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Crossing Probabilities of Multiple Ising Interfaces

Eveliina Peltola, Hao Wu

Abstract

We prove that in the scaling limit, the crossing probabilities of multiple interfaces in the critical planar Ising model with alternating boundary conditions are conformally invariant expressions given by the pure partition functions of multiple SLE(κ) with κ=3. In particular, this identifies the scaling limits with ratios of specific correlation functions of conformal field theory.

Crossing Probabilities of Multiple Ising Interfaces

Abstract

We prove that in the scaling limit, the crossing probabilities of multiple interfaces in the critical planar Ising model with alternating boundary conditions are conformally invariant expressions given by the pure partition functions of multiple SLE(κ) with κ=3. In particular, this identifies the scaling limits with ratios of specific correlation functions of conformal field theory.

Paper Structure

This paper contains 19 sections, 22 theorems, 125 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Partition Functions of Multiple SLEs
  3. Schramm-Loewner Evolutions
  4. Partition Functions of Multiple SLEs
  5. Useful Properties and Bounds
  6. Analyzing the Ising Partition Function
  7. Properties of the Bound Functions
  8. Cascade Asymptotics --- Proof of Proposition \ref{['prop::Z_total_asy_refined']}
  9. Upper and Lower Bounds --- Proof of Proposition \ref{['prop::Z_total_ineq']}
  10. Boundary Behavior
  11. Loewner Chains Associated to Partition Functions When $\kappa = 3$
  12. Cascade Relation for Partition Functions
  13. Continuity of the Loewner Chain --- Proof of Theorem \ref{['thm::loewner_Ztotal_continuity']}
  14. Crossing Probabilities in the Critical Ising Model
  15. Ising Model
  16. ...and 4 more sections

Key Result

Theorem \oldthetheorem

Let $(\Omega; x_1, \ldots, x_{2N})$ be a bounded polygon with $N \geq 1$, and suppose that discrete polygons $(\Omega^{\delta}; x_1^{\delta}, \ldots, x_{2N}^{\delta})$ on $\delta\mathbb{Z}^2$ converge to $(\Omega; x_1, \ldots, x_{2N})$ as $\delta\to 0$ in the close-Carathéodory sense. Consider the c and $\{\mathcal{Z}_{\alpha} \colon \alpha \in \mathrm{LP}_N\}$ is the collection of functions uniqu

Figures (2)

  • Figure 1.1: Critical Ising model configurations on a $100\times100$ square with alternating boundary conditions: there are six marked boundary points at which the boundary condition changes from $\oplus$ (black) to $\ominus$ (yellow). In this case, there are five possible planar topological connectivities of the three chordal interfaces, labeled by $\alpha$ (see Figure \ref{['fig::patterns']}). The total partition function has the symmetric form $\mathcal{Z}_{\mathrm{Ising}} = \sum_\alpha \mathcal{Z}_{\alpha}$, see Equation \ref{['eq::TotalPartPos']}, also featuring rotational symmetry of the boundary conditions under cyclic permutations of the marked boundary points (combined with a global spin flip $\oplus \leftrightarrow \ominus$ if necessary): the crossing probabilities remain invariant under such a cyclic permutation.
  • Figure 1.2: Illustration of the five possible planar topological connectivities $\alpha \in \mathrm{LP}_N$ of the three chordal interfaces in Figure \ref{['fig::Ising']} transformed to the upper half-plane (serving as a common reference domain by conformal invariance). Here, the first marked point on the left corresponds to the lower corner of the rotated square in Figure \ref{['fig::Ising']}, and we follow the marked points counterclockwise. We call the labels $\alpha \in \mathrm{LP}_N$ "link patterns".

Theorems & Definitions (42)

  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 32 more