Zero-one law for directional transience of one-dimensional random walks in dynamic random environments. (English) Zbl 1338.60102

Orenshtein, Tal; Soares dos Santos, Renato

Electron. Commun. Probab. 21, Paper No. 15, 11 p. (2016).

Summary: We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.

MSC:

60F20	Zero-one laws
60K37	Processes in random environments
82B41	Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82C44	Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics

Keywords:

random walk; dynamic random environment; zero-one law; transience; recurrence