Contact processes on random regular graphs

Steven Lalley; Wei Su

doi:10.1214/16-AAP1249

August 2017 Contact processes on random regular graphs

Steven Lalley, Wei Su

Ann. Appl. Probab. 27(4): 2061-2097 (August 2017). DOI: 10.1214/16-AAP1249

Abstract

We show that the contact process on a random $d$-regular graph initiated by a single infected vertex obeys the “cutoff phenomenon” in its supercritical phase. In particular, we prove that, when the infection rate is larger than the lower critical value of the contact process on the infinite $d$-regular tree, there are positive constants $C$, $p$ depending on the infection rate such that for any $\varepsilon>0$, when the number $n$ of vertices is large then (a) at times $t<(C-\varepsilon)\log n$ the fraction of infected vertices is vanishingly small, but (b) at time $(C+\varepsilon)\log n$ the fraction of infected vertices is within $\varepsilon$ of $p$, with probability $p$.

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Steven Lalley and Wei Su "Contact processes on random regular graphs," The Annals of Applied Probability 27(4), 2061-2097, (August 2017). https://doi.org/10.1214/16-AAP1249

Received: 1 April 2015; Published: August 2017

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