Conformal Measure Ensembles for Percolation and the FK–Ising Model

Federico Camia*, René Conijn, Demeter Kiss

*Corresponding author for this work

Research output: Chapter in Book / Report / Conference proceedingConference contributionAcademicpeer-review

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
Title of host publicationSojourns in Probability Theory and Statistical Physics - II
Subtitle of host publicationBrownian Web and Percolation, A Festschrift for Charles M. Newman
EditorsVladas Sidoravicius
PublisherSpringer
Pages44-89
Number of pages46
Volume2
ISBN (Electronic)9789811502989
ISBN (Print)9789811502972
DOIs
Publication statusPublished - 2019
EventInternational Conference on Probability Theory and Statistical Physics, 2016 - Shanghai, China
Duration: 25 Mar 201627 Mar 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
Country/TerritoryChina
CityShanghai
Period25/03/1627/03/16

Funding

The work of the first author was supported in part by the Netherlands Organization for Scientific Research (NWO) through grant Vidi 639.032.916. The work of the second author was partly supported by NWO Top grant 613.001.403. The second author was at VU University Amsterdam while most of the research was carried out. The third author thanks NWO for its financial support and Centrum Wiskunde & Informatica (CWI) for its hospitality during the time when he was a PhD student, when the project was initiated. All three authors thank Rob van den Berg for fruitful discussions. The first author thanks Chuck Newman for his friendship and invaluable guidance during many years, and for being a constant inspiration.

Keywords

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

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