We express the finite 3D Dimer partition function as a linear combination of determinants of oriented adjacency matrices, and the finite 3D Ising partition sum as a linear combination of products over aperiodic closed walks. The methodology we use is embedding of cubic lattice on 2D surfaces of large genus.
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Taxonomy
TopicsGraph theory and applications · Topological and Geometric Data Analysis · Markov Chains and Monte Carlo Methods