Electronic Resource
Springer
Journal of statistical physics
27 (1982), S. 457-472
ISSN:
1572-9613
Keywords:
Cayley tree
;
iteration
;
fixed point
;
spin glass
;
Gaussian distribution
;
local mean-field theory
;
SK equations
;
TAP equations.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract In this paper we study rigorously the random Ising model on a Cayley tree in the limit of infinite coordination numberz → 8. An iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice. In this way we obtain local properties of the model in the (thermodynamic) limit of an infinite number of shells. When the coupling constants are independent Gaussian random variables the SK expressions emerge as stable fixed points of our scheme and provide a valid local mean-field theory of spin glasses in which negative local entropy (at low temperatures) while perfectly possible mathematically may still perhaps be physically undesirable. Finally we examine the TAP equations and show that if the average over bond disorder and the limitz → 8 are actually performed, one recovers our iterative scheme and hence the SK equations also in the thermodynamic limit.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01011086
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