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Phase transitions for rates of convergence in the Blume-Emery-Griffiths model

  • We derive rates of convergence for limit theorems that reveal the intricate structure of the phase transitions in a mean-field version of the Blume-Emery-Griffith model. The theorems consist of scaling limits for the total spin. The model depends on the inverse temperature β and the interaction strength K. The rates of convergence results are obtained as (β,K) converges along appropriate sequences (βn,Kn) to points belonging to various subsets of the phase diagram which include a curve of second-order points and a tricritical point. We apply Stein's method for normal and non-normal approximation avoiding the use of transforms and supplying bounds, such as those of Berry-Esseen quality, on approximation error. We observe an additional phase transition phenomenon in the sense that depending on how fast Kn and βn are converging to points in various subsets of the phase diagram, different rates of convergences to one and the same limiting distribution occur.

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Document Type:Preprint
Author:Peter Eichelsbacher, Bastian Martschink
ArXiv Id:http://arxiv.org/abs/1304.2791
Date of first publication:2013/04/09
Departments, institutes and facilities:Fachbereich Elektrotechnik, Maschinenbau, Technikjournalismus
Dewey Decimal Classification (DDC):5 Naturwissenschaften und Mathematik / 51 Mathematik / 519 Wahrscheinlichkeiten, angewandte Mathematik
Entry in this database:2015/04/02