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Critical Ising model, Multiple SLE$_κ\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$ and $β$-Jacobi Ensemble

Mingchang Liu

TL;DR

This work establishes a precise link between multiple SLE curves with drift parameters $\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$ and $β$-Jacobi ensembles via two equivalent Gaussian free field–based constructions, and it proves a rigidity result: the joint hitting points at the target boundary have a density that matches Jacobi-type laws with $β=8/κ$. By showing uniqueness of the multiple $N$-SLE laws, the authors connect conditional SLE dynamics to these $β$-ensembles, and they extend the framework to critical Ising interfaces, deriving the scaling limit of key hitting events and proving convergence to the special case $κ=3$. The results illuminate how eigenvalue statistics from random matrix theory arise naturally in GFF/SLE couplings and provide a rigorous bridge between continuum SLE descriptions and lattice Ising models at criticality. Collectively, the paper advances the understanding of how conformal field theory, stochastic Loewner evolution, and random matrix ensembles interrelate in two-dimensional statistical mechanics.

Abstract

Fix $N\ge 1$ and suppose that $(Ω;x_1,\ldots, x_{N}; x_{N+1}, x_{N+2})$ is a polygon, i.e. $Ω$ is a simply connected domain with locally connected boundary and $x_1,\ldots,x_{N+2}$ are $N+2$ different points located counterclockwisely on $\partialΩ$. Fix $κ\in (0,4)$. In this paper, we will give two different constructions of multiple $N$-SLE$_κ\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$ on $(Ω;x_1,\ldots,x_{N}; x_{N+1},x_{N+2})$ and prove that they give the same law on random curves. Then, by establishing the uniqueness of multiple $N$-SLE$_κ\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$, we can obtain the joint law of the hitting points of multiple $N$-SLE$_κ\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$ with odd (resp. even) indices on $(x_{N+1}x_{N+2})$. After shrinking $x_1,\ldots,x_N$ to one point, the law of hitting points with odd (resp. even) indices converge to $β$-Jacobi ensemble with the conjectured relation $β=\frac{8}κ$. We will establish a direct connection between SLE-type curves and $β$-Jacobi ensemble. As an application, we consider critical Ising model on a discrete polygon $(Ω^δ_δ;x^δ_1,\ldots,x^δ_{N}; x^δ_{N+1},x^δ_{N+2})$ with alternating boundary $(x^δ_{N+2}x^δ_{N+1})$ and free boundary $(x^δ_{N+1}x^δ_{N+2})$. Motivated by the partition function of multiple $N$-SLE$_κ\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$, we derive the scaling limit of the probability of the event that the interface $γ_j^δ$ starting from $x^δ_j$ ends at $(x^δ_{N+1}x^δ_{N+2})$ for all $1\le j\le N$. Moreover, we prove that given this event, the interface $(γ_1^δ,\ldots,γ_N^δ)$ converges to multiple $N$-SLE$_κ\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$ with $κ=3$.

Critical Ising model, Multiple SLE$_κ\left(\frac{κ-6}{2},\frac{κ-6}{2}\right)$ and $β$-Jacobi Ensemble

TL;DR

This work establishes a precise link between multiple SLE curves with drift parameters and -Jacobi ensembles via two equivalent Gaussian free field–based constructions, and it proves a rigidity result: the joint hitting points at the target boundary have a density that matches Jacobi-type laws with . By showing uniqueness of the multiple -SLE laws, the authors connect conditional SLE dynamics to these -ensembles, and they extend the framework to critical Ising interfaces, deriving the scaling limit of key hitting events and proving convergence to the special case . The results illuminate how eigenvalue statistics from random matrix theory arise naturally in GFF/SLE couplings and provide a rigorous bridge between continuum SLE descriptions and lattice Ising models at criticality. Collectively, the paper advances the understanding of how conformal field theory, stochastic Loewner evolution, and random matrix ensembles interrelate in two-dimensional statistical mechanics.

Abstract

Fix and suppose that is a polygon, i.e. is a simply connected domain with locally connected boundary and are different points located counterclockwisely on . Fix . In this paper, we will give two different constructions of multiple -SLE on and prove that they give the same law on random curves. Then, by establishing the uniqueness of multiple -SLE, we can obtain the joint law of the hitting points of multiple -SLE with odd (resp. even) indices on . After shrinking to one point, the law of hitting points with odd (resp. even) indices converge to -Jacobi ensemble with the conjectured relation . We will establish a direct connection between SLE-type curves and -Jacobi ensemble. As an application, we consider critical Ising model on a discrete polygon with alternating boundary and free boundary . Motivated by the partition function of multiple -SLE, we derive the scaling limit of the probability of the event that the interface starting from ends at for all . Moreover, we prove that given this event, the interface converges to multiple -SLE with .

Paper Structure

This paper contains 12 sections, 19 theorems, 90 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Multiple $N$-$\mathrm{SLE}_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$
  3. Critical Ising model
  4. Preliminaries
  5. Loewner chains and SLE
  6. GFF flow lines
  7. $\beta$-Jacobi ensemble
  8. Ising model
  9. Notations
  10. Proof of Theorem \ref{['thm::multiple_odd']}--Corollary \ref{['coro::hitting_law']}
  11. Proof of Theorem \ref{['thm::same_point_odd']}--Corollary \ref{['coro::same_point_hitting_law']}
  12. Proof of Theorem \ref{['thm::Ising_limit']}

Key Result

Theorem 1.1

Fix $N\ge 1$ and $\kappa\in (0,4)$, and fix $x_1<\cdots<x_N<x_{N+1}$. We construct a probability measure on $(\ell_1,\ldots,\ell_N)\in X_N(\mathbb{H};x_1,\ldots,x_{N};x_{N+1},\infty)$ as follows. Then, the law of random curves $(\ell_1,\ldots,\ell_N)$ is a multiple $N$-$\mathrm{SLE}_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$. See Figure fig::ell for an illustration.

Figures (5)

  • Figure 1.1: This is an illustration of the setup of Ising model in this paper when $N=3$. Along $(x_5^\delta x_4^\delta)$, the boundary conditions are alternating boundary conditions; along $(x_4^\delta x_5^\delta)$, the boundary condition is free boundary condition. In this picture, the event $A_\delta$ occurs and the discrete curves in red are interfaces starting from $x_1^\delta,x_2^\delta$ and $x_3^\delta$.
  • Figure 1.2: This is an illustration of the Theorem \ref{['thm::multiple_odd']} when $N=3$. Note that $\ell_2$ hits $(x_4,+\infty)$ at $z_1$.
  • Figure 1.3: This is an illustration of the Theorem \ref{['thm::multiple_even']} when $N=3$. Note that $\eta_1$ hits $(x_4,+\infty)$ at $w_1$ and $\eta_3$ hits $(x_4,+\infty)$ at $w_2$.
  • Figure 1.4: This is an illustration of the Theorem \ref{['thm::multiple_odd']} when $N=3$. Note that $\ell_2$ hits $(x_4,+\infty)$ at $z_1$.
  • Figure 1.5: This is an illustration of the Theorem \ref{['thm::multiple_odd']} when $N=3$. Note that $\eta_1$ hits $(x_4,+\infty)$ at $w_1$ and $\eta_3$ hits $(x_4,+\infty)$ at $w_2$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 25 more