The Annals of Applied Probability

Multi-scale Lipschitz percolation of increasing events for Poisson random walks

Peter Gracar and Alexandre Stauffer

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Abstract

Consider the graph induced by $\mathbb{Z}^{d}$, equipped with uniformly elliptic random conductances. At time $0$, place a Poisson point process of particles on $\mathbb{Z}^{d}$ and let them perform independent simple random walks. Tessellate the graph into cubes indexed by $i\in\mathbb{Z}^{d}$ and tessellate time into intervals indexed by $\tau$. Given a local event $E(i,\tau)$ that depends only on the particles inside the space time region given by the cube $i$ and the time interval $\tau$, we prove the existence of a Lipschitz connected surface of cells $(i,\tau)$ that separates the origin from infinity on which $E(i,\tau)$ holds. This gives a directly applicable and robust framework for proving results in this setting that need a multi-scale argument. For example, this allows us to prove that an infection spreads with positive speed among the particles.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 1 (2019), 376-433.

Dates
Received: July 2017
Revised: May 2018
First available in Project Euclid: 5 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1544000432

Digital Object Identifier
doi:10.1214/18-AAP1420

Mathematical Reviews number (MathSciNet)
MR3910007

Zentralblatt MATH identifier
07039128

Subjects
Primary: 60G55: Point processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Multi-scale percolation Lipschitz surface spread of infection

Citation

Gracar, Peter; Stauffer, Alexandre. Multi-scale Lipschitz percolation of increasing events for Poisson random walks. Ann. Appl. Probab. 29 (2019), no. 1, 376--433. doi:10.1214/18-AAP1420. https://projecteuclid.org/euclid.aoap/1544000432


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References

  • [1] Barlow, M. T. (2004). Random walks on supercritical percolation clusters. Ann. Probab. 32 3024–3084.
  • [2] Barlow, M. T. and Hambly, B. M. (2009). Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Probab. 14 1–27.
  • [3] Candellero, E. and Teixeira, A. (2015). Percolation and isoperimetry on transitive graphs. Available at arXiv:1507.07765.
  • [4] Dirr, N., Dondl, P. W., Grimmett, G. R., Holroyd, A. E. and Scheutzow, M. (2010). Lipschitz percolation. Electron. Commun. Probab. 15 14–21.
  • [5] Gracar, P. and Stauffer, A. (2017). Random walks in random conductances: Decoupling and spread of infection. Available at arXiv:1701.08021.
  • [6] Grimmett, G. R. and Holroyd, A. E. (2012). Geometry of Lipschitz percolation. Ann. Inst. Henri Poincaré Probab. Stat. 48 309–326.
  • [7] Kesten, H. and Sidoravicius, V. (2005). The spread of a rumor or infection in a moving population. Ann. Probab. 33 2402–2462.
  • [8] Kesten, H. and Sidoravicius, V. (2006). A phase transition in a model for the spread of an infection. Illinois J. Math. 50 547–634.
  • [9] Peres, Y., Sinclair, A., Sousi, P. and Stauffer, A. (2013). Mobile geometric graphs: Detection, coverage and percolation. Probab. Theory Related Fields 156 273–305.
  • [10] Sidoravicius, V. and Stauffer, A. (2016). Multi-particle diffusion limited aggregation. Available at arXiv:1603.03218.
  • [11] Sidoravicius, V. and Sznitman, A.-S. (2009). Percolation for the vacant set of random interlacements. Comm. Pure Appl. Math. 62 831–858.
  • [12] Sidoravicius, V. and Teixeira, A. (2017). Absorbing-state transition for stochastic sandpiles and activated random walks. Electron. J. Probab. 22 Paper No. 33, 35.
  • [13] Stauffer, A. (2015). Space-time percolation and detection by mobile nodes. Ann. Appl. Probab. 25 2416–2461.
  • [14] Sznitman, A.-S. (2012). Decoupling inequalities and interlacement percolation on $G\times\mathbb{Z}$. Invent. Math. 187 645–706.
  • [15] Teixeira, A. (2016). Percolation and local isoperimetric inequalities. Probab. Theory Related Fields 165 963–984.