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Probability Seminar: Zoe Huang, Duke
December 5, 2019 @ 4:15 pm - 5:15 pm
The contact process on Galton-Watson trees
Abstract: The contact process describes an epidemic model where each
infected individual recovers at rate 1 and infects its healthy neighbors
at rate $\lambda$. We show that for the contact process on Galton-Watson
trees, when the offspring distribution (i) is subexponential the
critical value for local survival $\lambda_2=0$ and (ii) when it is
Geometric($p$) we have $\lambda_2 \le C_p$, where the $C_p$ are much
smaller than previous estimates. This is based on an improved (and in a
sense sharp) understanding of the survival time of the contact process
on star graphs. Recently it is proved by Bhamidi, Nam, Nguyen and Sly
(2019) that when the offspring distribution of the Galton-Watson tree
has exponential tail, the first critical value $\lambda_1$ of the
contact process is strictly positive. We prove that if the contact
process survives then the number of infected sites grows exponentially
fast. As a consequence we show that the contact process dies out at the
critical value $\lambda_1$ and does not survive strongly at $\lambda_2$.
Based on joint work with Rick Durrett.
Refreshments will be served at 3:45 in the 3rd floor lounge of Hanes Hall