Random curves, scaling limits and Loewner evolutions

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http://hdl.handle.net/10138/182421

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Kemppainen , A & Smirnov , S 2017 , ' Random curves, scaling limits and Loewner evolutions ' , Annals of Probability , vol. 45 , no. 2 , pp. 698-779 . https://doi.org/10.1214/15-AOP1074

Title: Random curves, scaling limits and Loewner evolutions
Author: Kemppainen, Antti; Smirnov, Stanislav
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2017-03
Language: eng
Number of pages: 82
Belongs to series: Annals of Probability
ISSN: 0091-1798
URI: http://hdl.handle.net/10138/182421
Abstract: In this paper, we provide a framework of estimates for describing 2D scaling limits by Schramm's SLE curves. In particular, we show that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical mechanics model will have scaling limits and those will be well described by Loewner evolutions with random driving forces. Interestingly, our proofs indicate that existence of a nondegenerate observable with a conformally- invariant scaling limit seems sufficient to deduce the required condition. Our paper serves as an important step in establishing the convergence of Ising and FK Ising interfaces to SLE curves; moreover, the setup is adapted to branching interface trees, conjecturally describing the full interface picture by a collection of branching SLEs.
Subject: Random curve
lattice model
weak convergence of probability measures
Schramm-Loewner evolution
Brownian motion
conformal mapping
UNIFORM SPANNING-TREES
ERASED RANDOM-WALKS
CONFORMAL-INVARIANCE
CRITICAL PERCOLATION
RANDOM-CLUSTER
ISING-MODEL
CONVERGENCE
SLE
CONTINUITY
PLANE
111 Mathematics
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