Limit of the environment viewed from Sinaï's walk
Résumé
For Sinai's walk we show that the empirical measure of the environment seen from the particle converges in law to some random measure. This limit measure is explicitly given in terms of the infinite valley, which construction goes back to Golosov (1984). As a consequence an "in law" ergodic theorem holds. When the limit is deterministic, it holds in probability. This allows some extensions to the recurrent case of the ballistic "environment's method" dating back to Kozlov and Molchanov (1984).
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