Critical Ising model on random triangulations of the disk: enumeration and local limits

Research output: Contribution to journalArticleScientific

Abstract

We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paper computes explicitly the partition function of this model by solving its Tutte's equation, extending a previous result by Bernardi and Bousquet-M\'elou to the model with Dobrushin boundary conditions. We show that the perimeter exponent of the model is 7/3 in contrast to the exponent 5/2 for uniform triangulations. In the second part, we show that the model has a local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other. The local limit is constructed explicitly using the peeling process along an Ising interface. Moreover, we show that the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of finite clusters are also obtained.
Original languageEnglish
JournalarXiv.org
ISSN2331-8422
Publication statusPublished - Jun 2018
MoE publication typeB1 Journal article

Fields of Science

  • math.PR
  • math-ph
  • math.CO
  • math.MP
  • 05C80, 60K35, 60K37

Cite this

@article{0cca53f61df84220811e39f4b6df9e96,
title = "Critical Ising model on random triangulations of the disk: enumeration and local limits",
abstract = "We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paper computes explicitly the partition function of this model by solving its Tutte's equation, extending a previous result by Bernardi and Bousquet-M\'elou to the model with Dobrushin boundary conditions. We show that the perimeter exponent of the model is 7/3 in contrast to the exponent 5/2 for uniform triangulations. In the second part, we show that the model has a local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other. The local limit is constructed explicitly using the peeling process along an Ising interface. Moreover, we show that the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of finite clusters are also obtained.",
keywords = "math.PR, math-ph, math.CO, math.MP, 05C80, 60K35, 60K37",
author = "Linxiao Chen and Joonas Turunen",
year = "2018",
month = "6",
language = "English",
journal = "arXiv.org",
issn = "2331-8422",
publisher = "Cornell University",

}

Critical Ising model on random triangulations of the disk : enumeration and local limits. / Chen, Linxiao; Turunen, Joonas.

In: arXiv.org , 06.2018.

Research output: Contribution to journalArticleScientific

TY - JOUR

T1 - Critical Ising model on random triangulations of the disk

T2 - enumeration and local limits

AU - Chen, Linxiao

AU - Turunen, Joonas

PY - 2018/6

Y1 - 2018/6

N2 - We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paper computes explicitly the partition function of this model by solving its Tutte's equation, extending a previous result by Bernardi and Bousquet-M\'elou to the model with Dobrushin boundary conditions. We show that the perimeter exponent of the model is 7/3 in contrast to the exponent 5/2 for uniform triangulations. In the second part, we show that the model has a local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other. The local limit is constructed explicitly using the peeling process along an Ising interface. Moreover, we show that the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of finite clusters are also obtained.

AB - We consider Boltzmann random triangulations coupled to the Ising model on their faces, under Dobrushin boundary conditions and at the critical point of the model. The first part of this paper computes explicitly the partition function of this model by solving its Tutte's equation, extending a previous result by Bernardi and Bousquet-M\'elou to the model with Dobrushin boundary conditions. We show that the perimeter exponent of the model is 7/3 in contrast to the exponent 5/2 for uniform triangulations. In the second part, we show that the model has a local limit in distribution when the two components of the Dobrushin boundary tend to infinity one after the other. The local limit is constructed explicitly using the peeling process along an Ising interface. Moreover, we show that the main interface in the local limit touches the (infinite) boundary almost surely only finitely many times, a behavior opposite to that of the Bernoulli percolation on uniform maps. Some scaling limits closely related to the perimeters of finite clusters are also obtained.

KW - math.PR

KW - math-ph

KW - math.CO

KW - math.MP

KW - 05C80, 60K35, 60K37

M3 - Article

JO - arXiv.org

JF - arXiv.org

SN - 2331-8422

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