Section 55
Chapter 54,969

Phase transitions in number theory: from the birthday problem to Sidon sets

Luque, B.; Torre, Ián.G.; Lacasa, L.

Physical Review. E Statistical Nonlinear and Soft Matter Physics 88(5): 052119


ISSN/ISBN: 1539-3755
PMID: 24329226
DOI: 10.1103/physreve.88.052119
Accession: 054968853

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In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed.

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