Mean-field theory for percolation models of the Ising type

zbMATH Open0934.60095arXivcond-mat/9803121MaRDI QIDQ1301613

Publication date: 17 April 2000

Published in: Journal of Statistical Physics (Search for Journal in Brave)

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

i l d e g a m m a^{p} r i m e

which characterises the divergence of the average size of finite clusters is 1/2 and

i l d e u^{p} r i m e

, 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 (

u^{p} r i m e = 1 / 2

,

g a m m a^{p} r i m e = 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.

Full work available at URL: https://arxiv.org/abs/cond-mat/9803121

zbMATH Keywords

percolation Ising model mean-field theory

Mathematics Subject Classification ID

Interacting random processes; statistical mechanics type models; percolation theory (60K35) Percolation (82B43) Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics (82B20)