Absence of infinite loops in the loop-weight-two mirror model

Prove that the infinite-volume limiting measure \mu^{mir}_{p} for the loop-weight-two Lorentz mirror model in the regime p_{NW},p_{NE}\ge p_\varnothing has no infinite loops almost surely.

Background

Theorem 1.3 establishes convergence of connection probabilities to zero, but this does not by itself exclude the existence of infinite loops in the limiting configuration. The paper identifies the missing conclusion and attributes it to the absence of a suitable monotonicity property for the mirror model, while stating that the absence of infinite loops is expected.

References

Note that while we do have that connection probabilities decay to 0 by Part 2 of Theorem \ref{thm:main-mirror}, we do not prove that the limiting measure \mu{mir}_{p} exhibits no infinite loops almost surely (although this is indeed expected).

— The spin-1/2 Heisenberg XXZ chain and the Lorentz mirror model with loop weight 2  (2608.21306 - Ryan, 21 Aug 2026) in Remark following Theorem 1.3, subsection “Mirror model results”