# SLE Boundary Visits

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**SLE Boundary Visits.** / Jokela, Niko; Järvinen, Matti; Kytölä, Kalle.

Research output: Contribution to journal › Article › Scientific › peer-review

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*ANNALES HENRI POINCARÉ*, vol. 17, no. 6, pp. 1263-1330. https://doi.org/10.1007/s00023-015-0452-7

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*ANNALES HENRI POINCARÉ*,

*17*(6), 1263-1330. https://doi.org/10.1007/s00023-015-0452-7

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TY - JOUR

T1 - SLE Boundary Visits

AU - Jokela, Niko

AU - Järvinen, Matti

AU - Kytölä, Kalle

PY - 2016/6

Y1 - 2016/6

N2 - 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.

AB - 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.

KW - SCHRAMM-LOEWNER EVOLUTION

KW - ERASED RANDOM-WALKS

KW - UNIFORM SPANNING-TREES

KW - RANDOM-CLUSTER MODEL

KW - CONFORMAL-INVARIANCE

KW - CRITICAL PERCOLATION

KW - SCALING LIMITS

KW - NATURAL PARAMETRIZATION

KW - CONNECTED DOMAINS

KW - MINKOWSKI CONTENT

UR - http://dx.doi.org/10.1007/s00023-015-0452-7

U2 - 10.1007/s00023-015-0452-7

DO - 10.1007/s00023-015-0452-7

M3 - Article

VL - 17

SP - 1263

EP - 1330

JO - ANNALES HENRI POINCARÉ

JF - ANNALES HENRI POINCARÉ

SN - 1424-0637

IS - 6

ER -

ID: 1982711