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Percolation and $O(1)$ loop model

Mikhail Khristoforov, Stanislav Smirnov

TL;DR

The article provides an 'ultimate' proof of Cardy’s formula for critical percolation on the hexagonal lattice by constructing a spinor/loop framework and a discrete holomorphic observable with exact holomorphicity. The approach identifies the scaling limit of crossing probabilities as the conformal map $φ$ to the equilateral triangle, proving that $\lim_{δ\to0}\mathbb P^{\mathrm{perc}}_{Ω^δ}[∂_{A^δ B^δ}Ω^δ \leftrightarrow ∂_{C^δ D^δ}Ω^δ] = \frac{φ(C^{\bullet})-φ(D^{\bullet})}{φ(C^{\bullet})-φ(A^{\bullet})}$ for Jordan domains, and extends to general Carathéodory-convergent domains. The method relies on a parafermionic observable with discrete holomorphicity, Russo–Seymour–Welsh estimates for compactness, and a compactness/limit identification argument showing the limit must be the conformal map, enabling straightforward generalizations to related observables and link-pattern probabilities. This framework integrates discrete holomorphicity with conformal invariance and offers a robust path toward broader generalizations in percolation and related models.

Abstract

We present an "ultimate" proof of Cardy's formula for the critical percolation on the hexagonal lattice \cite{Smirnov01criticalpercolation}, showing the existence of the universal and conformally invariant scaling limit of crossing probabilities. The new approach is more conceptual, less technically demanding, and is amenable to generalizations.

Percolation and $O(1)$ loop model

TL;DR

The article provides an 'ultimate' proof of Cardy’s formula for critical percolation on the hexagonal lattice by constructing a spinor/loop framework and a discrete holomorphic observable with exact holomorphicity. The approach identifies the scaling limit of crossing probabilities as the conformal map to the equilateral triangle, proving that for Jordan domains, and extends to general Carathéodory-convergent domains. The method relies on a parafermionic observable with discrete holomorphicity, Russo–Seymour–Welsh estimates for compactness, and a compactness/limit identification argument showing the limit must be the conformal map, enabling straightforward generalizations to related observables and link-pattern probabilities. This framework integrates discrete holomorphicity with conformal invariance and offers a robust path toward broader generalizations in percolation and related models.

Abstract

We present an "ultimate" proof of Cardy's formula for the critical percolation on the hexagonal lattice \cite{Smirnov01criticalpercolation}, showing the existence of the universal and conformally invariant scaling limit of crossing probabilities. The new approach is more conceptual, less technically demanding, and is amenable to generalizations.
Paper Structure (8 sections, 6 theorems, 20 equations, 4 figures)

This paper contains 8 sections, 6 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Percolation model
  3. Loop representation
  4. Spinor percolation model
  5. Discrete holomorphicity
  6. Theorem \ref{['Thm1']} for the Jordan case
  7. A priori estimates
  8. Theorem \ref{['Thm1']} for the general case

Key Result

Theorem 1

If $\{(Ω^δ, A^δ, B^δ, C^δ, D^δ) \}_δ$ approaches $(Ω^{\bullet}, A^{\bullet}, B^{\bullet}, C^{\bullet}, D^{\bullet})$ (in the sense of Definition definition:NiceApproximation) then where $φ$ is the conformal map from $Ω^{\bullet}$ to an equilateral triangle, mapping $A^{\bullet}, B^{\bullet}, C^{\bullet}$ to vertices.

Figures (4)

  • Figure 1: Here $σ$ is drawn in blue and yellow and $ξ$ in red; $\mathcal{IP}(ξ)$ is thick and outer boundaries are dashed.
  • Figure 2: Link patterns $[{ z \leftrightsquigarrow u_1}]$, $[{ z \leftrightsquigarrow u_2}]$, $[{ z \leftrightsquigarrow u_3}]$.
  • Figure 3: Graphical proof of Lemma \ref{['lemma:DiscreteHolomorphicity']}. Mid-edges $z_1, z_2, z_3$ are marked with diamonds and $u_1, u_2, u_3$ with circles. Configurations are grouped horizontally.
  • Figure 4: Two examples of a $\hexagon$-path $[xy]$ (in blue), sets $S$ (in orange) and $\tilde{S}$ (in green) and $\hexagon$-paths $α(β(ξ)), β(ξ)$ (in red, only on the left). In general, $S$ either surrounds $x$ and $y$ as on the left, or cuts away the part of $\Omega$ containing them, as on the right. On the right we sketch how such cuts would look if $\Omega^{\bullet}$ is non-Jordan.

Theorems & Definitions (15)

  • Theorem 1: Smirnov'01, Smirnov01criticalpercolation
  • Lemma 2
  • proof
  • Definition 3
  • Lemma 4: Discrete holomorphicity
  • proof
  • Corollary 5
  • proof
  • Remark 6
  • Definition 7
  • ...and 5 more