Gumbel Laws in the Symmetric Exclusion Process

Michael Conroy, Sunder Sethuraman

Research output: Contribution to journalArticlepeer-review


We consider the symmetric exclusion particle system on Z starting from an infinite particle step configuration in which there are no particles to the right of a maximal one. We show that the scaled position Xt/ (σbt) - at of the right-most particle at time t converges to a Gumbel limit law, where bt=t/logt , at=log(t/(2πlogt)) , and σ is the standard deviation of the random walk jump probabilities. This work solves a problem left open in Arratia (Ann Probab 11(2):362–373, 1983). Moreover, to investigate the influence of the mass of particles behind the leading one, we consider initial profiles consisting of a block of L particles, where L→ ∞ as t→ ∞ . Gumbel limit laws, under appropriate scaling, are obtained for Xt when L diverges in t. In particular, there is a transition when L is of order bt , above which the displacement of Xt is the same as that under an infinite particle step profile, and below which it is of order tlogL . Proofs are based on recently developed negative dependence properties of the symmetric exclusion system. Remarks are also made on the behavior of the right-most particle starting from a step profile in asymmetric nearest-neighbor exclusion, which complement known results.

Original languageEnglish (US)
Pages (from-to)723-764
Number of pages42
JournalCommunications in Mathematical Physics
Issue number1
StatePublished - Aug 2023

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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