TY - JOUR

T1 - Gumbel Laws in the Symmetric Exclusion Process

AU - Conroy, Michael

AU - Sethuraman, Sunder

N1 - Funding Information: The authors were supported partially by grant ARO W911NF-181-0311. Publisher Copyright: © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/8

Y1 - 2023/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85160450881&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85160450881&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s00220-023-04746-1

DO - https://doi.org/10.1007/s00220-023-04746-1

M3 - Article

SN - 0010-3616

VL - 402

SP - 723

EP - 764

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -