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We give a direct probabilistic construction for correlation functions in alogarithmic conformal field theory (log-CFT) of central charge $-2$.Specifically, we show that scaling limits of Peano curves in the uniformspanning tree in topological polygons with general boundary conditions aregiven by certain variants of the SLE$_\kappa$ with $\kappa=8$. We also provethat the associated crossing probabilities have conformally invariant scalinglimits, given by ratios of explicit SLE$_8$ partition functions. Thesepartition functions are interpreted as correlation functions in a log-CFT.Remarkably, it is clear from our results that this theory is not a minimalmodel and exhibits logarithmic phenomena --- indeed, the limit functions havelogarithmic asymptotic behavior, that we calculate explicitly. General fusionrules for them could be inferred from the explicit formulas.

Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT