Polychromatic Arm Exponents for the Critical Planar FK-Ising model

TL;DR

This paper derives the arm exponents for SLE$_{kappa}$ with $kappa$ in (4,8) and applies these results to determine boundary and interior arm exponents for the critical planar FK-Ising model, confirming conformal invariance.

Contribution

It provides the first derivation of polychromatic arm exponents for SLE$_{kappa}$ in the range (4,8) and applies these to the critical FK-Ising model, extending understanding of its scaling limits.

Findings

Derived six boundary arm exponent patterns.

Derived three interior arm exponent patterns.

Confirmed conformal invariance of the FK-Ising model's scaling limit.

Abstract

Schramm Loewner Evolution (SLE) is a one-parameter family of random planar curves introduced by Oded Schramm in 1999 as the candidates for the scaling limits of the interfaces in the planar critical lattice models. This is the only possible process with conformal invariance and a certain "domain Markov property". In 2010, Chelkak and Smirnov proved the conformal invariance of the scaling limits of the critial planar FK-Ising model which gave the convergence of the interface to SLE$_{16/3}$. We derive the arm exponents of SLE$_{\kappa}$ for $\kappa\in (4,8)$. Combining with the convergence of the interface, we derive the arm exponents of the critical FK-Ising model. We obtain six different patterns of boundary arm exponents and three different patterns of interior arm exponents of the critical planar FK-Ising model on the square lattice.