A Curie-Weiss Model of Self-Organized Criticality : The Gaussian Case

2013/06/06 by Matthias Gorny, Gorny, Matthias
Mathematics · Physics and Astronomy · #60F05 60K35 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60F05 #msc:60K35

paper · pdf · doi:10.48550/arxiv.1306.1561

The Gaussian case, initially in the first version of arXiv:1301.6911, removed from the second version and presented here

arxiv created 2013/06/06 · arxiv updated 2013/06/10

Abstract

We try to design a simple model exhibiting self-organized criticality, which is amenable to a rigorous mathematical analysis. To this end, we modify the generalized Ising Curie-Weiss model by implementing an automatic control of the inverse temperature. With the help of exact computations, we show that, in the case of a centered Gaussian measure with positive variance σ², the sum S_n of the random variables has fluctuations of order n^3/4 and that S_n/n^3/4 converges to the distribution C exp(-x⁴/(4σ⁴)) dx where C is a suitable positive constant.

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