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 $(\Omega ;x_1,\dots ,x_N;x_{N+1},x_{N+2})$ is a polygon, i.e. $\Omega$ is a simply connected domain with locally connected boundary and $x_1,\dots ,x_{N+2}$ 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 $(\Omega ;x_1,\dots ,x_N;x_{N+1},x_{N+2})$ 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 $x_1,\dots ,x_N$ 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 $({\Omega }_{\delta }^{\delta };x_1^{\delta },\dots ,x_N^{\delta };x_{N+1}^{\delta },x_{N+2}^{\delta })$ with alternating boundary $(x_{N+2}^{\delta }{x}_{N+1}^{\delta })$ and free boundary $(x_{N+1}^{\delta }{x}_{N+2}^{\delta })$. 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 $x_j^{\delta }$ ends at $(x_{N+1}^{\delta }{x}_{N+2}^{\delta })$ for all $1\le j\le N$. Moreover, we prove that given this event, the interface $({\gamma }_1^{\delta },\dots ,{\gamma }_N^{\delta })$ converges to multiple $N$-SLE${}_{\kappa }\left(\frac{\kappa -6}{2},\frac{\kappa -6}{2}\right)$ with $\kappa =3$.

29 pages, 5 figures