Exact Solution of the Six-Vertex Model on a Random Lattice

TL;DR

This paper provides an exact solution to the six-vertex model on a random lattice, revealing its connections to string theory and detailing key geometric and physical properties that vary with model parameters.

Contribution

It introduces an exact solution of the 6-vertex model on a dynamical lattice using matrix model techniques, linking it to c=1 string theory and calculating explicit geometric quantities.

Findings

Explicit disk amplitude expression

Fractal dimension of boundary varies with coupling

Average number of loops and vortex dimensions computed

Abstract

We solve exactly the 6-vertex model on a dynamical random lattice, using its representation as a large N matrix model. The model describes a gas of dense nonintersecting oriented loops coupled to the local curvature defects on the lattice. The model can be mapped to the c=1 string theory, compactified at some length depending on the vertex coupling. We give explicit expression for the disk amplitude and evaluate the fractal dimension of its boundary, the average number of loops and the dimensions of the vortex operators, which vary continuously with the vertex coupling.