The Annals of Applied Probability

Critical parameter of random loop model on trees

Jakob E. Björnberg and Daniel Ueltschi

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We give estimates of the critical parameter for random loop models that are related to quantum spin systems. A special case of the model that we consider is the interchange- or random-stirring process. We consider here the model defined on regular trees of large degrees, which are expected to approximate high spatial dimensions. We find a critical parameter that indeed shares similarity with existing numerical results for the cubic lattice. In the case of the interchange process, our results improve on earlier work by Angel and by Hammond, in that we determine the second-order term of the critical parameter.

Article information

Ann. Appl. Probab., Volume 28, Number 4 (2018), 2063-2082.

Received: October 2016
Revised: March 2017
First available in Project Euclid: 9 August 2018

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general) 82B31: Stochastic methods

Random loop model quantum Heisenberg


Björnberg, Jakob E.; Ueltschi, Daniel. Critical parameter of random loop model on trees. Ann. Appl. Probab. 28 (2018), no. 4, 2063--2082. doi:10.1214/17-AAP1315.

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