Phase separation in random cluster models III: circuit regularity

TL;DR

This paper proves that conditioned circuits in the subcritical Fortuin-Kasteleyn model exhibit high regularity, with regeneration sites evenly distributed, enabling precise fluctuation bounds crucial for understanding phase separation.

Contribution

It introduces a notion of regeneration sites in conditioned circuits and proves their widespread distribution, facilitating advanced fluctuation analysis in random cluster models.

Findings

Regeneration sites are evenly distributed along conditioned circuits.

Maximum distance between regeneration sites is logarithmic in circuit size.

The results enable precise control of circuit fluctuations.

Abstract

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn planar random cluster model on the presence of an open circuit Gamma_0 encircling the origin and enclosing an area of at least (or exactly) n^2. In this paper, we prove that the resulting circuit is highly regular: we define a notion of a regeneration site in such a way that, for any such element v of Gamma_0, the circuit Gamma_0 cuts through the radial line segment through v only at v. We show that, provided that the conditioned circuit is centred at the origin in a natural sense, the set of regeneration sites reaches into all parts of the circuit, with maximal distance from one such site to the next being at most logarithmic in n with high probability. The result provides a flexible control on the conditioned circuit that permits the use of surgical techniques to bound its fluctuations, and, as such, it plays a crucial role in the derivation of bounds on the local fluctuation of the circuit carried out in arXiv:1001.1527 and arXiv:1001.1528.