A. Coniglio

Clusters and Ising critical droplets: a renormalisation group approach

The Migdal-Kadanoff renormalisation group for two-dimensions is employed to obtain the global phase diagram for the site-bond correlated percolation problem. It is found that the Ising critical point (K=Kc,H=O) is a percolation point for a range of bond probability rho B such that 1>or= rho B>or=1-e-2Kc. In particular, as rho B approaches 1-e-2Kc, the percolation clusters become less compact and coincide with the Ising critical droplets.

Cluster structure near the percolation threshold

Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc. In particular, the author proves the relations p(dpij/dp)=( lambda ij) where p is the bond density, pij is the pair connectedness function and ( lambda ij) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-pc mod -1. The remaining (multiply connected) bonds in the percolating backbone, which lump together in blobs, diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the blobs can be controlled by varying a parameter, and study the influence on the critical exponents.

Exact Relations Between Damage Spreading and Thermodynamical Properties

We investigate the damage or Hamming distance between two configurations of Ising spins. We find an exact relation between the difference of the two possible types of damage and the spin-spin correlation function, which is generally valid. For the specific case of ferromagnetic interactions, heat bath dynamics and same sequence of random numbers, this relation involves only one type of damage. The susceptibility and the magnetization can also be expressed in terms of the damage. Numerical determination of the damage for the 2d Ising model is not only an efficient way to calculate correlation functions but also gives access to spin fluctuations visualized as clusters of damaged sites which have a fractal dimension d−β/ν at Tc and whose size distribution is also related to static exponents.

Kinetic Growth Walk: A New Model for Linear Polymers

To describe the irreversible growth of linear polymers, we introduce a new type of perturbed random walk, related to the zero initiator concentration limit of the kinetic gelation model. Our model simulates real polymer growth by permitting the initiator (walker) to form the next bond with an unsaturated monomer at one of the neighbouring sites of its present location. A heuristic kinetic self-consistent field argument along the lines introduced by Pietronero suggests a fractal dimensionality, df = (d + 1)/2, in agreement with our Monte Carlo and series expansion results (including the usually expected logarithmic correction at the upper critical dimension dc = 3.

Percolation Transition in Spin Glasses

The percolation properties of geometrical clusters related to spin fluctuations have been investigated for the 3d ± J Ising spin glass. The percolation transition is found at a temperature Tp 3.92, far from the spin glass critical temperature. The critical exponents are consistent with the random percolation exponents.

Clusters and droplets in the q-state Potts model

A Potts correlated polychromatic percolation is studied. The clusters are made of sites corresponding to a given value of the q-state Potts variables, connected by bonds being active with probability pB. To treat this problem an s-state Potts Hamiltonian diluted with q-state Potts variables (instead of lattice gas variables) is introduced to which the the Migdal-Kadanoff renormalisation group is applied. It is found for a particular choice of pB=1-e-K (where K is the Potts coupling constant divided by the Boltzmann factor) that these clusters, called droplets diverge at the Potts critical point with Potts exponents.

Percolation, gelation and dynamical behaviour in colloids

We review some results on the dynamics of gelation phenomena, obtained via a lattice model and via molecular dynamics using a DLVO potential. This study allowed us to make a connection between classical gelation and the phenomenology of colloidal systems, suggesting that gelation phenomena in attractive colloids at low temperature and low volume fraction can be described in terms of a two-line scenario.

Exact relations between droplets and thermal fluctuations in external field

The authors extend the definition of droplets in Ising and Potts models to the case of an external field different from zero. They also find exact relations between thermal properties and connectivity properties which show why, in mean field, the mean cluster size does not diverge as the susceptibility when the critical temperature is approached from below.

Damage Spreading in Spin Glasses

By studying the time evolution of two configurations submitted to the same thermal noise, we investigate phase space properties of the three-dimensional ±J Ising spin glass. Our results for the distance between states are consistent with the picture of phase space having a multivalley structure below Tc. By fixing the damage at one site, we also study the Hamming distance between the two configurations at its critical point, where the distribution of probability of getting damaged n times is found to be multifractal.

Nucleation and metastability in three-dimensional Ising models

We present Monte Carlo experiments on nucleation theory in the nearest-neighbor three-dimensional Ising model and in Ising models with long-range interactions. For the nearest-neighbor model, our results are compatible with the classical nucleation theory (CNT) for low temperatures, while for the long-range model a breakdown of the CNT was observed near the mean-field spinodal. A new droplet model and a zeroth-order theory of droplet growth are also presented.