$beta$-ensemble in the scaling limit of discrete models

Speaker: 刘明昶 博士，首都师范大学

Title: $\beta$-ensemble in the scaling limit of discrete models

Inviter: 随机分析研究中心

Time & Venue: 2025 年10月10日15:00–16:00 南楼613

Abstract: The $\beta$-ensemble is a mathematical model describing the distribution of charged particles with repulsive interactions, where the parameter $\beta>0$ controls the interaction strength. Schramm-Loewner Evolution (SLE) is a family of curves parameterised by $\kappa>0$ describing the scaling limits of discrete statistical models at criticality. It has been conjectured that $\beta$-ensemble and SLE$_\kappa$ curve are connected when $\beta=\frac{2}{\kappa}$ and $\beta=\frac{8}{\kappa}$. A known example is that the driving function of certain multiple SLEs, when parameterised by common parameterisation, coincides with Dyson’s Brownian motion—a dynamic version of $\beta$-ensemble. However, a direct connection between $\beta$-ensemble itself and SLE$_\kappa$ remains unclear. In this report, I will discuss two discrete models: loop-erased random walk and Ising model. Through these examples, I will explain that how $\beta$-ensembles describe the distribution of hitting points of certain multiple SLE$_\kappa$ curves. Inspired by this relation, I will also explain how to derive the scaling limit of certain connection probabilities in these two discrete models.