P. M. Duxbury

Floppy modes and the free energy: Rigidity and connectivity percolation on Bethe lattices

We show that the negative of the number of {ital floppy modes} behaves as a {ital free energy} for both connectivity and rigidity percolation, and we illustrate this result using Bethe lattices. The rigidity transition on Bethe lattices is found to be first order at a bond concentration close to that predicted by Maxwell constraint counting. We calculate the probability of a bond being on the infinite cluster and also on the overconstrained part of the infinite cluster, and show how a {ital specific heat }can be defined as the second derivative of the free energy. We demonstrate that the Bethe lattice solution is equivalent to that of the random bond model, where points are joined randomly (with equal probability at all length scales) to have a given coordination, and then subsequently bonds are randomly removed. {copyright} {ital 1999} {ital The American Physical Society}

GROUND STATE STRUCTURE OF RANDOM MAGNETS

Using exact optimization methods, we find all of the ground states of ({plus_minus}h) random-field Ising magnets (RFIM) and of dilute antiferromagnets in a field (DAFF). The degenerate ground states are usually composed of isolated clusters (two-level systems) embedded in a frozen background. We calculate the paramagnetic response (sublattice response) and the ground state entropy for the RFIM (DAFF) due to these clusters. In both two and three dimensions there is a broad regime in which these quantities are strictly positive, even at irrational values of h/J (J is the exchange constant). {copyright} {ital 1998} {ital The American Physical Society}

Permeability and conductivity of platelet-reinforced membranes and composites.

We present large-scale simulations of the diffusion constant D of a random composite consisting of aligned platelets with aspect ratio a/b>>1 in a matrix (with diffusion constant D0) and find that D/D(0)=1/(1+c(1)x+c(2)x(2)), where x=av(f)/b and v(f) is the platelet volume fraction. We demonstrate that for large aspect ratio platelets the pair term (x(2)) dominates suggesting large property enhancements for these materials. However, a small amount of face-to-face ordering of the platelets markedly degrades the efficiency of platelet reinforcement.

Ground state nonuniversality in the random-field Ising model.

Two attractive and often used ideas, namely, universality and the concept of a zero-temperature fixed point, are violated in the infinite-range random-field Ising model. In the ground state we show that the exponents can depend continuously on the disorder and so are nonuniversal. However, we also show that at finite temperature the thermal order-parameter exponent 1/2 is restored so that temperature is a relevant variable. Broader implications of these results are discussed.

Random manifolds in non-linear resistor networks: applications to varistors and superconductors

We show that current localization in polycrystalline varistors occurs on paths which are usually in the universality class of the directed polymer in a random medium. We also show that, in ceramic superconductors, voltage localizes on a surface which maps to an Ising domain wall. The emergence of these manifolds is explained and their structure is illustrated using direct solution of non-linear resistor networks.

Combinatorial optimization methods in disordered systems

We give an overview of the applications of methods from combinatorial optimization to problems in disordered systems. The optimization methods are efficient, for example it is possible to find the ground state of a random field Ising magnet containing one million sites in a couple of minutes on a high end workstation. Combinatorial algorithms for rigidity percolation and minimal energy domain walls in random exchange magnets are even more efficient.

Interface layering transitions in novel geometries

We show that a novel choice of boundary conditions leads to interfaces which unbind from a surface through a series of layering transitions. Even though the models have only nearest-neighbour interactions the transitions take place at zero temperature. Series expansions are used to probe the behaviour of the phase boundaries as the temperature is increased from zero.