## Optimal Tail Estimates for Directed Last Passage Site Percolation with Geometric Random Variables

**J. Baik Department of Mathematics, Princeton University
**

**P. Deift Courant Institute of Mathematical Sciences,
New York University
**

**K. T.-R. McLaughlin
Department of Mathematics, University of North
Carolina, Chapel Hill
**

**P. D. Miller Department of Mathematics, University of Michigan,
Ann Arbor
**

**X. Zhou Department of Mathematics, Duke University
**

### Abstract:

In this paper, we obtain optimal uniform lower tail estimates for the probability distribution of the properly scaled length of the longest up/right path of the last passage site percolation model considered by Johansson in [12]. The estimates are used to prove a lower tail moderate deviation result for the model. The estimates also imply the convergence of moments, and also provide a verification of the universal scaling law relating the longitudinal and the transversal fluctuations of the model.