Electronic Journal of Probability

Almost exponential decay for the exit probability from slabs of ballistic RWRE

Enrique Guerra and Alejandro Ramirez

Full-text: Open access

Abstract

It is conjectured that in dimensions d ≥ 2 any random walk in an i.i.d. uniformly elliptic random environment (RWRE) which is directionally transient is ballistic. The ballisticity conditions for RWRE somehow interpolate between directional transience and ballisticity and have served to quantify the gap which would need to be proven in order to answer affirmatively this conjecture. Two important ballisticity conditions introduced by Sznitman in 2001 and 2002 are the so called conditions (T′) and (T): given a slab of width L orthogonal to l, condition (T′) in direction l is the requirement that the annealed exit probability of the walk through the side of the slab in the half-space ${x : x \cdot l < 0}$, decays faster than $e(−CL^\gamma)$ for all $\gamma\in (0, 1)$ and some constant C &gt; 0, while condition (T) in direction l is the requirement that the decay is exponential e(−CL). It is believed that (T′) implies (T). In this article we show that (T′) implies at least an almost (in a sense to be made precise) exponential decay.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 24, 17 pp.

Dates
Accepted: 5 March 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067130

Digital Object Identifier
doi:10.1214/EJP.v20-3655

Mathematical Reviews number (MathSciNet)
MR3325093

Zentralblatt MATH identifier
1328.60225

Subjects
Primary: 60K37: Processes in random environments
Secondary: 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Random walk in random environment ballisticity conditions effective criterion

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

Citation

Guerra, Enrique; Ramirez, Alejandro. Almost exponential decay for the exit probability from slabs of ballistic RWRE. Electron. J. Probab. 20 (2015), paper no. 24, 17 pp. doi:10.1214/EJP.v20-3655. https://projecteuclid.org/euclid.ejp/1465067130


Export citation

References

  • Berger, Noam; Drewitz, Alexander; Ramirez, Alejandro F. Effective polynomial ballisticity conditions for random walk in random environment. Comm. Pure Appl. Math. 67 (2014), no. 12, 1947–1973.
  • Campos, David; Ramirez, Alejandro F. Ellipticity criteria for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 160 (2014), no. 1-2, 189–251.
  • Drewitz, A.; Ramirez, A. F. Ballisticity conditions for random walk in random environment. Probab. Theory Related Fields 150 (2011), no. 1-2, 61–75.
  • Drewitz, Alexander; Ramirez, Alejandro F. Quenched exit estimates and ballisticity conditions for higher-dimensional random walk in random environment. Ann. Probab. 40 (2012), no. 2, 459–534.
  • Smith, Walter L.; Wilkinson, William E. On branching processes in random environments. Ann. Math. Statist. 40 1969 814–827.
  • Solomon, Fred. Random walks in a random environment. Ann. Probability 3 (1975), 1–31.
  • Sznitman, Alain-Sol. On a class of transient random walks in random environment. Ann. Probab. 29 (2001), no. 2, 724–765.
  • Sznitman, Alain-Sol. An effective criterion for ballistic behavior of random walks in random environment. Probab. Theory Related Fields 122 (2002), no. 4, 509–544.