Backbone Exponent and Annulus Crossing Probability for Planar Percolation

Pierre Nolin; Wei Qian; Xin Sun; Zijie Zhuang

doi:10.1103/PhysRevLett.134.117101

Backbone Exponent and Annulus Crossing Probability for Planar Percolation

Pierre Nolin^*
, Wei Qian^*
, Xin Sun^*
, Zijie Zhuang^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

4 Citations (Scopus)

39 Downloads (CityUHK Scholars)

Abstract

We report the recent derivation of the backbone exponent for 2D percolation. In contrast to previously known exactly solved percolation exponents, the backbone exponent is a transcendental number, which is a root of an elementary equation. We also report an exact formula for the probability that there are two disjoint paths of the same color crossing an annulus. The backbone exponent captures the leading asymptotic, while the other roots of the elementary equation capture the asymptotic of the remaining terms. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Our approach is based on the coupling between Schramm-Loewner evolution curves and Liouville quantum gravity (LQG), and the integrability of Liouville CFT that governs the LQG surfaces. © 2025 American Physical Society.

Original language	English
Article number	117101
Journal	Physical Review Letters
Volume	134
Issue number	11
Online published	19 Mar 2025
DOIs	https://doi.org/10.1103/PhysRevLett.134.117101
Publication status	Published - 21 Mar 2025

Bibliographical note

Full text of this publication does not contain sufficient affiliation information. With consent from the author(s) concerned, the Research Unit(s) information for this record is based on the existing academic department affiliation of the author(s).

Funding

We thank B. Duplantier and two anonymous referees for helpful comments. P.\u2009N. is partially supported by a GRF grant from the Research Grants Council of the Hong Kong SAR (project CityU11318422). W.\u2009Q. and X.\u2009S. are supported by National Key R&D Program of China (No. 2023YFA1010700). Z.\u2009Z. is partially supported by NSF Grant No. DMS-1953848.

Publisher's Copyright Statement

COPYRIGHT TERMS OF DEPOSITED FINAL PUBLISHED VERSION FILE: Nolin, P., Qian, W., Sun, X., & Zhuang, Z. (2025). Backbone Exponent and Annulus Crossing Probability for Planar Percolation. Physical Review Letters, 134(11), Article 117101. https://doi.org/10.1103/PhysRevLett.134.117101 The copyright of this article is owned by American Physical Society.

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title = "Backbone Exponent and Annulus Crossing Probability for Planar Percolation",

abstract = "We report the recent derivation of the backbone exponent for 2D percolation. In contrast to previously known exactly solved percolation exponents, the backbone exponent is a transcendental number, which is a root of an elementary equation. We also report an exact formula for the probability that there are two disjoint paths of the same color crossing an annulus. The backbone exponent captures the leading asymptotic, while the other roots of the elementary equation capture the asymptotic of the remaining terms. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Our approach is based on the coupling between Schramm-Loewner evolution curves and Liouville quantum gravity (LQG), and the integrability of Liouville CFT that governs the LQG surfaces. {\textcopyright} 2025 American Physical Society.",

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N2 - We report the recent derivation of the backbone exponent for 2D percolation. In contrast to previously known exactly solved percolation exponents, the backbone exponent is a transcendental number, which is a root of an elementary equation. We also report an exact formula for the probability that there are two disjoint paths of the same color crossing an annulus. The backbone exponent captures the leading asymptotic, while the other roots of the elementary equation capture the asymptotic of the remaining terms. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Our approach is based on the coupling between Schramm-Loewner evolution curves and Liouville quantum gravity (LQG), and the integrability of Liouville CFT that governs the LQG surfaces. © 2025 American Physical Society.

AB - We report the recent derivation of the backbone exponent for 2D percolation. In contrast to previously known exactly solved percolation exponents, the backbone exponent is a transcendental number, which is a root of an elementary equation. We also report an exact formula for the probability that there are two disjoint paths of the same color crossing an annulus. The backbone exponent captures the leading asymptotic, while the other roots of the elementary equation capture the asymptotic of the remaining terms. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Our approach is based on the coupling between Schramm-Loewner evolution curves and Liouville quantum gravity (LQG), and the integrability of Liouville CFT that governs the LQG surfaces. © 2025 American Physical Society.

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Backbone Exponent and Annulus Crossing Probability for Planar Percolation

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GRF: Recoveries and Avalanches in Forest Fires and Related Models

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Backbone Exponent and Annulus Crossing Probability for Planar Percolation

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GRF: Recoveries and Avalanches in Forest Fires and Related Models

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