 98612 Chayes L., Coniglio A., Machta J., Shtengel K.
 The Mean Field Theory for
Percolation Models of the Ising Type
(38K, RevTeX)
Sep 19, 98

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Abstract. The $q=2$ random cluster model is studied in the context of two meanfield
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 highdensity side which vindicates the results of
earlier
studies. In particular, the exponent $\tilde \gamma^\prime$ which
characterizes 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 meanfield 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 lengthscales. 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.
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