Communications in Mathematical Physics

, Volume 143, Issue 3, pp 467–499 | Cite as

Exponent inequalities for the bulk conductivity of a hierarchical model

  • K. Golden


The bulk conductivityσ*(p) of the bond lattice in ℤ d is considered, where the bonds have conductivity 1 with probabilityp or ε≧0 with probability 1-p Various representations of the derivatives ofσ*(p) are developed. These representations are used to analyze the behavior ofσ*(p) for ε=0 near the percolation thresholdp c , when the conducting backbone is assumed to have a hierarchical node-link-blob (NLB) structure. This model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds. Exact asymptotics of\(\frac{{d^2 \sigma ^* }}{{dp^2 }}\) for the NLB model are proven under some technical assumptions. The proof employs a novel technique whereby\(\frac{{d^2 \sigma ^* }}{{dp^2 }}\) for the NLB model with ε=0 andp nearp c is computed using perturbation theory forσ*(p) (for two- and three-component resistor lattices) aroundp=1 with a sequence of ε′s converging to 1 as one goes deeper in the hierarchy. These asymptotics establish convexity ofσ*(p) (for the NLB model) nearp c , and that its critical exponentt obeys the inequalities 1≦t≦2 ford=2,3, while 2≦t≦3 ford≧4. The upper boundt=2 ind=3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model.


Neural Network Statistical Physic Complex System Perturbation Theory Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • K. Golden
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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