Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-24T13:17:35.594Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential decay for constrained-degree percolation

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  03 December 2024

Diogo C. dos Santos  and
Roger W. C. Silva
Diogo C. dos Santos*
Affiliation:
Universidade Federal de Alagoas
Roger W. C. Silva*
Affiliation:
Universidade Federal de Minas Gerais
*
*Postal address: Instituto de Matemática, Universidade Federal de Alagoas, Av. Lourival Melo Mota, CEP 57072-900, Maceió, AL, Brazil. Email: [email protected]
**Postal address: Departamento de Estatística, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, CEP 30123-970, Belo Horizonte, MG, Brazil. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the constrained-degree percolation model in a random environment (CDPRE) on the square lattice. In this model, each vertex v has an independent random constraint $\kappa_v$ which takes the value $j\in \{0,1,2,3\}$ with probability $\rho_j$. The dynamics is as follows: at time $t=0$ all edges are closed; each edge e attempts to open at a random time $U(e)\sim \mathrm{U}(0,1]$, independently of all the other edges. It succeeds if at time U(e) both its end vertices have degrees strictly smaller than their respective constraints. We obtain exponential decay of the radius of the open cluster of the origin at all times when its expected size is finite. Since CDPRE is dominated by Bernoulli percolation, this result is meaningful only if the supremum of all values of t for which the expected size of the open cluster of the origin is finite is larger than $\frac12$. We prove this last fact by showing a sharp phase transition for an intermediate model.

Keywords

Dependent percolationrandom environmentinfinite range

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation 82B26: Phase transitions (general)
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 2 , June 2025 , pp. 795 - 807
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108, 489–526.CrossRefGoogle Scholar
Aizenman, M. and Grimmett, G. R. (1991). Strict monotonicity for critical points in percolation and ferromagnetic models. J. Statist. Phys. 63, 817–835.CrossRefGoogle Scholar
Amir, G. and Baldasso, R. (2020). Percolation in majority dynamics. Electron. J. Prob. 25, 1–18.CrossRefGoogle Scholar
Balister, P., Bollobás, B. and Riordan, O. Essential enhancements revisited. Preprint, arxiv:1402.0834v1.Google Scholar
de Lima, B. N. B, Sanchis, R., dos Santos, D. C., Sidoravicius, V. and Teodoro, R. (2020). The constrained-degree percolation model. Stoch. Process. Appl. 130, 5492–5509.CrossRefGoogle Scholar
Duminil-Copin, H. and Tassion, V. (2016). A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^d$ . Enseign. Math. 62, 199–206.CrossRefGoogle Scholar
Duminil-Copin, H., Raoufi, A. and Tassion, V. (2019). Exponential decay of connection probabilities for subcritical Voronoi percolation in $\mathbb{R}^d$ . Prob. Theory Relat. Fields 173, 479–490.CrossRefGoogle Scholar
Garet, O., Marchand, R. and Marcovici, I. (2018). Does Eulerian percolation on ${\mathbb{Z}}^2$ percolate? ALEA 15, 279–294.CrossRefGoogle Scholar
Grimmett, G. R. and Janson, S. (2010). Random graphs with forbidden vertex degrees. Random Structures Algorithms 37, 137–175.CrossRefGoogle Scholar
Grimmett, G. R. and Li, Z. (2017). The 1–2 model. Contemp. Math. 969, 139–152.CrossRefGoogle Scholar
Hartarsky, I. and de Lima, B. N. B. (2022). Weakly constrained-degree percolation on the hypercubic lattice. Stoch. Process. Appl. 153, 128–144.CrossRefGoogle Scholar
Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys. 74, 41–59.CrossRefGoogle Scholar
Li, Z. (2020). Constrained percolation, Ising model, and XOR Ising model on planar lattices. Random Structures Algorithms 57, 474–525.CrossRefGoogle Scholar
O’Donnell, R., Saks, M., Schramm, O. and Servedio, R. A. (2005). Every decision tree has an influential variable. In Proc. 46th Ann. IEEE Symp. Foundations of Computer Science. IEEE, Piscataway, NJ, pp. 31–39.CrossRefGoogle Scholar
Sanchis, R., dos Santos, D. C. and Silva, R. W. C. (2022). Constrained-degree percolation in random environment. Ann. Inst. H. Poincaré Prob. Statist. 58, 1887–1899.Google Scholar