Journal of Statistical Physics

, Volume 102, Issue 5–6, pp 1085–1132 | Cite as

Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models

  • Janko Gravner
  • Craig A. Tracy
  • Harold Widom
Article

Abstract

We introduce a class of one-dimensional discrete space-discrete time stochastic growth models described by a height function ht(x) with corner initialization. We prove, with one exception, that the limiting distribution function of ht(x) (suitably centered and normalized) equals a Fredholm determinant previously encountered in random matrix theory. In particular, in the universal regime of large x and large t the limiting distribution is the Fredholm determinant with Airy kernel. In the exceptional case, called the critical regime, the limiting distribution seems not to have previously occurred. The proofs use the dual RSK algorithm, Gessel's theorem, the Borodin–Okounkov identity and a novel, rigorous saddle point analysis. In the fixed x, large t regime, we find a Brownian motion representation. This model is equilvalent to the Seppäläinen–Johansson model. Hence some of our results are not new, but the proofs are.

growth processes shape fluctuations limit theorems digital boiling random matrix theory Airy kernel Painlevé II saddle point analysis invariance principle 

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Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • Janko Gravner
    • 1
  • Craig A. Tracy
    • 2
  • Harold Widom
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaDavis
  2. 2.Department of Mathematics, Institute of Theoretical DynamicsUniversity of CaliforniaDavis
  3. 3.Department of MathematicsUniversity of CaliforniaSanta Cruz

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