## The Annals of Applied Probability

### Contact processes on random regular graphs

#### 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 $p$.

#### Article information

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

Dates
Revised: June 2016
First available in Project Euclid: 30 August 2017

https://projecteuclid.org/euclid.aoap/1504080026

Digital Object Identifier
doi:10.1214/16-AAP1249

Mathematical Reviews number (MathSciNet)
MR3693520

Zentralblatt MATH identifier
1373.60159

#### Citation

Lalley, Steven; Su, Wei. Contact processes on random regular graphs. Ann. Appl. Probab. 27 (2017), no. 4, 2061--2097. doi:10.1214/16-AAP1249. https://projecteuclid.org/euclid.aoap/1504080026

#### References

• [1] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. Universality for first passage percolation on sparse random graphs. Available at arXiv:1210.6839.
• [2] Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin. 1 311–316.
• [3] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
• [4] Bollobás, B. and Fernandez de la Vega, W. (1982). The diameter of random regular graphs. Combinatorica 2 125–134.
• [5] Chatterjee, S. and Durrett, R. (2013). A first order phase transition in the threshold $\theta\geq2$ contact process on random $r$-regular graphs and $r$-trees. Stochastic Process. Appl. 123 561–578.
• [6] Cranston, M., Mountford, T., Mourrat, J.-C. and Valesin, D. (2014). The contact process on finite homogeneous trees revisited. ALEA Lat. Am. J. Probab. Math. Stat. 11 385–408.
• [7] Ding, J., Sly, A. and Sun, N. Maximum independent sets on random regular graphs. Available at arXiv:1310.4787.
• [8] Durrett, R. and Jung, P. (2007). Two phase transitions for the contact process on small worlds. Stochastic Process. Appl. 117 1910–1927.
• [9] Harris, T. E. (1978). Additive set-valued Markov processes and graphical methods. Ann. Probab. 6 355–378.
• [10] Hoory, S., Linial, N. and Wigderson, A. (2006). Expander graphs and their applications. Bull. Amer. Math. Soc. (N.S.) 43 439–561 (electronic).
• [11] Lalley, S. P. and Sellke, T. (1998). Limit set of a weakly supercritical contact process on a homogeneous tree. Ann. Probab. 26 644–657.
• [12] Liggett, T. (1996). Multiple transition points for the contact process on the binary tree. Ann. Probab. 24 1675–1710.
• [13] Liggett, T. M. (2005). Interacting Particle Systems. Springer, Berlin.
• [14] Lubetzky, E. and Sly, A. (2010). Cutoff phenomena for random walks on random regular graphs. Duke Math. J. 153 475–510.
• [15] Madras, N. and Schinazi, R. (1992). Branching random walks on trees. Stochastic Process. Appl. 42 255–267.
• [16] Morrow, G., Schinazi, R. and Zhang, Y. (1994). The critical contact process on a homogeneous tree. J. Appl. Probab. 31 250–255.
• [17] Mountford, T. S. (1993). A metastable result for the finite multidimensional contact process. Canad. Math. Bull. 36 216–226.
• [18] Mourrat, J.-C. and Valesin, D. Phase transition of the contact process on random regular graphs. Available at arXiv:1405.0865.
• [19] Pemantle, R. (1992). The contact process on trees. Ann. Probab. 20 2089–2116.
• [20] Stacey, A. (1996). The existence of an intermediate phase for the contact process on trees. Ann. Probab. 24 1711–1726.
• [21] Stacey, A. (2001). The contact process on finite homogeneous trees. Probab. Theory Related Fields 121 551–576.
• [22] Wormald, N. C. (1999). Models of random regular graphs. In Surveys in Combinatorics, 1999 (Canterbury). London Mathematical Society Lecture Note Series 267 239–298. Cambridge Univ. Press, Cambridge.