The Annals of Probability

Local time on the exceptional set of dynamical percolation and the incipient infinite cluster

Alan Hammond, Gábor Pete, and Oded Schramm

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Abstract

In dynamical critical site percolation on the triangular lattice or bond percolation on $\mathbb{Z}^{2}$, we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten’s incipient infinite cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is different. We believe that the two laws are mutually singular, but do not show this. We also study the collapse of the infinite cluster near typical exceptional times and establish a relation between static and dynamic exponents, analogous to Kesten’s near-critical relation.

Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 2949-3005.

Dates
Received: April 2013
Revised: June 2014
First available in Project Euclid: 11 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1449843624

Digital Object Identifier
doi:10.1214/14-AOP950

Mathematical Reviews number (MathSciNet)
MR3433575

Zentralblatt MATH identifier
1341.60128

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 82B43: Percolation [See also 60K35]
Secondary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Keywords
Dynamical percolation critical phenomena Palm measure

Citation

Hammond, Alan; Pete, Gábor; Schramm, Oded. Local time on the exceptional set of dynamical percolation and the incipient infinite cluster. Ann. Probab. 43 (2015), no. 6, 2949--3005. doi:10.1214/14-AOP950. https://projecteuclid.org/euclid.aop/1449843624


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References

  • [1] Ahlberg, D. (2013). The asymptotic shape, large deviations and dynamical stability in first-passage percolation on cones. Preprint. Available at arXiv:1107.2280.
  • [2] Angel, O., Goodman, J., den Hollander, F. and Slade, G. (2008). Invasion percolation on regular trees. Ann. Probab. 36 420–466.
  • [3] Benjamini, I., Häggström, O., Peres, Y. and Steif, J. E. (2003). Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31 1–34.
  • [4] Benjamini, I., Kalai, G. and Schramm, O. (1999). Noise sensitivity of Boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90 5–43 (2001).
  • [5] Benjamini, I. and Schramm, O. (1998). Exceptional planes of percolation. Probab. Theory Related Fields 111 551–564.
  • [6] Broman, E. I., Garban, C. and Steif, J. E. (2013). Exclusion sensitivity of Boolean functions. Probab. Theory Related Fields 155 621–663.
  • [7] Broman, E. I. and Steif, J. E. (2006). Dynamical stability of percolation for some interacting particle systems and $\varepsilon$-movability. Ann. Probab. 34 539–576.
  • [8] Camia, F. and Newman, C. M. (2007). Critical percolation exploration path and $\mathrm{SLE}_{6}$: A proof of convergence. Probab. Theory Related Fields 139 473–519.
  • [9] Damron, M., Sapozhnikov, A. and Vágvölgyi, B. (2009). Relations between invasion percolation and critical percolation in two dimensions. Ann. Probab. 37 2297–2331.
  • [10] Duminil-Copin, H., Garban, C. and Pete, G. (2014). The near-critical planar FK-Ising model. Comm. Math. Phys. 326 1–35.
  • [11] Fontes, L. R. G., Newman, C. M., Ravishankar, K. and Schertzer, E. (2009). Exceptional times for the dynamical discrete web. Stochastic Process. Appl. 119 2832–2858.
  • [12] Garban, C., Pete, G. and Schramm, O. (2013). The scaling limits of near-critical and dynamical percolation. Preprint. Available at arXiv:1305.5526.
  • [13] Garban, C., Pete, G. and Schramm, O. (2010). The Fourier spectrum of critical percolation. Acta Math. 205 19–104.
  • [14] Garban, C., Pete, G. and Schramm, O. (2013). Pivotal, cluster, and interface measures for critical planar percolation. J. Amer. Math. Soc. 26 939–1024.
  • [15] Garban, C. and Steif, J. E. (2012). Noise sensitivity and percolation. In Probability and Statistical Physics in Two and More Dimensions (D. Ellwood, C. Newman, V. Sidoravicius and W. Werner, eds.). Clay Math. Proc. 15 49–154. Amer. Math. Soc., Providence, RI.
  • [16] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [17] Häggström, O., Peres, Y. and Steif, J. E. (1997). Dynamical percolation. Ann. Inst. Henri Poincaré Probab. Stat. 33 497–528.
  • [18] Hammond, A., Mossel, E. and Pete, G. (2012). Exit time tails from pairwise decorrelation in hidden Markov chains, with applications to dynamical percolation. Electron. J. Probab. 17 no. 68, 16.
  • [19] Hara, T. and Slade, G. (2000). The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. J. Stat. Phys. 99 1075–1168.
  • [20] Hara, T. and Slade, G. (2000). The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phys. 41 1244–1293.
  • [21] Hoffman, C. (2006). Recurrence of simple random walk on $\mathbb{Z}^{2}$ is dynamically sensitive. ALEA Lat. Am. J. Probab. Math. Stat. 1 35–45.
  • [22] Járai, A. A. (2003). Incipient infinite percolation clusters in 2D. Ann. Probab. 31 444–485.
  • [23] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [24] Kesten, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369–394.
  • [25] Kesten, H. (1987). Scaling relations for $2$D-percolation. Comm. Math. Phys. 109 109–156.
  • [26] Kesten, H. (1987). A scaling relation at criticality for $2$D-percolation. In Percolation Theory and Ergodic Theory of Infinite Particle Systems (Minneapolis, Minn., 19841985). IMA Vol. Math. Appl. 8 203–212. Springer, New York.
  • [27] Khoshnevisan, D. (2008). Dynamical percolation on general trees. Probab. Theory Related Fields 140 169–193.
  • [28] Lawler, G. F., Schramm, O. and Werner, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7 no. 2, 13 pp. (electronic).
  • [29] Liggett, T. M. (2002). Tagged particle distributions or how to choose a head at random. In In and Out of Equilibrium (Mambucaba, 2000). Progress in Probability 51 133–162. Birkhäuser, Boston, MA.
  • [30] Liggett, T. M. (2005). Interacting Particle Systems. Springer, Berlin. Reprint of the 1985 original.
  • [31] Lyons, R. and Peres, Y. (2015). Probability on Trees and Networks. Book in preparation, Cambridge Univ. Press. Current version available at http://pages.iu.edu/~rdlyons/.
  • [32] Nolin, P. (2008). Near-critical percolation in two dimensions. Electron. J. Probab. 13 1562–1623.
  • [33] Peres, Y., Schramm, O. and Steif, J. E. (2009). Dynamical sensitivity of the infinite cluster in critical percolation. Ann. Inst. Henri Poincaré Probab. Stat. 45 491–514.
  • [34] Sapozhnikov, A. (2011). The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions. Electron. Commun. Probab. 16 775–780.
  • [35] Schramm, O. (2000). Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 221–288.
  • [36] Schramm, O. and Smirnov, S. (2011). On the scaling limits of planar percolation. Ann. Probab. 39 1768–1814. With an appendix by Christophe Garban.
  • [37] Schramm, O. and Steif, J. E. (2010). Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. (2) 171 619–672.
  • [38] Smirnov, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239–244.
  • [39] Smirnov, S. (2006). Towards conformal invariance of 2D lattice models. In International Congress of Mathematicians. Vol. II 1421–1451. Eur. Math. Soc., Zürich.
  • [40] Smirnov, S. and Werner, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729–744.
  • [41] Steif, J. E. (2009). A survey of dynamical percolation. In Fractal Geometry and Stochastics IV. Progress in Probability 61 145–174. Birkhäuser, Basel.
  • [42] Sznitman, A.-S. (2012). Topics in Occupation Times and Gaussian Free Fields. European Mathematical Society (EMS), Zürich.
  • [43] Werner, W. (2009). Lectures on two-dimensional critical percolation. In Statistical Mechanics. IAS/Park City Math. Ser. 16 297–360. Amer. Math. Soc., Providence, RI.