Optimal tail estimates for directed last passage site percolation with geometric random variables. (English) Zbl 1016.15022

Baik, Jinho; Deift, Percy; McLaughlin, Kenneth T.-R.; Miller, Peter; Zhou, Xin

Adv. Theor. Math. Phys. 5, No. 6, 1207-1250 (2001).

Summary: 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 K. Johansson [Commun. Math. Phys. 209, No. 2, 437-476 (2000; Zbl 0969.15008)]. 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.

MSC:

15B52	Random matrices (algebraic aspects)
82B43	Percolation
60K35	Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

random matrices; optimal uniform lower tail estimates; probability distribution; last passage site percolation model; convergence of moments; universal scaling law