Conformal Measure Ensembles for Percolation and the FK–Ising Model

Federico Camia, René Conijn, Demeter Kiss

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Under some general assumptions, we construct the scaling limit of open clusters and their associated counting measures in a class of two dimensional percolation models. Our results apply, in particular, to critical Bernoulli site percolation on the triangular lattice and to the critical FK–Ising model on the square lattice. Fundamental properties of the scaling limit, such as conformal covariance, are explored. As an application to Bernoulli percolation, we obtain the scaling limit of the largest cluster in a bounded domain. We also apply our results to the critical, two-dimensional Ising model, obtaining the existence and uniqueness of the scaling limit of the magnetization field, as well as a geometric representation for the continuum magnetization field which can be seen as a continuum analog of the FK representation of the Ising model.

Original languageEnglish (US)
Title of host publicationSojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman
EditorsVladas Sidoravicius
PublisherSpringer
Pages44-89
Number of pages46
ISBN (Print)9789811502972
DOIs
StatePublished - 2019
EventInternational Conference on Probability Theory and Statistical Physics, 2016 - Shanghai, China
Duration: Mar 25 2016Mar 27 2016

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume299
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

ConferenceInternational Conference on Probability Theory and Statistical Physics, 2016
CountryChina
CityShanghai
Period3/25/163/27/16

Keywords

  • Critical cluster
  • Ising model
  • Magnetization field
  • Percolation
  • Scaling limit
  • Schramm–Smirnov topology

ASJC Scopus subject areas

  • Mathematics(all)

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