Integrable Combinatorics

TL;DR

This paper investigates integrable combinatorial problems from physics, demonstrating how their conservation laws enable exact solutions across models like random surfaces, lattice models, and representation theory structures.

Contribution

It introduces a unified perspective on integrable combinatorics, highlighting new connections and exact solvability in diverse physical and mathematical models.

Findings

Identification of integrable structures in combinatorial physics problems

Demonstration of exact solutions enabled by conservation laws

Application to random surfaces, lattice models, and representation theory

Abstract

We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems exactly solvable. We illustrate this with: random surfaces, lattice models, and structure constants in representation theory.