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Durham University

Department of Mathematical Sciences

Academic Staff

Publication details for Mikhail Menshikov

Hryniv, Ostap, Menshikov, Mikhail V. & Wade, Andrew R. (2013). Random walk in mixed random environment without uniform ellipticity. Proceedings of the Steklov Institute of Mathematics 282(1): 106-123.

Author(s) from Durham


We study a random walk in random environment on ℤ+. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i) points endowed with probabilities drawn from a symmetric distribution with heavy tails at 0 and 1, and (ii) “fast points” with a fixed systematic drift. Without these fast points, the model is related to the diffusion in heavy-tailed (“stable”) random potential studied by Schumacher and Singh; the fast points perturb that model. The two components compete to determine the behaviour of the random walk; we identify phase transitions in terms of the model parameters. We give conditions for recurrence and transience and prove almost sure bounds for the trajectories of the walk.