The infinite valley for a recurrent random walk in random environment

Nina Gantert, Yuval Peres, Zhan Shi

Research output: Contribution to journalArticlepeer-review

11 Scopus citations


We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov, see Comm. Math. Phys. 92 (1984) 491-506. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.

Original languageEnglish
Pages (from-to)525-536
Number of pages12
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number2
StatePublished - May 2010
Externally publishedYes


  • Empirical distribution
  • Local time
  • Random walk in random environment
  • Self-intersection local time


Dive into the research topics of 'The infinite valley for a recurrent random walk in random environment'. Together they form a unique fingerprint.

Cite this