The Phase Transition in Statistical Models Defined on Farey Fractions

TL;DR

This paper studies statistical models on Farey fractions, revealing they share the same free energy and exhibit a second-order phase transition characterized by a specific heat divergence, with implications for spin chains and multifractals.

Contribution

It demonstrates the equivalence of free energy among different models on Farey fractions and characterizes their phase transition using spectral analysis of the transfer operator.

Findings

Models have identical free energy.

Second-order phase transition with specific heat divergence.

Discontinuity in magnetization at transition.

Abstract

We consider several statistical models defined on the Farey fractions. Two of these models may be regarded as "spin chains", with long-range interactions, while another arises in the study of multifractals associated with chaotic maps exhibiting intermittency. We prove that these models all have the same free energy. Their thermodynamic behavior is determined by the spectrum of the transfer operator (Ruelle-Perron-Frobenius operator), which is defined using the maps (presentation functions) generating the Farey "tree". The spectrum of this operator was completely determined by Prellberg. It follows that these models have a second-order phase transition with a specific heat divergence of the form [t (ln t)^2]^(-1). The spin chain models are also rigorously known to have a discontinuity in the magnetization at the phase transition.