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Backbone Exponent and Annulus Crossing Probability for Planar Percolation
Pierre Nolin1, Wei Qian1, Xin Sun2
1City University of Hong Kong, China.
Physical Review Letters
|April 7, 2025
Summary
We derived the backbone exponent for 2D percolation, revealing it as a transcendental number. This finding advances understanding of critical phenomena and connects to conformal field theory via Liouville quantum gravity.
Area of Science:
- Statistical Physics
- Probability Theory
- Conformal Field Theory
Background:
- Percolation theory studies systems near a phase transition.
- Exactly solved percolation exponents are typically rational numbers.
- The asymptotic behavior of percolation systems is crucial for understanding critical phenomena.
Purpose of the Study:
- To derive the backbone exponent for 2D percolation.
- To investigate the mathematical nature of this exponent.
- To explore connections between percolation theory and conformal field theory.
Main Methods:
- Utilizing Schramm-Loewner evolution (SLE) curves.
- Coupling SLE with Liouville quantum gravity (LQG).
- Leveraging the integrability of Liouville conformal field theory (CFT).
Main Results:
- The backbone exponent for 2D percolation is a transcendental number.
- This exponent is a root of an elementary equation.
- An exact formula for the probability of two disjoint paths crossing an annulus was derived.
Conclusions:
- The backbone exponent governs the leading asymptotic behavior.
- Other roots of the equation capture remaining asymptotic terms.
- This suggests the backbone exponent is part of a CFT with a bulk spectrum related to these roots.
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