Critical parameter of random loop model on trees

TL;DR

This paper estimates the critical parameter for random loop models on large-degree regular trees, providing improved results and insights relevant to quantum spin systems and high-dimensional lattices.

Contribution

It offers new estimates for the critical parameter on regular trees, including the second-order term for the interchange process, advancing understanding of these models in high dimensions.

Findings

Critical parameter estimates for random loop models on large-degree trees.

Improved second-order approximation for the interchange process critical point.

Results align with numerical data for cubic lattices.

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.