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

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 $κ=3$.