A C N de Magalhaes

Pure and random Potts-like models: real-space renormalization-group approach

Abstract The present status of knowledge on the pure and random Potts models and various related systems (resistor network, directed bond percolation, Ising-like frustrated models, Z(q) model and the discrete cubic model) is reviewed. The available real-space renormalization group techniques which can further enlighten this picture are tutorially presented and discussed.

APERIODIC INTERACTIONS ON HIERARCHICAL LATTICES : AN EXACT CRITERION FOR THE POTTS FERROMAGNET CRITICALITY

We discuss the critical behaviour of the q-state Potts model on a diamond-like hierarchical lattice with ferromagnetic interactions according to an aperiodic two-letter substitutional sequence. We show that the geometric (deterministic) fluctuations become relevant for , where is the wandering exponent of the sequence, D is the fractal dimension of the lattice, and is the critical exponent associated with the specific heat of the uniform model. We also point out that the criteria for analysing the relevance of deterministic and random fluctuations are generically different.

Duality relation for potts multispin correlation functions

The duality relation for multispin correlation functions of the Potts model on a plane graph is derived using the formulation of a percolation average. It is shown that in simple cases the correlation functions are related to those of the dual graph. But more generally the correlation functions are expressible as linear combinations of ratios of certain percolation averages, the dual partitioned equivalent transmissivities, and dual correlation functions. An overall multiplication factor arises in these considerations, and the author elucidate the graph-theoretic meaning of this function.

Z(λ) model and flows: Subgraph break-collapse method

A recursive algorithm previously developed for carrying out renormalisation calculations for the lambda -state Potts model is generalised to the Z(λ) model. The relations used are based on an expression the authors derive for the pair correlation function in terms of mod- lambda flows, which represents an extension of a similar result for the partition function previously obtained by Biggs (1976). The use of flows enables them to prove and extend the formulae which appear in the break-collapse method of Mariz and co-workers (1989). It is argued that the use of fixed-flow bonds rather than the precollapsed bonds used by the latter authors leads to a more efficient algorithm.

The n-component cubic model and flows : subgraph break-collapse method

We specialize to then-component cubic model the subgraph break-collapse method which we recently developed for theZ(λ) model. The cubic model has less symmetry than the Potts model, for which the method was originally developed, but nevertheless it is still possible to reduce considerably the computational complexity of the generalZ(λ) model. Our recursive algorithm is similar, forn=2, to the break-collapse method for theZ(4) model proposed by Mariz and co-workers. It allows the exact calculation for the partition function and correlation functions forn-component cubic clusters, withn as a variable, without the need to examine all of the spin configurations. An important application is therefore in real-space renormalization-group calculations.

Critical frontiers associated with the bond-diluted Ising ferromagnet on triangular and honeycomb lattices

Within a real space renormalisation group framework (12 different procedures, all of them using star-triangle and duality-type transformations) the authors calculate accurate approximations for the critical frontiers associated with the quenched bond-diluted first-neighbour, spin-1/2 Ising ferromagnet on triangular and honeycomb lattices. All of them provide, in both pure bond percolation and pure Ising limits, the exact critical points and exact or almost exact derivatives in the p-t space (p is the bond-independent occupancy probability and t identical to tanh(J/kBT)). The authors best numerical proposals lead to the exact derivative in the pure percolation limit (p=pc) and, in what concerns the pure Ising limit (p=1) derivative, to a 0.15% error for the triangular lattice and to a 0.96% error for the honeycomb one; in the intermediate region (pc<p<1), where the exact critical frontiers are still unknown, the worst error in the variable t (for fixed p) is estimated to be less than 0.27% for the triangular lattice and 0.14% for the honeycomb one.

The Discrete N-Vector Ferromagnet: Connection to a Percolation with Frustration Features

We extend the Kasteleyn–Fortuin formalism to the discrete N-vector ferromagnet. We show that the free energy and the correlation functions of this model are related, when the number of states tends to 1, to the mean number of clusters and to the pair connectedness of a polychromatic bond percolation type problem which combines frustration and connectivity features.