TY - JOUR
T1 - Phase transition of the contact process on random regular graphs
AU - Mourrat, Jean Christophe
AU - Valesin, Daniel
N1 - Publisher Copyright:
© 2016, University of Washington. All Rights Reserved.
PY - 2016
Y1 - 2016
N2 - We consider the contact process with infection rate λ on a random (d + 1)-regular graph with n vertices, Gn. We study the extinction time τGn (that is, the random amount of time until the infection disappears) as n is taken to infinity. We establish a phase transition depending on whether λ is smaller or larger than λ1((image found)d), the lower critical value for the contact process on the infinite, (d+1)-regular tree: if λ < λ1((image found)d), τGn grows logarithmically with n, while if λ > λ1((image found)d), it grows exponentially with n. This result differs from the situation where, instead of Gn, the contact process is considered on the d-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on Td.
AB - We consider the contact process with infection rate λ on a random (d + 1)-regular graph with n vertices, Gn. We study the extinction time τGn (that is, the random amount of time until the infection disappears) as n is taken to infinity. We establish a phase transition depending on whether λ is smaller or larger than λ1((image found)d), the lower critical value for the contact process on the infinite, (d+1)-regular tree: if λ < λ1((image found)d), τGn grows logarithmically with n, while if λ > λ1((image found)d), it grows exponentially with n. This result differs from the situation where, instead of Gn, the contact process is considered on the d-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on Td.
KW - Configuration model
KW - Contact process
KW - Interacting particle systems
KW - Random graph
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U2 - 10.1214/16-EJP4476
DO - 10.1214/16-EJP4476
M3 - Article
AN - SCOPUS:84963819022
SN - 1083-6489
VL - 21
JO - Electronic Journal of Probability
JF - Electronic Journal of Probability
ER -