Loop percolation versus link percolation in the random loop model

TL;DR

This paper quantitatively compares loop and link percolation transitions in the random loop model, showing they cannot coincide and providing bounds on their separation, with implications for weighted models.

Contribution

It provides a quantitative analysis of the separation between loop and percolation phase transitions in the random loop model, extending results to weighted models.

Findings

Loop and percolation phase transitions are separated by a minimal gap.

The results apply to both unweighted and weighted loop models.

A quantitative bound on the gap between the two transitions.

Abstract

In [Muhl2019], Peter M\"uhlbacher showed that in the random loop model without loop weights, a loop phase transition (assuming it exists) cannot occur at the same parameter as the percolation phase transition of the occupied edges. In this work, we give a quantitative version of this result, specifying a minimal gap between the percolation phase transition and a possible loop phase transition. A substantial part of our argument also works for weighted loop models.