Infinite light rays and infinite clusters with infinitely many pivots
Abstract
It is conjectured that in the Lorentz mirror model on Z2 all trajectories are finite. Quas showed that, almost surely, the two halves of any infinite trajectory, cut at an edge, meet at infinitely many vertices, each of which is pivotal: changing its state can make the trajectory finite. We construct plane graphs on which infinite trajectories exist, and all of them have this property. Given p∈(0,1), we replace each edge of a random one-ended subtree of Z2 by a number of parallel edges, depending on p, that is at most logarithmic in the height of the corresponding descendant tree. The medial graph of the resulting plane multigraph is a factor of the tree, and for the uniform spanning tree it has finite vertex intensity. In the mirror model on this medial graph, with probability p for mirrors along primal edges and any probability r∈(0,1−p] for mirrors along dual edges, infinite trajectories exist, and the two halves of each of them meet infinitely often. The construction rests on Bernoulli percolation: every random one-ended locally finite tree can be decorated in this way such that bond percolation with parameter p is critical and percolates, and every vertex of the infinite cluster has infinitely many pivotal edges. We characterize this property for bond and site percolation and for mirrors.
Related
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BibTeX
@misc{zerner2026infinite,
title = {{Infinite light rays and infinite clusters with infinitely many pivots}},
author = {Martin P. W. Zerner},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12460},
eprint = {2610.12460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12460},
}
RIS
TY - UNPB
TI - Infinite light rays and infinite clusters with infinitely many pivots
AU - Martin P. W. Zerner
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12460
UR - https://arxiv.org/abs/2610.12460
AB - It is conjectured that in the Lorentz mirror model on Z^2 all trajectories are finite. Quas showed that, almost surely, the two halves of any infinite trajectory, cut at an edge, meet at infinitely many vertices, each of which is pivotal: changing its state can make the trajectory finite. We construct plane graphs on which infinite trajectories exist, and all of them have this property. Given p∈(0,1), we replace each edge of a random one-ended subtree of Z^2 by a number of parallel edges, depending on p, that is at most logarithmic in the height of the corresponding descendant tree. The medial graph of the resulting plane multigraph is a factor of the tree, and for the uniform spanning tree it has finite vertex intensity. In the mirror model on this medial graph, with probability p for mirrors along primal edges and any probability r∈(0,1−p] for mirrors along dual edges, infinite trajectories exist, and the two halves of each of them meet infinitely often. The construction rests on Bernoulli percolation: every random one-ended locally finite tree can be decorated in this way such that bond percolation with parameter p is critical and percolates, and every vertex of the infinite cluster has infinitely many pivotal edges. We characterize this property for bond and site percolation and for mirrors.
ER -