Journal of Statistical Physics

, Volume 177, Issue 5, pp 1009–1021 | Cite as

Criticality of Measures on 2-d Ising Configurations: From Square to Hexagonal Graphs

  • Valentina Apollonio
  • Roberto D’Autilia
  • Benedetto ScoppolaEmail author
  • Elisabetta Scoppola
  • Alessio Troiani


On the space of Ising configurations on the 2-d square lattice, we consider a family of non Gibbsian measures introduced by using a pair Hamiltonian, depending on an additional inertial parameter q. These measures are related to the usual Gibbs measure on \({\mathbb Z}^2\) and turn out to be the marginal of the Gibbs measure of a suitable Ising model on the hexagonal lattice. The inertial parameter q tunes the geometry of the system. The critical behaviour and the decay of correlation functions of these measures are studied thanks to relation with the Random Cluster model. This measure turns out to be interesting also because it is the stationary measure of a class of Probabilistic Cellular Automata (PCA). Such PCA can be used to obtain a fast sample of the Ising measures on 2-d lattices.


Ising model Random cluster model Phase transitions Correlation functions 



We are grateful to Hugo Duminil–Copin for useful and interesting discussions. B.S. acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. E.S. has been supported by the PRIN 20155PAWZB “Large Scale Random Structures”. A.T. has been supported by Project FARE 2016 Grant R16TZYMEHN. B.S. and E.S. thank the support of the A*MIDEX Project (n. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR).


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Authors and Affiliations

  1. 1.Dipartimento di Matematica e FisicaUniversità Roma TreRomeItaly
  2. 2.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomeItaly
  3. 3.Dipartimento di Matematica “Tullio Levi–Civita”Università di PadovaPaduaItaly

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