Critical parameter of random loop model on trees

Abstract

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.

arXiv comment14 pages, 5 figures

References in corpus (9)

• Geometric Aspects of Quantum Spin States
• Thermodynamic Limit for the Mallows Model on $S_n$
• Random loop representations for quantum spin systems
• Infinite cycles in the random stirring model on trees
• The probability of long cycles in interchange processes
• The free energy in a class of quantum spin systems and interchange processes
• A numerical study of the 3D random interchange and random loop models
• The random interchange process on the hypercube
• Large cycles in random permutations related to the Heisenberg model

Cited by in corpus (4)

• The interchange process with reversals on the complete graph
• Phase transition for the interchange and quantum Heisenberg models on the Hamming graph
• Phase transition for loop representations of Quantum spin systems on trees
• Critical temperature of Heisenberg models on regular trees, via random loops