We derive probabilistic limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume-Emery-Griffiths model [4]. These probabilistic limit theorems consist of scaling limits for the total spin and moderate deviation principles (MDPs) for the total spin. The model under study is defined by a probability distribution that depends on the parameter...

J.P Emery, A.W. Harris, M. Kaasalainen, and J. Berthier, SETI Institute, Carl Sagan Center, 515 N. Whisman Road, Mountain view CA 94043, USA, [email protected]. UC-Berkeley, Dept of Astronomy, 601 Campbell Hall, Berkeley CA 94720, USA. Institut de Mécanique Céleste et de Calcul des Éphémérides, Observatoire de Paris, France, [email protected]. Astronomical Institute, Charles University, Prague,...

We use the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. This function can be used to recover the Ricci tensor on smooth Riemannian manifolds by the formula Ric(γ′ (0) , γ′ (0)) = 1 2 d ds2 Ric4g (x, γ (s)) for any curve γ(s).

In this paper, we develop a general theory for the estimation of the transition probabilities of reversible Markov chains using the maximum entropy principle. A broad range of physical models can be studied within this approach. We use one-dimensional classical spin systems to illustrate the theoretical ideas. The examples studied in this paper are: the Ising model, the Potts model and the Blum...