Renormalized (1σ) expansion for lattice animals and localization

Abstract

The use of the ($\frac{1}{\sigma }$) 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 $\sigma =z-1\mathrm{}$, 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 ($\frac{1}{\sigma }$) solution and is exact for the Cayley tree. We obtain correction terms by using a bond renormalized interaction so that to order ${\left(\frac{1}{\sigma }\right)}^5$ 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) ($\frac{t}{E_c}$), where $E_c$ is the mobility energy and $t$ is the magnitude of the hopping matrix element whose sign is random. At $d=8\mathrm{}$ our results appear to be accurate to within about 0.01% for both animals and localization. We also obtain an expansion for $\frac{Q\left(K_c\right)}{\left(z{K}_c\right)}$ whose divergence near spatial dimensionality $d=4\mathrm{}$ supports the idea that the order-parameter exponent $\beta$ for lattice animals passes through zero at $d=4\mathrm{}$.