Infinite loops in the unweighted Lorentz mirror model

Determine whether the unweighted Lorentz mirror model on Z^2 with positive no-mirror probability p_>0 contains infinite loops, thereby resolving the stated open problem for the model outside the case p_\varnothing=0.

Background

The paper introduces the Lorentz mirror model as a random-loop model on the square lattice in which each vertex receives a northwest mirror, a northeast mirror, or no mirror. The existence of infinite loops is identified as the principal question for this model. The case with no empty vertices is related to critical Bernoulli percolation and is known not to have infinite loops, whereas the positive no-mirror-probability regime remains unresolved.

References

For p_>0 the problem is open.

— The spin-1/2 Heisenberg XXZ chain and the Lorentz mirror model with loop weight 2  (2608.21306 - Ryan, 21 Aug 2026) in Section 1.1, subsection “The Lorentz mirror model with loop weight 2”