Phase Separation in Random Cluster Models III: Circuit Regularity

Phase separation in random cluster models III: circuit regularity

We study the droplet that results from conditioning the subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Γ encircling the origin and enclosing an area of at least (or exactly) n . 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 site v ∈ Γ, the circuit Γ cuts thro...

Phase separation in random cluster models II: the droplet at equilibrium, and local deviation lower bounds

We study the droplet that results from conditioning the planar subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Γ0 encircling the origin and enclosing an area of at least (or exactly) n . We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR ( Γ0 ) , this being the maximum distance from a point in...

Phase separation in random cluster models I: uniform upper bounds on local deviation

This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the planar Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit Γ0 encircling the origin and enclosing an area of at least (or exactly) n . (By the Fortuin-Kasteleyn representation, the model is a close relative of the drop...

Extension on Regularity Condition in DEA Models

Eigenvalue Separation in Some Random Matrix Models

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secul...