The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spin models with O n continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized o...

In this paper we study phase transitions for the Potts model with four competing interactions (external field, the nearest neighbor, the second neighbors and triples of neighbors) on Cayley tree of order two. We prove that for some parameter values of the model there is phase transition. We reduce the problem of describing limiting Gibbs measures to the problem of solving a system of nonlinear ...

We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal properties of which are related to the critical singularities of the random-Potts model. The optimization problem of finding the dominant graph, is studied on the squ...

In this article, we study the q-state Potts random matrix models extended to branched polymers, by the equations of motion method. We obtain a set of loop equations valid for any arbitrary value of q. We show that, for q = 2− 2 cos l r π (l, r mutually prime integers with l < r ), the resolvent satisfies an algebraic equation of degree 2r− 1 if l+ r is odd and r− 1 if l+ r is even. This general...

The partition function of the Potts model can be expressed in terms of famous combinatorial graph polynomials for restricted Hamiltonians. We investigate the Hamiltonian with variable edge interaction and external magnetic field, for which the partition function is the V-polynomial. Because the partition function is a polynomial, we can use motivic methods from algebraic geometry to characteriz...

Klippel-Feil syndrome (KFS) is characterized by the failure in segmentation of two or more vertebrae due to an abnormal division of the mesodermal somites and has been reported to be associated with cardiac and central nervous system anomalies. We report a rare occurrence of concurrent Klippel-Feil anomaly, tethering and dermoid cyst of dorsal spine masquerading Potts’ disease in an eighteen ye...

We introduce a Potts model with quenched, frustrated disorder, that enjoys of a gauge symmetry that forbids spontaneous magnetization, and allows the glassy phase to extend from Tc down to T = 0. We study numerical the 4 dimensional model with q = 4 states. We show the existence of a glassy phase, and we characterize it by studying the probability distributions of an order parameter, the binder...

The 3-D Z(3) Potts model is a model for finite temperature QCD with heavy quarks. The chemical potential in QCD becomes an external magnetic field in the Potts model. Following Alford et al.[3], we revisit this mapping, and determine the phase diagram for an arbitrary chemical potential, real or imaginary. Analytic continuation of the phase transition line between real and imaginary chemical po...

We restudy the phase diagram of the 2D-Ising model with competing interactions J1 on nearest neighbour and J2 on next-nearest neighbour bonds via Monte-Carlo simulations. We present the finite temperature phase diagram and introduce computational methods which allow us to calculate transition temperatures close to the critical point at J2 = 0.5J1. Further on we investigate the character of the ...

The influence of surface tension and size on rounding of cell aggregates are studied using chick embryonic cells and numerical simulations of the cellular Potts model. Our results show exponential relaxation in both cases as verified in previous studies using 2D Hydra cell aggregates. The relaxation time decreases with higher surface tension as expected from hydrodynamics laws. However, it incr...