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Phase transitions in long-range Ising models and an optimal condition for factors of $g$-measures

Published online by Cambridge University Press:  07 September 2017

ANDERS JOHANSSON ,
ANDERS ÖBERG  and
MARK POLLICOTT
ANDERS JOHANSSON
Affiliation:
Department of Mathematics, University of Gävle, 801 76 Gävle, Sweden email ajj@hig.se
ANDERS ÖBERG
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, 751 06 Uppsala, Sweden email anders@math.uu.se
MARK POLLICOTT
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK email mpollic@maths.warwick.ac.uk
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Abstract

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$-measures. Indag. Math. (N.S.)22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1317 - 1330
Copyright
© Cambridge University Press, 2017 

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