Fulvio Peruggi

Percolation points and critical point in the Ising model

Rigorous inequalities are proved, which relate percolation probability, mean cluster size and pair connectedness respectively with magnetization, susceptibility and pair correlation function in ferromagnetic Ising models. In two dimensions the critical point is shown to be a percolation point, while in three dimensions this is not true.

Percolation and phase transitions in the Ising model

We give a description of the mechanism of phase transitions in the Ising model, pointing out the connection between the spontaneous magnetization and the existence of infinite clusters of “up” and “down” spins. The picture is more complete in the two-dimensional Ising model, where we can also use a generalized version of a result by Miyamoto.

The Potts model on Bethe lattices. I. General results

The q-state ferromagnetic Potts model (FPM) and antiferromagnetic Potts model (APM) are solved on Bethe lattices for all values of the external magnetic field and temperature. The exact expressions of all thermodynamic functions of interest in the FPM and APM are calculated. The authors find the complete phase diagrams for both systems. In the FPM there are first-order phase transitions at the critical point for every q>2. In the APM they find second-order phase transitions along a critical line for every q>or=2.

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.

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.

Probability measures and Hamiltonian models on Bethe lattices. I. Properties and construction of MRT probability measures

The properties of one‐step Markov, rotationally and m‐step (m=1 or 2) translationally invariant (MRT) probability measures on q‐state‐site (qSS) Bethe lattices are studied. A theorem is proven, which completely defines such measures in terms of m(q2+q) fundamental probabilities. These are explicitly calculated for any MRT–qSS Hamiltonian model. As a consequence of our approach, the dychotomy between alternative solutions of Hamiltonian models on Bethe lattices is solved.

Phase diagrams of the q-state potts model on Bethe lattices

We use a rigorous, recently developed, method of solution of spin models on Bethe lattices (Cayley trees). It allows us to detect all the phases of the ferromagnetic and antiferromagnetic q-state Potts models in a field (FPM and APM respectively). The phase diagrams of the FPM are characterized by three critical lines of first-order transitions. In the APM the ordered phases (which appear always for h > 0, and under certain conditions for h= 0 and/or h<0) are separated from the disordered one by phase boundaries characterized by second-order transitions, and are separated one from the other by critical lines characterized by first-order and second-order transitions. Several results are found, which hold true on standard lattices, too.

Probability measures and Hamiltonian models on Bethe lattices. II. The solution of thermal and configurational problems

In a previous paper we introduced a method for the construction of rotationally and translationally invariant probability measures generated by one‐step Markov Hamiltonian models on q‐state‐site Bethe lattices. Here, the corresponding thermal problems are solved by finding the relative free energy, which gives complete information on the properties of the models under study. Configurational problems also can be solved with the present tools. As an example, the solution of polychromatic correlated‐site/random‐bond percolation models is found.

Critical behaviour in three-state Potts antiferromagnets on a Bethe lattice

A rigorous method recently introduced for the solution of spin models on Bethe lattices is used to solve the three-state antiferromagnetic Potts model in an external field, and to find its phase diagram. Critical lines where the system undergoes first-order or second-order phase transitions are found, which are in complete qualitative agreement with recent results by Racz and Vicsek. Also new unexpected ordered phases are found.