Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=2$ Logarithmic CFT
Abstract
We give a direct probabilistic construction for correlation functions in a logarithmic conformal field theory (logCFT) of central charge $2$. Specifically, we show that scaling limits of Peano curves in the uniform spanning tree in topological polygons with general boundary conditions are given by certain variants of the SLE$_\kappa$ with $\kappa=8$. We also prove that the associated crossing probabilities have conformally invariant scaling limits, given by ratios of explicit SLE$_8$ partition functions. These partition functions are interpreted as correlation functions in a logCFT. Remarkably, it is clear from our results that this theory is not a minimal model and exhibits logarithmic phenomena  indeed, the limit functions have logarithmic asymptotic behavior, that we calculate explicitly. General fusion rules for them could be inferred from the explicit formulas.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.04421
 Bibcode:
 2021arXiv210804421L
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 82B20;
 60J67;
 60K35
 EPrint:
 70 pages, 13 figures