Geometrically Constrained Statistical Models on Fixed and Random Lattices: From Hard Squares to Meanders

TL;DR

This paper reviews how field theory and matrix models help analyze combinatorial problems in statistical physics, focusing on lattice configurations, universality classes, and exact asymptotics for complex structures like meanders.

Contribution

It demonstrates the application of quantum gravity and matrix model techniques to predict critical behaviors and asymptotics in lattice-based combinatorial models.

Findings

Identification of universality classes for lattice models

Exact asymptotic formulas for configuration counts

Insights into the role of lattice colorability in models

Abstract

We review various combinatorial applications of field theoretical and matrix model approaches to equilibrium statistical physics involving the enumeration of fixed and random lattice model configurations. We show how the structures of the underlying lattices, in particular their colorability properties, become relevant when we consider hard-particles or fully-packed loop models on them. We show how a careful back-and-forth application of results of two-dimensional quantum gravity and matrix models allows to predict critical universality classes and consequently exact asymptotics for various numbers, counting in particular hard object configurations on fixed or random lattices and meanders.