Limit theorems for a random directed graph
Vortrag von Prof. Dr. Sergey Foss
Datum: 10.04.13 Zeit: 17.15 - 19.00 Raum: Y27H25
We consider a stochastic directed graph on the integers whereby a directed edge between i and a larger integer j exists with probability p that may depend on the distance j-i, and there is no edges from bigger to smaller integers. Edge lengths L(i,j) may be constants or i.i.d. random variables. We introduce also a complementary "infinite bin" model. We study the asymptotics for the maximal path length in a long chunk of the graph. Under certain assumptions, the model has a regenerative structure, and the SLLN and the CLT follow. Otherwise, we obtain scaling laws and asymptotic distributions expressed in terms of a "continuous last-passage percolation" model on [0,1]. If time allows, we introduce multi-dimensional extensions of the models. The talk is based on joint papers with T. Konstantopoulos (2003, MPRF), D. Denisov and T. Konstantopoulos (2012, AnnAP), J. Martin and Ph. Schmidt (AnnAP, to appear) and S. Zachary (Adv/JAP, to appear).