Planar UST Branches and $c=-2$ Degenerate Boundary Correlations

TL;DR

This paper establishes a conformal field theory framework for boundary effects in the wired uniform spanning tree at central charge c=-2, connecting probabilistic boundary probabilities with algebraic CFT structures.

Contribution

It provides a rigorous CFT description of boundary connection probabilities in the wired UST and demonstrates the validity of BPZ equations and algebraic structures at c=-2.

Findings

Boundary-to-boundary connection probabilities converge to explicit CFT quantities.

BPZ PDEs of arbitrary order are satisfied in the model.

An underlying Temperley-Lieb algebra acts on boundary correlation functions.

Abstract

We provide a conformal field theory (CFT) description of the probabilistic model of boundary effects in the wired uniform spanning tree (UST) and its algebraic content, concerning the entire first row of the Kac table with central charge $c=-2$. Namely, we prove that all boundary-to-boundary connection probabilities for (potentially fused) branches in the wired UST converge in the scaling limit to explicit CFT quantities, expressed in terms of determinants, which can also be viewed as conformal blocks of degenerate primary fields in a boundary CFT with central charge $c=-2$. Moreover, we verify that the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs (i.e., Virasoro degeneracies) of arbitrary orders hold, and we also reveal an underlying valenced Temperley-Lieb algebra action on the space of boundary correlation functions of primary fields in this model. To prove these results, we combine probabilistic techniques with representation theory.