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

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

2003-07-09

arxiv.org/abs/cond-mat/0307189v3

Phys. Rev. E vol-69, 011103 (2004)

Physics

Condensed Matter

Statistical Mechanics

5 pages Revtex, 3 .eps figures included, new references added

Scientific paper

10.1103/PhysRevE.69.011103

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.

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