The Mean Field Theory for Percolation Models of the Ising Type

TL;DR

This paper investigates the mean field theory of percolation models related to the Ising model, revealing anomalous critical exponents, two diverging length scales, and stability of these exponents under disorder, with implications for high-dimensional systems.

Contribution

It provides a detailed analysis of critical exponents in mean field percolation models of the Ising type, highlighting anomalous values and the presence of two diverging length scales.

Findings

Critical exponents $ ilde eta$, $ ilde u^\prime$, and $ ilde u$ are characterized.

Anomalous critical exponents indicate an upper critical dimension of 6.

Finite cluster exponents are stable under disorder.

Abstract

The $q=2$ random cluster model is studied in the context of two mean field models: The Bethe lattice and the complete graph. For these systems, the critical exponents that are defined in terms of finite clusters have some anomalous values as the critical point is approached from the high density side which vindicates the results of earlier studies. In particular, the exponent $\tilde \gamma^\prime$ which characterises the divergence of the average size of finite clusters is 1/2 and $\tilde\nu^\prime$, the exponent associated with the length scale of finite clusters is 1/4. The full collection of exponents indicates an upper critical dimension of 6. The standard mean field exponents of the Ising system are also present in this model ($\nu^\prime = 1/2$, $\gamma^\prime = 1$) which implies, in particular, the presence of two diverging length scales. Furthermore, the finite cluster exponents are stable to the addition of disorder which, near the upper critical dimension, may have interesting implications concerning the generality of the disordered system/correlation length bounds.