1,720,977 research outputs found
The thermodynamic limit on Bethe lattices
No full textThe author emphasises the fact that, to obtain the exact solution of Hamiltonian models on Bethe lattices one can apply explicitly the formal method of rigorous statistical mechanics, i.e. the thermodynamic limit of probability measures. This approach solves the well known dichotomy between clashing alternative solutions, and provides a very simple analytic solution for a large class of Hamiltonian models
First-order transitions in percolation models
No full textWe describe the analytic solution of a Potts-correlated-site/random-bond percolation model which exhibits a first-order percolative transition at the Potts critical point. The subject is discussed in connection with the droplet theory in lattice spin models
Probability measures and Hamiltonian models on Bethe lattices. II. The solution of thermal and configurational problems
No full textIn 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
Probability measures and Hamiltonian models on Bethe lattices. I. Properties and construction of MRT probability measures
No full textThe 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
Clusters and droplets in the q-state Potts model
No full textA 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
The Potts model on Bethe lattices: I. General results
No full textThe 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
A-B site-bond correlated percolation for antiferromagnetic Ising model: The Bethe cluster approximation
No full textWe study the site-bond percolation problem for clusters of holes and particles with antiferromagnetic order by means of the Bethe cluster approximation. We find that the droplets (i.e. P B =1- e -| K|/2) diverge at the antiferromagnetic critical point H=0, T= T c; however for H≠0 they diverge along a percolation line which is different from the Antiferromagnetic Phase Boundary except at T=0
The Potts model on Bethe lattices: II. Special topics
No full textWe analyze the properties of the q-state ferromagnetic Potts model for real q. The nature of the phase transition at the critical point is first-order for q=/=2, and second-order for q=2. The random-bond percolation limit q-->1, and its second-order-like transition, are not related to the previous behaviour since they arise from non-stable phases of the system. It is suggested that this property characterizes the model on high-dimensional lattices, too
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