Conformal covariance of connection probabilities in the 2D critical FK-Ising model

TL;DR

This paper proves that connection probabilities in the critical FK-Ising model are conformally covariant in the scaling limit, advancing understanding of conformal invariance in statistical physics models.

Contribution

It establishes the conformal covariance of connection probabilities in the critical FK-Ising model and provides new exact formulas for Ising boundary spin correlations.

Findings

Connection probabilities have nontrivial conformally covariant limits.

Provides an alternative proof of conformal covariance of Ising correlations.

Derives new exact formulas for boundary spin correlation functions.

Abstract

We study connection probabilities between vertices of the square lattice for the critical random-cluster (FK) model with cluster weight 2, which is related to the critical Ising model. We consider the model on the plane and on domains conformally equivalent to the upper half-plane. We prove that, when appropriately rescaled, the connection probabilities between vertices in the domain or on the boundary have nontrivial limits, as the mesh size of the square lattice is sent to zero, and that those limits are conformally covariant. This provides an important step in the proof of the Delfino-Viti conjecture for FK-Ising percolation as well as an alternative proof of the conformal covariance of the Ising spin correlation functions. In an appendix, we also derive new exact formulas for some Ising boundary spin correlation functions.