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Kempner Colloquium
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\begin{Huge}\textsc{The Contact Process on Power Law \\ \vskip .1in Random Graphs}
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\begin{Large}\textbf{Daniel Valesin}
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(UBC)
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The contact process on a graph $G$ is an interacting particle system that
models the spread of an infection in a population. Every infected node
transmits the infection to each neighbour with rate $\lambda > 0$ and recovers from the infection with rate one. If $G$ is the
$d$-dimensional integer lattice, the process exhibits a phase transition as
$\lambda$ varies: if $\lambda$ is smaller than a certain threshold, then the
infection disappears quickly, but if it is larger, then there is a chance
for a massive epidemic. In this talk we consider the case when $G$ is a
uniformly chosen random graph on $n$ vertices with a fixed heavy-tailed
degree distribution. It is a result due to Chatterjee and Durrett (2009)
that on such graphs, for any positive value of $\lambda$, the epidemic
prevails for a long time. We study both the amount of time for the epidemic
to halt and the typical density $\rho(\lambda)$ of infected vertices at
times when the process is still active. We show that the small-$\lambda$
behaviour of $\rho(\lambda)$ depends sensitively on the exponent $a$ of the
power law of the degree distribution of $G$: the decay of $\rho(\lambda)$ as
$\lambda$ goes to zero exhibits three different phases if $2 < a \leq 2.5$,
$2.5 < a \leq 3$ or $3 < a$. Talk based on joint work with Thomas Mountford
and Qiang Yao.




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January 10, 2014

4:00 p.m.

MATH 350

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