Loop content of bond percolation on hyperbolic triangulations: an exact identity and a 1/λ expansion
Abstract
We study the homology of percolation clusters on disk-shaped patches of the regular {3,q} triangle tilings for q≥7. Varying q gives a one-parameter family of hyperbolic tilings, from mildly curved (q=7) to deeply hyperbolic (q=20), and this paper measures and derives loop content as a function of curvature. A single growth-rate parameter λ(q), the asymptotic ratio of successive ring sizes, governs every result. Total persistent H1 of the bond-percolation complex, per vertex, is measured across q=7,…,20 and found to be a nearly linear function of 1/λ(q). An exact identity reduces total loop persistence to the weight of the lattice's minimum spanning tree minus a boundary correction. The boundary correction follows from the ring recursion, and the spanning-tree weight is derived to first order in 1/λ(q) by attaching rings one at a time to a contracted interior, with a correction for loops that close through the ring outside. The resulting formula has no free parameters, gives the large-curvature intercept in closed form as 5/4−2π/(3√3), and agrees with the measurements to within the size of the next-order term.
Related
- Geometric ideal triangulations of hyperbolic 3-manifolds
- Continuity for random walks in hyperbolic percolation: drift, entropy, and dimension
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- Veering triangulations of once-punctured torus bundles are unique
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BibTeX
@misc{treisman2026loop,
title = {{Loop content of bond percolation on hyperbolic triangulations: an exact identity and a $1/λ$ expansion}},
author = {Zachary Treisman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30347},
eprint = {2609.30347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30347},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Loop content of bond percolation on hyperbolic triangulations: an exact identity and a 1/λ expansion
AU - Zachary Treisman
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30347
UR - https://arxiv.org/abs/2609.30347
AB - We study the homology of percolation clusters on disk-shaped patches of the regular {3,q} triangle tilings for q≥7. Varying q gives a one-parameter family of hyperbolic tilings, from mildly curved (q=7) to deeply hyperbolic (q=20), and this paper measures and derives loop content as a function of curvature. A single growth-rate parameter λ(q), the asymptotic ratio of successive ring sizes, governs every result. Total persistent H_1 of the bond-percolation complex, per vertex, is measured across q=7,…,20 and found to be a nearly linear function of 1/λ(q). An exact identity reduces total loop persistence to the weight of the lattice's minimum spanning tree minus a boundary correction. The boundary correction follows from the ring recursion, and the spanning-tree weight is derived to first order in 1/λ(q) by attaching rings one at a time to a contracted interior, with a correction for loops that close through the ring outside. The resulting formula has no free parameters, gives the large-curvature intercept in closed form as 5/4−2π/(3√3), and agrees with the measurements to within the size of the next-order term.
ER -