Speaker: Manjunath Krishnapur, IISc
January 5, 2024 04:00 PM
- 05:00 PM
Abstract:
Label the vertices of Z2 by i.i.d. Exponential random variables. For a path in Z2, its weight is the sum of the exponentials along the path. The last passage time Tn is the maximum weight among all oriented paths (those with exactly 2n steps) in Z2 that connect (0, 0) and (n, n). It is a well-known theorem of Johansson that Qn = (Tn − 4n)/n1/3 converges in distribution to the Tracy-Widom distribution with parameter 2. The LIL problem asks for the limsup and liminf behaviour of Qn.
In earlier works of Ledoux and of Basu-Ganguly-Hegde-K., it was shown that the limsup of Qn/(log log n)2/3 and liminf of Qn/(log log n)1/3 are finite constants. In joint work with Jnaneshwar Baslingker, Riddhipratim Basu and Sudeshna Bhattacharjee we show that these limiting constants agree with what was expected the tails of the Tracy-Widom distribution. In the lecture, we give an overview of this and related results and give a sketch of the proof.