Mathematics & Physics 2015, 8(2), 165–172 УДК 517.98 The Uniqueness of the Translation-invariant Gibbs Measure for Four State HC-models on a Cayley Tree Rustam M.Khakimov∗ Institute of Mathematics Do’rmon Yo’li str., 29, Tashkent, 100125 Uzbekistan Received 04.01.2015, received in revised form 17.02.2015, accepted 06.03.2015 We consider fertile Hard-Core (HC) models with activity parameter λ > 0 and four states on the Cayley tree of order two. <...> In this paper for each of these models the uniqueness of the translation-invariant Gibbs measure is proved. <...> The hard core (HC) model arises in the study of random independent sets of a graph ( [1]), the study of gas molecules on a lattice [2], and in the analysis of multi-casting in telecommunication networks [3]. <...> A HC model on d-dimensional lattice Zd, was introduced and studied by Mazel and Suhov in [4]. <...> In [8] a HC (Hard Core) model with two states on a Cayley tree was studied and it was proved that the translation-invariant Gibbs measure for this model is unique. <...> In [12] the fertile three-state HC-models corresponding to graphs "the hinge", "the pipe", "the wand", "the key" and four-state HC-models corresponding to graphs "the stick", "the key", "the gun" are introduce. <...> In [10] translation-invariant and periodic Gibbs measures for HC-model in the case "the key" on a Cayley tree is studied and it was proved that the translation-invariant measure is unique for any positive activity λ. <...> In this paper we consider fertile four states HC-models corresponding to graphs "the stick", "the key" and "the gun" on a Cayley tree of order two. <...> In each case it is proved that the translation-invariant Gibbs measure is unique. ∗rustam-7102@rambler.ru Siberian Federal University. <...> All rights reserved c – 165 – Rustam M.Khakimov The Uniqueness of the Translation-invariant Gibbs Measure for Four State . 1. <...> Definitions and known facts The Cayley tree ℑk of order k 1 is an infinite tree, i.e., a graph without cycles, such that exactly k +1 edges originate from each vertex. <...> The set of all configurations in V denote by Ω. <...> A vertex y is called a direct descendant of x if y > x and x, y are neighbors. <...> We let S(x) denote the set of direct descendants of x. <...> We note that in ℑk, any vertex x = x0 has k direct <...>