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

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http://hdl.handle.net/10138/314993

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Chen , L & Turunen , J 2020 , ' Critical Ising model on random triangulations of the disk : enumeration and local limits ' , Communications in Mathematical Physics , vol. 374 , no. 3 , pp. 1577-1643 . https://doi.org/10.1007/s00220-019-03672-5

Julkaisun nimi: Critical Ising model on random triangulations of the disk : enumeration and local limits
Tekijä: Chen, Linxiao; Turunen, Joonas
Muu tekijä: University of Helsinki, Department of Mathematics and Statistics
University of Helsinki, Department of Mathematics and Statistics
Päiväys: 2020-03
Kieli: eng
Sivumäärä: 67
Kuuluu julkaisusarjaan: Communications in Mathematical Physics
ISSN: 0010-3616
URI: http://hdl.handle.net/10138/314993
Tiivistelmä: 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-Melou (J Combin Theory Ser B 101(5):315-377, 2011) 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.
Avainsanat: math.PR
math-ph
math.CO
math.MP
05C80, 60K35, 60K37
PLANAR LATTICE
EQUATIONS
MAP
111 Mathematics
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