Critical Ising model, Multiple SLE$_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$ and $\beta$-Jacobi Ensemble

Abstract

Fix $N\ge 1$ and suppose that $(\Omega;x_1,\ldots, x_{N}; x_{N+1}, x_{N+2})$ is a polygon, i.e. $\Omega$ is a simply connected domain with locally connected boundary and $x_1,\ldots,x_{N+2}$ are $N+2$ different points located counterclockwisely on $\partial\Omega$. Fix $\kappa\in (0,4)$. In this paper, we will give two different constructions of multiple $N$-SLE$_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$ on $(\Omega;x_1,\ldots,x_{N}; x_{N+1},x_{N+2})$ and prove that they give the same law on random curves. Then, by establishing the uniqueness of multiple $N$-SLE$_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$, we can obtain the joint law of the hitting points of multiple $N$-SLE$_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$ 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 $\beta$-Jacobi ensemble with the conjectured relation $\beta=\frac{8}{\kappa}$. We will establish a direct connection between SLE-type curves and $\beta$-Jacobi ensemble. As an application, we consider critical Ising model on a discrete polygon $(\Omega^\delta_\delta;x^\delta_1,\ldots,x^\delta_{N}; x^\delta_{N+1},x^\delta_{N+2})$ with alternating boundary $(x^\delta_{N+2}x^\delta_{N+1})$ and free boundary $(x^\delta_{N+1}x^\delta_{N+2})$. Motivated by the partition function of multiple $N$-SLE$_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$, we derive the scaling limit of the probability of the event that the interface $\gamma_j^\delta$ starting from $x^\delta_j$ ends at $(x^\delta_{N+1}x^\delta_{N+2})$ for all $1\le j\le N$. Moreover, we prove that given this event, the interface $(\gamma_1^\delta,\ldots,\gamma_N^\delta)$ converges to multiple $N$-SLE$_\kappa\left(\frac{\kappa-6}{2},\frac{\kappa-6}{2}\right)$ with $\kappa=3$.