Mod-Gaussian Convergence and Its Applications for Models of Statistical Mechanics
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In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by Marc Yor. After recalling our recent result which interprets the limiting function as a measure of “breaking of symmetry” in the Gaussian approximation in the framework of general central limit theorems type results, we introduce the framework of L1-mod-Gaussian convergence in which the residue function is obtained as (up to a normalizing factor) the probability density of some sequences of random variables converging in law after a change of probability measure. In particular we recover some celebrated results due to Ellis and Newman on the convergence in law of dependent random variables arising in statistical mechanics. We complete our results by giving an alternative approach to the Stein method to obtain the rate of convergence in the Ellis-Newman convergence theorem and by proving a new local limit theorem. More generally we illustrate our results with simple models from statistical mechanics.
KeywordsLimit Theorem Ising Model Moment Generate Function Transfer Matrix Method Dependent Random Variable
The authors would like to thank M. Carston and C. Newman for fruitful discussions on the models coming from statistical mechanics; and P.-O. Dehaye and V. Féray for comments on the combinatorics of the cumulants of the one-dimensional Ising model. We would also like to thank the anonymous referee for his valuable comments, that have allowed us in particular to correct a false statement of Theorem 13 that appeared in an earlier version of this paper.
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