# Sharp phase transition for random loop models on trees

## TL;DR

This paper proves a sharp phase transition for the emergence of infinite loops in the random loop model on d-ary trees, providing bounds and an asymptotic expansion for the critical parameter.

## Contribution

It establishes a precise phase transition and derives an asymptotic expansion for the critical parameter in the random loop model on trees, with explicit coefficient calculations.

## Key findings

- Proves sharp phase transition for infinite loops on trees
- Provides bounds for the critical parameter
- Calculates asymptotic expansion coefficients up to order 6

## Abstract

We investigate the random loop model on the $d$-ary tree. For $d \geq 3$, we establish a (locally) sharp phase transition for the existence of infinite loops. Moreover, we derive rigorous bounds that in principle allow to determine the value of the critical parameter with arbitrary precision. Additionally, we prove the existence of an asymptotic expansion for the critical parameter in terms of $d^{-1}$. The corresponding coefficients can be determined in a schematic way and we calculated them up to order $6$.