Large Deviations on a Cayley Tree I: Rate Functions

2015/12/27 by Anatoly E. Patrick, Patrick, Anatoly E.
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #cond-mat.stat-mech #math.PR

paper · pdf · doi:10.48550/arxiv.1512.08234

29 pages, 4 figures

arxiv created 2015/12/27 · arxiv updated 2015/12/29

Abstract

We study the spherical model of a ferromagnet on a Cayley tree and show that in the case of empty boundary conditions the ferromagnetic phase transition takes place at the critical temperature T_c=(6√(2))/(5)J, where J is the interaction strength. For any temperature the equilibrium magnetization, m_n, tends to zero in the thermodynamic limit, and the true order parameter is the renormalized magnetization r_n=n^3/2m_n, where n is the number of generations in the Cayley tree. Below T_c, the equilibrium values of the order parameter are given by ρ^* = ±\frac2π (√(2)-1)² √(1-(T)/(T_c)). There is one more notable temperature, T_{\rm p}, in the model. Below that temperature the influence of homogeneous boundary field penetrates throughout the tree. We call T_{\rm p} the penetration temperature, and it is given by T_{\rm p}= \fracJ W_{\rm Cayley} (3/2) (1-(1)/(√(2)) ( (h)/(2J) )² ). The main new technical result of the paper is a complete set of orthonormal eigenvectors for the discrete Laplace operator on a Cayley tree.

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