1211.5850 (Murray T. Batchelor)

The surprising connection between exactly solved lattice models and
discrete holomorphicity    [PDF]

Murray T. Batchelor

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.

View original: http://arxiv.org/abs/1211.5850