The Annals of Applied Probability

Contact processes on random regular graphs

Steven Lalley and Wei Su

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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 $p$.

Article information

Ann. Appl. Probab., Volume 27, Number 4 (2017), 2061-2097.

Received: April 2015
Revised: June 2016
First available in Project Euclid: 30 August 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K37: Processes in random environments 05C80: Random graphs [See also 60B20]

Contact process random regular graph cutoff phenomenon


Lalley, Steven; Su, Wei. Contact processes on random regular graphs. Ann. Appl. Probab. 27 (2017), no. 4, 2061--2097. doi:10.1214/16-AAP1249.

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