An Anisotropic Ballistic Deposition Model with Links to the Ulam Problem and the Tracy-Widom Distribution

Satya Majumdar ¹, Sergei K. Nechaev ^2, 3

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 69 (2004) 011103

We compute exactly the asymptotic distribution of scaled height in a (1+1)–dimensional anisotropic ballistic deposition model by mapping it to the Ulam problem of finding the longest nondecreasing subsequence in a random sequence of integers. Using the known results for the Ulam problem, we show that the scaled height in our model has the Tracy-Widom distribution appearing in the theory of random matrices near the edges of the spectrum. Our result supports the hypothesis that various growth models in $(1+1)$ dimensions that belong to the Kardar-Parisi-Zhang universality class perhaps all share the same universal Tracy-Widom distribution for the suitably scaled height variables.

• 1. Laboratoire de Physique Théorique – IRSAMC (LPT),
  CNRS : UMR5152 – Université Paul Sabatier – Toulouse III
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
  CNRS : UMR8626 – Université Paris XI – Paris Sud
• 3. Landau Institute for Theoretical Physics,
  Landau Institute for Theoretical Physics