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A new computation of pairing probabilities in several multiple-curve models

Alex Karrila

TL;DR

The core of the argument are the convexity and uniqueness properties of local multiple SLE measures, valid for all κ > 0 and thus in principle applicable for any underlying random curve model, once suitably connected to localmultiple SLEs.

Abstract

We give a new, short computation of pairing probabilities for multiple chordal interfaces in the critical Ising model, the harmonic explorer, and for multiple level lines of the Gaussian free field. The core of the argument are the known convexity property and a new uniqueness property of local multiple SLE$(κ)$ measures, valid for all $κ> 0$. In particular, the proof is directly is applicable for any underlying random curve model, once it is identified as a local multiple SLE$(κ)$ both conditionally and unconditionally on the pairing topology.

A new computation of pairing probabilities in several multiple-curve models

TL;DR

The core of the argument are the convexity and uniqueness properties of local multiple SLE measures, valid for all κ > 0 and thus in principle applicable for any underlying random curve model, once suitably connected to localmultiple SLEs.

Abstract

We give a new, short computation of pairing probabilities for multiple chordal interfaces in the critical Ising model, the harmonic explorer, and for multiple level lines of the Gaussian free field. The core of the argument are the known convexity property and a new uniqueness property of local multiple SLE measures, valid for all . In particular, the proof is directly is applicable for any underlying random curve model, once it is identified as a local multiple SLE both conditionally and unconditionally on the pairing topology.
Paper Structure (19 sections, 8 theorems, 35 equations, 2 figures)

This paper contains 19 sections, 8 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Solving the pairings probabilities
  4. The Ising model
  5. Notation and domain approximations
  6. External inputs
  7. Checking the inputs
  8. Proof
  9. The multiple harmonic explorer
  10. The level lines of the Gaussian free field
  11. The SLE theory lemmas
  12. Proof of Lemma \ref{['lem:ptt fcn conv space ppty']}
  13. Proof of Lemma \ref{['lem:dr fcns <-> measures']}
  14. Proof of Lemma \ref{['lem:ex of density']}
  15. Precise definitions of the models
  16. ...and 4 more sections

Key Result

Lemma 1.1

Fix launching points $V_0^1 < \ldots < V^{2N}_0$, an index $1 \leq j \leq 2N$, and a localization neighbourhood of $V^j_0$. In this geometry, let ${\sf P}$ be a convex combination of finitely many laws $({\sf P}_\alpha)_\alpha$ of some multiple $\mathsf{SLE}(\kappa)$ driving functions with respectiv Then, also ${\sf P}$ is a local multiple $\mathsf{SLE}(\kappa)$ driving function, and the correspon

Figures (2)

  • Figure 1: Left, middle: Simulations of the critical Ising model with alternating boundary conditions on $100 \times 100$ (left) and $400 \times 400$ (middle) square grid graphs. The model with $2N$ boundary condition alternation points naturally gives rise to $N$ chordal interfaces (here $N=4$), pairing up the alternation points in some random manner. Labelling the alternation points in these two examples by the points of compass in the natural manner, the pairings of the alternation points by the interfaces are here $\{ \{ \mathrm{sw}, \mathrm{s} \}, \{ \mathrm{se}, \mathrm{e} \}, \{ \mathrm{ne}, \mathrm{n} \}, \{ \mathrm{nw}, \mathrm{w} \} \}$ (left) and $\{ \{ \mathrm{sw}, \mathrm{s} \}, \{ \mathrm{se}, \mathrm{w} \}, \{ \mathrm{e}, \mathrm{ne} \}, \{ \mathrm{n}, \mathrm{nw} \} \}$ (middle). Right: A schematic illustration of the setup where local multiple SLEs are studied in this note: $2N$ fixed boundary points $V_0^1 < \ldots < V^{2N}_0$, a fixed index $1 \leq j \leq 2N$, and a fixed neighbourhood of $V^j_0$ in $\mathbb{H}$ not containing any other marked boundary points on its boundary. The Loewner growth process \ref{['eq:Loewner ODE']}--\ref{['eq:NSLE SDE']} starting from $V^j_0$ is then considered up to the stopping time when $K_t$ first hits the boundary (in $\mathbb{H}$) of this localization neighbourhood.
  • Figure 2: The boundary conditions in the conditional law of Proposition \ref{['prop:GFF lvl lines']}(ii) extend the original boundary condition in a natural "cliff line" manner.

Theorems & Definitions (9)

  • Lemma 1.1: Convexity and uniqueness properties of multiple SLEs
  • Theorem 2.1: Pairing probabilities in the critical Ising model
  • Theorem 2.2: Pairing probabilities of multiple harmonic explorers
  • Theorem 2.3: Pairing probabilities of GFF level lines
  • Remark 2.4
  • Lemma 3.1: Uniqueness of the multiple SLE partition function
  • Lemma 3.2
  • Proposition A.1: GFF level lines
  • Corollary A.2: GFF level lines are global multiple SLEs