Description
Mathematics Colloquium.
Speaker: Parthanil Roy.
Affiliation: ISI Bangalore.
Date and Time: Wednesday 11 September, 4:30 pm - 5:30 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: How to tell a tale of two tails?
Abstract: We study the extremes of branching random walks under the
assumption that underlying Galton-Watson tree has infinite progeny mean.
It is assumed that the displacements are either regularly varying or have
lighter tails. In the regularly varying case, it is shown that the point
process sequence of normalized extremes converges to a Poisson random
measure. In the lighter-tailed case, however, the behaviour is much more
subtle, and the scaling of the position of the rightmost particle in the
n-th generation depends on the family of stepsize distribution, not just
its parameter(s). In all of these cases, we discuss the convergence in
probability of the scaled maxima sequence. Our results and methodology are
applied to study the almost sure convergence in the context of cloud speed
for branching random walks with infinite progeny mean. The exact cloud
speed constants are calculated for regularly varying displacements and
also for stepsize distributions having a nice exponential decay.
This talk is based on a joint work with Souvik Ray (Stanford University),
Rajat Subhra Hazra (ISI Kolkata) and Philippe Soulier (Univ of Paris
Nanterre). We will first review the literature (mainly, the PhD thesis
work of Ayan Bhattacharya) and then talk about the current work. Special
care will be taken so that a significant portion of the talk remains
accessible to everyone.