Daniel Gandolfo

On a Model of Random Cycles

Abstract We consider a model of random permutations of the sites of the cubic lattice. Permutations are weighted so that sites are preferably sent onto neighbors. We present numerical evidence for the occurrence of a transition to a phase with infinite, macroscopic cycles.

New Phase Transitions of the Ising Model on Cayley Trees

We show that the nearest neighbor Ising model on the Cayley tree exhibits new temperature–driven phase transitions. These transitions occur at various inverse temperatures different from the critical one. They are characterised by a change in the number of Gibbs states as well as by a drastic change of the behavior of free energies at these new transition points.We also consider the model in the presence of an external field and compute the free energies of translation invariant and some alternating boundary conditions.

Cooperative Phenomenon in B/Si(111) Segregation

In the frame of a phenomenological model we show that sound cooperative effect (filament structure of typical patterns) discovered in the B/Si(111) segregation (Thibaudau F., Roge T. P., Mathiez Ph., Dumas P. and Salvan F., Europhys. Lett., 25 (1994) 353) could be explained by elastic tension creating a non-local interaction of triplet type. As a criterium we use the fitting of concentration equilibrium curve in combination with histogram treatment of B/Si(111) patterns based on a linear stratification of images.

A lattice model for the line tension of a sessile drop

Within a semi-infinite three-dimensional lattice gas model describing the coexistence of two phases on a substrate, we study, by cluster expansion techniques, the free energy (line tension) associated with the contact line between the two phases and the substrate. We show that this line tension, is given at low temperature by a convergent series whose leading term is negative, and equals 0 at zero temperature.

Limit theorems and coexistence probabilities for the Curie–Weiss Potts model with an external field

The Curie-Weiss Potts model is a mean field version of the well-known Potts model. In this model, the critical line [beta]=[beta]c(h) is explicitly known and corresponds to a first-order transition when q>2. In the present paper we describe the fluctuations of the density vector in the whole domain [beta][greater-or-equal, slanted]0 and h[greater-or-equal, slanted]0, including the conditional fluctuations on the critical line and the non-Gaussian fluctuations at the extremity of the critical line. The probabilities of each of the two thermodynamically stable states on the critical line are also computed. Similar results are inferred for the random-cluster model on the complete graph.

Memory capacity in neural networks with spatial correlations between attractors

We consider the neural network model (Miyashita, 1988. Griniasty et al., 1993) proposed to describe neurophysiological experiments in which structurally uncorrelated patterns are converted into spatially correlated attractors. For such a network of N neurons and for values of the coupling constant a between succeeding patterns, taken in the interval [0,12), we prove the existence of a threshold storage capacity αc(a) such that there exists a local minima (in the energy function) near each of the M(N)(

A Manifold of Pure Gibbs States of the Ising Model on the Lobachevsky Plane

In this paper we construct many ‘new’ Gibbs states of the Ising model on the Lobachevsky plane, the millefeuilles. Unlike the usual states on the integer lattices, our foliated states have infinitely many interfaces. The interfaces are rigid and fill the Lobachevsky plane with positive density. We also construct analogous states on the Cayley trees.

Retrieval Properties of Bidirectional Associative Memories

We study a two-layer neural network made of N and M(N) neurons, producing a two-way association search for a family of p(N) patterns, where each pattern is a pair of two independent sub-categories of information having respectively N and M(N) components. In terms of the ratio γ=limN→∞M(N)/N, we study the retrieval capability of this network and show that there exists, at least, three regimes of association for which we determine the evolution of the threshold αc(γ) of the storage capacity α=limN→∞p(N)/N.

On the parallel dynamics of the diluted clock neural network

We present explicit formulae for one-step evolution of the overlaps in a clock neural network for the parallel dynamics. These formulae constitute recursion relations for the exact dynamics in the case of extreme dilution. The fixed point equation for this exactly soluble model is investigated for varying temperature T, capacity parameter α and the number q of the clock states.

Cellular automata approach to site percolation on ℤ2. A numerical study

We present a cellular automata model as a new approach to Bernoulli site percolation on the square lattice. A new macroscopic quantity is defined and numerically computed at each level step of the automata dynamics. Its limit manifests a critical behavior at a value of the site occupancy probability quite close to those obtained for site percolation on ℤ2 with the best-known numerical methods.