Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Statistical physics on a product of trees

Tom Hutchcroft

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $G$ be the product of finitely many trees $T_{1}\times T_{2}\times\cdots\times T_{N}$, each of which is regular with degree at least three. We consider Bernoulli bond percolation and the Ising model on this graph, giving a short proof that the model undergoes a second order phase transition with mean-field critical exponents in each case. The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new.

We also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function along a random walk, as well as some generalizations of this lemma.

Résumé

Soit $G$ le produit d’un nombre fini d’arbres $T_{1}\times T_{2}\times\cdots\times T_{N}$ ayant chacun un degré constant supérieur à trois, nous étudions la percolation de Bernoulli et le modèle d’Ising sur ce graphe et présentons une preuve simple de l’existence d’une transition de phase de second ordre ayant les mêmes exposants critiques que le modèle en champs moyen. Le résultat pour la percolation est une preuve alternative d’un résultat de Kozma (2013), tandis que le résultat pour le modèle d’Ising est nouveau.

Nous présentons également une nouvelle preuve, reposant sur des techniques similaires, du lemme de Schramm concernant la vitesse de décroissance de la fonction à deux points le long de la marche aléatoire, ainsi que des généralisations de ce lemme.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 1001-1010.

Dates
Received: 13 December 2017
Revised: 17 March 2018
Accepted: 13 April 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820839

Digital Object Identifier
doi:10.1214/18-AIHP906

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60B99: None of the above, but in this section

Keywords
Percolation Ising model Triangle condition Bubble diagram Nonuniqueness Nonamenable groups Mean-field Nonunimodular

Citation

Hutchcroft, Tom. Statistical physics on a product of trees. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 1001--1010. doi:10.1214/18-AIHP906. https://projecteuclid.org/euclid.aihp/1557820839


Export citation

References

  • [1] M. Aizenman. Geometric analysis of $\varphi^{4}$ fields and Ising models. I, II. Comm. Math. Phys. 86 (1) (1982) 1–48.
  • [2] M. Aizenman and D. J. Barsky. Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 (3) (1987) 489–526.
  • [3] M. Aizenman, D. J. Barsky and R. Fernández. The phase transition in a general class of Ising-type models is sharp. J. Stat. Phys. 47 (3–4) (1987) 343–374.
  • [4] M. Aizenman, H. Duminil-Copin and V. Sidoravicius. Random currents and continuity of Ising model’s spontaneous magnetization. Comm. Math. Phys. 334 (2) (2015) 719–742.
  • [5] M. Aizenman and R. Fernández. On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44 (3–4) (1986) 393–454.
  • [6] M. Aizenman and C. M. Newman. Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys. 36 (1–2) (1984) 107–143.
  • [7] D. Aldous and R. Lyons. Processes on unimodular random networks. Electron. J. Probab. 12 (54) (2007) 1454–1508.
  • [8] D. J. Barsky and M. Aizenman. Percolation critical exponents under the triangle condition. Ann. Probab. 19 (4) (1991) 1520–1536.
  • [9] I. Benjamini, R. Lyons, Y. Peres and O. Schramm. Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab. 27 (3) (1999) 1347–1356.
  • [10] H. Duminil-Copin. Lectures on the Ising and Potts Models on the Hypercubic Lattice, 2017.
  • [11] H. Duminil-Copin, A. Raoufi and V. Tassion. Exponential decay of connection probabilities for subcritical Voronoi percolation in $\mathbb{R}^{d}$. 2017.
  • [12] H. Duminil-Copin, A. Raoufi and V. Tassion. Sharp phase transition for the random-cluster and Potts models via decision trees. 2017.
  • [13] H. Duminil-Copin and V. Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Comm. Math. Phys. (2015) 1–21.
  • [14] M. Fekete. Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17 (1) (1923) 228–249.
  • [15] G. R. Grimmett. Percolation (grundlehren der mathematischen wissenschaften). 2010.
  • [16] M. Heydenreich and R. van der Hofstad. Progress in high-dimensional percolation and random graphs.
  • [17] T. Hutchcroft. Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. C. R., Math. 354 (9) (2016) 944–947.
  • [18] T. Hutchcroft. Non-uniqueness and mean-field criticality for percolation on nonunimodular transitive graphs. 2017.
  • [19] T. Hutchcroft. Self-avoiding walk on nonunimodular transitive graphs. 2017.
  • [20] G. Kozma. Percolation on a product of two trees. Ann. Probab. 39 (5) (2011) 1864–1895.
  • [21] R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge University Press, New York, 2016. Available at http://pages.iu.edu/~rdlyons/.
  • [22] L. Onsager. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev. 2 (65) (1944) 117–149.
  • [23] Y. Peres. Percolation on nonamenable products at the uniqueness threshold. Ann. Inst. Henri Poincaré B, Probab. Stat. 36 (3) (2000) 395–406.
  • [24] A. Raoufi. A note on continuity of magnetization at criticality for the ferromagnetic Ising model on amenable quasi-transitive graphs with exponential growth. 2016.
  • [25] R. H. Schonmann. Multiplicity of phase transitions and mean-field criticality on highly non-amenable graphs. Comm. Math. Phys. 219 (2) (2001) 271–322.
  • [26] A. D. Sokal. A rigorous inequality for the specific heat of an Ising or $\varphi^{4}$ ferromagnet. Phys. Lett. A 71 (5–6) (1979) 451–453.
  • [27] C. N. Yang. The spontaneous magnetization of a two-dimensional Ising model. Phys. Rev. 2 (85) (1952) 808–816.