Entropy and singularity of spectral and pivotal sets in planar percolation
Abstract
The spectral sample and pivotal set of a percolation crossing have the same one- and two-coordinate inclusion probabilities under the uniform product measure. We prove that both Shannon entropies are comparable to total influence for critical square-lattice bond and triangular-lattice site crossings, with matching estimates at every retained nonroot spatial resolution. The triangular entropy bounds are uniform over a class of inhomogeneous near-critical product measures. For critical triangular-site square crossings, the two laws are asymptotically mutually singular: an elementary triangular face is forbidden in the pivotal set and occurs with positive density in a large spectral sample. Every fixed face density below an explicit threshold has nonempty spectral probability comparable to R2α4(R)2, while the nonempty overlap of the two laws has decay exponent 11/12. Their minimum expected symmetric-difference distance over all couplings is comparable to total influence. Geometric separation persists after independent thinning whenever the retention probability ρR satisfies ρR3R2α4(R)→∞; the face detector fails when this quantity tends to zero. The proofs combine spatial encoding, arm estimates, local Fourier cancellation, and an exact six-cycle coercivity inequality preserved under exterior conditioning.
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BibTeX
@misc{shmalo2026entropy,
title = {{Entropy and singularity of spectral and pivotal sets in planar percolation}},
author = {Yitzchak Shmalo},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15388},
eprint = {2609.15388},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15388},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Entropy and singularity of spectral and pivotal sets in planar percolation
AU - Yitzchak Shmalo
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15388
UR - https://arxiv.org/abs/2609.15388
AB - The spectral sample and pivotal set of a percolation crossing have the same one- and two-coordinate inclusion probabilities under the uniform product measure. We prove that both Shannon entropies are comparable to total influence for critical square-lattice bond and triangular-lattice site crossings, with matching estimates at every retained nonroot spatial resolution. The triangular entropy bounds are uniform over a class of inhomogeneous near-critical product measures. For critical triangular-site square crossings, the two laws are asymptotically mutually singular: an elementary triangular face is forbidden in the pivotal set and occurs with positive density in a large spectral sample. Every fixed face density below an explicit threshold has nonempty spectral probability comparable to R^2α_4(R)^2, while the nonempty overlap of the two laws has decay exponent 11/12. Their minimum expected symmetric-difference distance over all couplings is comparable to total influence. Geometric separation persists after independent thinning whenever the retention probability ρ_R satisfies ρ_R^3R^2α_4(R)→∞; the face detector fails when this quantity tends to zero. The proofs combine spatial encoding, arm estimates, local Fourier cancellation, and an exact six-cycle coercivity inequality preserved under exterior conditioning.
ER -