Radial BPZ equations and partition functions of FK-Ising interfaces conditional on one-arm event

TL;DR

This paper constructs positive solutions to radial BPZ equations and demonstrates that FK-Ising interface partition functions, conditioned on a one-arm event, satisfy these equations, linking probabilistic events with conformal field theory.

Contribution

It establishes that FK-Ising interface partition functions conditioned on a one-arm event are solutions to radial BPZ equations, providing a new connection between probability and conformal field theory.

Findings

Partition functions are positive solutions to radial BPZ equations.

Conditional FK-Ising interface partition functions satisfy BPZ equations.

Links probabilistic conditioning with conformal invariance principles.

Abstract

Radial BPZ equations come naturally when one solves Dub\'{e}dat's commutation relation in the radial setting. We construct positive solutions to radial BPZ equations and show that partition functions of FK-Ising interfaces in a polygon conditional on a one-arm event are positive solutions to radial BPZ equations.