SLE Boundary Visits

Research output: Contribution to journalArticleScientificpeer-review

Details

Original languageEnglish
Pages (from-to)1263-1330
Number of pages68
JournalANNALES HENRI POINCARÉ
Volume17
Issue number6
Publication statusPublished - Jun 2016
MoE publication typeA1 Journal article-refereed

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Research units

Abstract

We study the probabilities with which chordal Schramm-Loewner evolutions (SLE) visit small neighborhoods of boundary points. We find formulas for general chordal SLE boundary visiting probability amplitudes, also known as SLE boundary zig-zags or order refined SLE multi-point Green's functions on the boundary. Remarkably, an exact answer can be found to this important SLE question for an arbitrarily large number of marked points. The main technique employed is a spin chain-Coulomb gas correspondence between tensor product representations of a quantum group and functions given by Dotsenko-Fateev type integrals. We show how to express these integral formulas in terms of regularized real integrals, and we discuss their numerical evaluation. The results are universal in the sense that apart from an overall multiplicative constant the same formula gives the amplitude for many different formulations of the SLE boundary visit problem. The formula also applies to renormalized boundary visit probabilities for interfaces in critical lattice models of statistical mechanics: we compare the results with numerical simulations of percolation, loop-erased random walk, and Fortuin-Kasteleyn random cluster models at Q = 2 and Q = 3, and find good agreement.

    Research areas

  • SCHRAMM-LOEWNER EVOLUTION, ERASED RANDOM-WALKS, UNIFORM SPANNING-TREES, RANDOM-CLUSTER MODEL, CONFORMAL-INVARIANCE, CRITICAL PERCOLATION, SCALING LIMITS, NATURAL PARAMETRIZATION, CONNECTED DOMAINS, MINKOWSKI CONTENT

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