Multiple phases and vicious walkers in a wedge

TL;DR

This paper extends an exact field theoretical approach to analyze phase separation in a wedge, revealing that interfaces behave as vicious walkers and uncovering a wedge covariance involving self-Fourier functions.

Contribution

It generalizes previous models to include an intermediate phase and demonstrates the vicious walker behavior of interfaces in a wedge geometry.

Findings

Interfaces act as vicious walkers with calculable passage probabilities.

The theory reveals a wedge covariance relating wedge and half-plane properties.

Self-Fourier functions emerge in the wedge covariance relation.

Abstract

We consider a statistical system in a planar wedge, for values of the bulk parameters corresponding to a first order phase transition and with boundary conditions inducing phase separation. Our previous exact field theoretical solution for the case of a single interface is extended to a class of systems, including the Blume-Capel model as the simplest representative, allowing for the appearance of an intermediate layer of a third phase. We show that the interfaces separating the different phases behave as trajectories of {\it vicious} walkers, and determine their passage probabilities. We also show how the theory leads to a remarkable form of wedge covariance, i.e. a relation between properties in the wedge and in the half plane, which involves the appearance of self-Fourier functions.