Bethe approximation for self-interacting lattice trees

¹ Institut de Physique Theorique, Université de Fribourg Chemin du Musée 3, CH-1700 Fribourg, Switzerland
² Department of Mathematics, Imperial College 180 Queen's Gate, London SW7 2BZ, UK
³ Dipartimento di Fisica, Politecnico di Torino - c. Duca degli Abruzzi 24, 10129 Torino and INFM Unità Torino Politecnico - Torino, Italy

Corresponding authors: Paolo.DeLosRios@unifr.ch lises@ic.ac.uk alex@athena.polito.it

Received: 25 September 2000
Accepted: 13 November 2000

Abstract

In this paper we develop a Bethe approximation, based on the cluster variation method, which is apt to study lattice models of branched polymers. We show that the method is extremely accurate in cases where exact results are known as, for instance, in the enumeration of spanning trees. Moreover, the expressions we obtain for the asymptotic number of spanning trees and lattice trees on a graph coincide with analogous expressions derived through different approaches. We study the phase diagram of lattice trees with nearest-neighbour attraction and branching energies. We find a collapse transition at a tricritical θ point, which separates an expanded phase from a compact phase. We compare our results for the θ transition in two and three dimensions with available numerical estimates.