On extreme Gibbs measures of the Vannimenus model

Abstract

One of the main problems of statistical physics is that of describing all Gibbs measures corresponding to a given Hamiltonian. It is well known that such measures form a nonempty convex compact subset in the set of all probability measures. The purpose of this paper is to investigate extreme Gibbs measures of the Vannimenus model.

Description of Extreme Gibbs Measures for the Ising Model with Three Interactions

“…In previous works [4,5], for given Hamiltonians we obtained the Gibbs states that correspond in probability theory to what are called Markov chains with memory length 2. In this paper, we have studied the extreme Gibbs measures corresponding to Hamiltonian (1).…”

“…In [4], we have studied phase diagram and extreme Gibbs measures of the Ising model on a Cayley tree in the presence of competing binary and ternary interactions. In [5], we have obtained the extreme Gibbs measures of the Vannimenus model [6].…”

“…In this paper, we combine the results obtained in [4] and [5]. Our model with competing nearest-neighbors, next-nearest neighbors, and prolonged next-nearest-neighbors ternary interactions is dened by the following Hamiltonian:…”

Gibbs measures of an Ising model with competing interactions on the triangular chandelier-lattice

“…The theory of probability is one of the basic branches of mathematics lying at the base of the theory of statistical mechanics [8,[15][16][17][18][19][20]. As is known, one of the fundamental problems of statistical mechanics is to specify the set of all Gibbs measures associated to the given Hamiltonian [21][22][23][24][25]. A Gibbs measure is a probability measure frequently used in many problems of probability theory and statistical mechanics.…”

On Gibbs Measures of the Potts Model with Three Competing Interactions on Cayley Tree of Order 3

“…It is also well known that the structure of the lattice plays an important role in examining the spin systems. The Potts model, as a generalization of the Ising model on Cayley tree with competing interactions appeared in a pioneering work of Vannimenus [4], has recently been studied extensively (see [1,3,5,6]). On the Cayley tree one can consider two types of next-nearest-neighbors (triple neighbors): prolonged and one-level next-nearest-neighbors (respectively two-level triple neighbors).…”