The surprising connection between exactly solved lattice models and discrete holomorphicity

TL;DR

This paper explores the unexpected link between discrete holomorphicity and exactly solved lattice models, showing how discrete Cauchy-Riemann equations help determine Boltzmann weights satisfying integrability conditions, with applications to self-avoiding walks.

Contribution

It reviews the connection between discrete holomorphicity and solvable lattice models, and demonstrates how this approach can be explicitly applied to the Z_N model and self-avoiding walks.

Findings

Discrete holomorphicity leads to solvable linear equations for Boltzmann weights.

Boltzmann weights satisfying star-triangle equations emerge from discrete holomorphicity.

Application to rigorous proofs in planar self-avoiding walks.

Abstract

Over the past few years it has been discovered that an "observable" can be set up on the lattice which obeys the discrete Cauchy-Riemann equations. The ensuing condition of discrete holomorphicity leads to a system of linear equations which can be solved to yield the Boltzmann weights of the underlying lattice model. Surprisingly, these are the well known Boltzmann weights which satisfy the star-triangle or Yang-Baxter equations at criticality. This connection has been observed for a number of exactly solved models. I briefly review these developments and discuss how this connection can be made explicit in the context of the Z_N model. I also discuss how discrete holomorphicity has been used in recent breakthroughs in the rigorous proof of some key results in the theory of planar self-avoiding walks.