Observation of SLE$(\kappa,\rho)$ on the Critical Statistical Models

TL;DR

This paper verifies the hydrodynamically normalized SLE(,) formalism for critical statistical models, demonstrating its effectiveness in analyzing domain wall curves in models like ASM and percolation.

Contribution

It provides the first direct verification of hydrodynamically normalized SLE(,) for critical models and applies it to domain wall curves, improving numerical analysis accuracy.

Findings

Hydrodynamically normalized SLE(,) is more reliable for interface loop analysis.

The method successfully applied to ASM () and critical percolation ().

Verification supports the hypothesis that this SLE formalism governs such curves.

Abstract

Schramm-Loewner Evolution (SLE) is a stochastic process that helps classify critical statistical models using one real parameter $\kappa$. Numerical study of SLE often involves curves that start and end on the real axis. To reduce numerical errors in studying the critical curves which start from the real axis and end on it, we have used hydrodynamically normalized SLE($\kappa,\rho$) which is a stochastic differential equation that is hypothesized to govern such curves. In this paper we directly verify this hypothesis and numerically apply this formalism to the domain wall curves of the Abelian Sandpile Model (ASM) ($\kappa=2$) and critical percolation ($\kappa=6$). We observe that this method is more reliable for analyzing interface loops.