Solvable statistical models on a random lattice

TL;DR

This work presents a unified, solvable framework for $ADE$ and $\\hat{A}\\hat{D}\\hat{E}$ height models on random 2D lattices, connecting Dynkin-diagram target spaces, coupled random-matrix ensembles, Coulomb-gas representations, and soliton $\\tau$-functions. It derives a comprehensive loop-diagram formalism with genus-expanded amplitudes, including a Fredholm-determinant representation for $\\hat{A}_r$ models and explicit loop equations that relate to Virasoro constraints and topological gravity. The authors develop both bosonic and fermionic vertex-operator constructions that realize the partition functions as $\\tau$-functions of integrable hierarchies (KdV/mKdV and affine sinh-Gordon) and provide explicit propagators and vertices for loop computations on arbitrary topology. The framework clarifies the role of the ADE classification in critical phenomena on fluctuating surfaces and offers practical tools for computing genus expansions and correlation functions in these models, with potential implications for discretized quantum gravity and noncritical string theory.

Abstract

We give a sequence of equivalent formulations of the $ADE$ and $\hat A\hat D\hat E$ height models defined on a random triangulated surface: random surfaces immersed in Dynkin diagrams, chains of coupled random matrices, Coulomb gases, and multicomponent Bose and Fermi systems representing soliton $τ$-functions. We also formulate a set of loop-space Feynman rules allowing to calculate easily the partition function on a random surface with arbitrary topology. The formalism allows to describe the critical phenomena on a random surface in a unified fashion and gives a new meaning to the $ADE$ classification.