# Mathematics Colloquia and Seminars

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### A Disordered Growth Model

**Mathematical Physics & Probability**

Speaker: | Janko Gravner, UC Davis |

Location: | 693 Kerr |

Start time: | Mon, Nov 5 2001, 3:10PM |

The talk will be an overview of recent work with Craig Tracy and Harold Widom. We study what is arguably the simplest nontrivial model of disordered interface growth in two dimensions. The interface is given by a height function on the sites of the one--dimensional integer lattice and grows in discrete time: (1) the height above the site x adopts the height above the site to its left if the latter height is larger, (2) otherwise, the height above x increases by 1 with probability p_x. We assume that p_x are chosen independently at random with a common distribution F, and that the initial state is such that the origin is far above the other sites. If the tails of F are sufficiently thin, the growing surface divides into three regimes. The simplest one is deterministic, with no fluctuations. In the pure regime, the quenched fluctuations scale as the third root of time and approach a Tracy-Widom distribution. In the composite regime, the quenched fluctuation scale as the square root of time, and satisfy a central limit theorem. The annealed fluctuations can be determined as well.

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