## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

7-1-1982

#### Publication Source

Physical Review B

#### Volume

26

#### Issue

1

#### Start Page

337

#### Last Page

366

#### DOI

10.1103/PhysRevB.26.337

#### Abstract

The use of the (1/σ) expansion to calculate the thermodynamic properties of systems such as the Ising model or percolation whose diagrammatic expansion contains only diagrams with no free ends is reviewed. Here σ=z−1, where z is the coordination number of the lattice. For more general problems we formulate a self-consistency condition for a site potential h, so that diagrams with free ends are eliminated. Construction of h gives the leading order in (1/σ) solution and is exact for the Cayley tree. We obtain correction terms by using a bond renormalized interaction so that to order (1/σ)^{5} we need only consider two-site problems. Results are given for (1) K_{c}, the critical fugacity for animals, when either H, the fugacity for free ends, or Q, the density of free ends, is fixed, and (2) (t/E_{c}), where E_{c} is the mobility energy and t is the magnitude of the hopping matrix element whose sign is random. At d=8 our results appear to be accurate to within about 0.01% for both animals and localization. We also obtain an expansion for Q(K_{c})/(zK_{c}) whose divergence near spatial dimensionality d=4 supports the idea that the order-parameter exponent β for lattice animals passes through zero at d=4.

#### Recommended Citation

Harris, A.
(1982).
Renormalized (1/σ) Expansion for Lattice Animals and Localization.
*Physical Review B,*
*26*
(1),
337-366.
http://dx.doi.org/10.1103/PhysRevB.26.337

**Date Posted:** 12 August 2015

This document has been peer reviewed.