TY - GEN

T1 - Conformal Measure Ensembles for Percolation and the FK–Ising Model

AU - Camia, Federico

AU - Conijn, René

AU - Kiss, Demeter

N1 - Funding Information:
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.

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Critical cluster

KW - Ising model

KW - Magnetization field

KW - Percolation

KW - Scaling limit

KW - Schramm–Smirnov topology

UR - http://www.scopus.com/inward/record.url?scp=85077866799&partnerID=8YFLogxK

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U2 - 10.1007/978-981-15-0298-9_2

DO - 10.1007/978-981-15-0298-9_2

M3 - Conference contribution

AN - SCOPUS:85077866799

SN - 9789811502972

T3 - Springer Proceedings in Mathematics and Statistics

SP - 44

EP - 89

BT - Sojourns in Probability Theory and Statistical Physics - II - Brownian Web and Percolation, A Festschrift for Charles M. Newman

A2 - Sidoravicius, Vladas

PB - Springer

T2 - International Conference on Probability Theory and Statistical Physics, 2016

Y2 - 25 March 2016 through 27 March 2016

ER -