Integrability of Limit Shapes of the Six Vertex Model

TL;DR

This paper demonstrates that the limit shape equation of the 6-vertex model on a cylinder is integrable, constructing infinitely many conserved quantities and connecting it to known integrable PDEs like the complex Burgers equation at special points.

Contribution

It constructs an infinite family of conserved quantities for the limit shape equation, indicating its integrability and linking it to well-known integrable PDEs.

Findings

Limit shape equation admits infinitely many conserved quantities.

The equation is likely an integrable PDE with gradient constraints.

At the free fermionic point, it reduces to the complex Burgers equation.

Abstract

The main result of this paper is the construction of infinitely many conserved quantities (corresponding to commuting transfer-matrices) for the limit shape equation for the 6-vertex model on a cylinder. This suggests that the limit shape equation is an integrable PDE with gradient constraints. At the free fermionic point this equation becomes the complex Burgers equation.