Limit Theorems for the Bipartite Potts Model
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Limit Theorems for the Bipartite Potts Model Qun Liu1 Received: 5 July 2020 / Accepted: 7 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We consider the bipartite Potts model in which interaction strengths depend only on the groups the particles belong to. We first rigorously compute the exact value of the free energy, then put forward two scaling limit theorems for the joint distribution of two empirical vectors which serve as order parameters. It turns out that the limit distribution can be Gaussian or the exponential distribution of higher order under different conditions. Keywords Spin glasses · Thermodynamic limits · Central limit theorems
1 Introduction As a mathematical model for ferromagnetism, the Ising model [1] has played a central role in statistical physics, and has applications far beyond the limitation of statistical physics. For example, to model social interactions such as political affinities, Banerjee et al. [2] proposed a method by using each spin to represent the vote of U.S. senators. Schneidman et al. [3] regarded the Ising model as an ideal model for neural activity. Since the corresponding Boltzmann-Gibbs measure is known to be the stationary distribution of the so called Glauber dynamics, Montanari and Saberi [4] used the Ising model to study the spread of information in social networks. As a special case of the Ising model, the Curie–Weiss model has attracted extensive attention since it can be interpreted as a spin system on complete graphs. By using some large deviation approaches, Ellis [5] constructs a beautiful probabilistic description for the thermodynamic limit of this model, and the analysis of the asymptotic behavior for the total magnetization was proposed early by Ellis and Newman [6]. In contrast to the Curie–Weiss model, the spins of the Curie–Weiss–Potts model can occupy more than two different states, so the Curie–Weiss model can be considered as a
Communicated by Aernout van Enter. The research of this paper is supported by the National Natural Science Foundation of China (Grant Nos. 11901275).
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Qun Liu [email protected] School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China
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much simpler special case of the Curie–Weiss–Potts model. In [7], Ellis and Wang provided some law of large numbers, as well as some central limit theorems for the empirical vectors in the Curie–Weiss–Potts model. These asymptotic results were utilized in [8] to obtain the limit theorems for the maximum likelihood estimators of the inverse temperature, as well as the external field. The complete analysis of the phase transitions for the Curie–Weiss–Potts model was studied by Costeniuc et al. [9]. To capture the large scale behavior of some socio economic systems, Contucci and Ghirlanda [10] purposed a multi-species extension of the Curie–Weiss model. This new particle system is also named bipartite mean field model since the interaction strength between different particles can be divided into two blo
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