The near-critical planar FK-Ising model

TL;DR

This paper investigates the near-critical behavior of the planar FK-Ising model, revealing how correlation length and edge addition dynamics differ from standard percolation, with implications for understanding phase transitions.

Contribution

It provides a precise determination of the correlation length via crossing probabilities and uncovers a self-organized edge addition phenomenon near criticality.

Findings

Correlation length is explicitly determined.

Edges arrive in a self-organized manner near criticality.

The behavior differs from standard percolation models.

Abstract

We study the near-critical FK-Ising model. First, a determination of the correlation length defined via crossing probabilities is provided. Second, a phenomenon about the near-critical behavior of FK-Ising is highlighted, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations $\omega_p$ (e.g., in the one introduced in [Gri95]), as one raises $p$ near $p_c$, the new edges arrive in a self-organized way, so that the correlation length is not governed anymore by the number of pivotal edges at criticality.