Electronic Journal of Probability

Asymptotics for Lipschitz percolation above tilted planes

Alexander Drewitz, Michael Scheutzow, and Maite Wilke-Berenguer

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Lipschitz percolation random surface $\rho$-percolation

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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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