Electronic Journal of Probability

Asymptotics for Lipschitz percolation above tilted planes

Alexander Drewitz, Michael Scheutzow, and Maite Wilke-Berenguer

Full-text: Open access

Abstract

We consider  Lipschitz percolation in $d+1$ dimensions above planes tilted by an angle $\gamma$ along one or several coordinate axes. In particular, we are interested in the asymptotics of the critical probability as $d \to \infty$ as well as $\gamma \uparrow \pi/4.$ Our principal results show that the convergence of the critical probability to 1 is polynomial as $d\to \infty$ and $\gamma \uparrow \pi/4.$ In addition, we identify the correct order of this polynomial convergence and in $d=1$ we also obtain the correct prefactor.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 117, 23 pp.

Dates
Received: 21 April 2015
Accepted: 1 November 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067223

Digital Object Identifier
doi:10.1214/EJP.v20-4251

Mathematical Reviews number (MathSciNet)
MR3425537

Zentralblatt MATH identifier
1328.60212

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B43: Percolation [See also 60K35]

Keywords
Lipschitz percolation random surface $\rho$-percolation

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Drewitz, Alexander; Scheutzow, Michael; Wilke-Berenguer, Maite. Asymptotics for Lipschitz percolation above tilted planes. Electron. J. Probab. 20 (2015), paper no. 117, 23 pp. doi:10.1214/EJP.v20-4251. https://projecteuclid.org/euclid.ejp/1465067223


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