## Electronic Communications in Probability

### On the largest component in the subcritical regime of the Bohman-Frieze process

Sanchayan Sen

#### Abstract

Kang, Perkins, and Spencer [7] conjectured that the size of the largest component of the Bohman-Frieze process at a fixed time $t$ smaller than $t_c$, the critical time for the process, is $L_1(t)=O(\log n/(t_c-t)^2)$ with high probability. Bhamidi, Budhiraja, and Wang [3] have shown that a bound of the form $L_1(t_n)=O((\log n)^4/(t_c-t_n)^2)$ holds with high probability for $t_n\leq t_c-n^{-\gamma }$ where $\gamma \in (0,1/4)$. In this paper, we improve the result in [3] by showing that for any fixed $\lambda >0$, $L_1(t_n)=O(\log n/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-\lambda n^{-1/3}$. In particular, this settles the conjecture in [7].

#### Article information

Source
Electron. Commun. Probab. Volume 21 (2016), paper no. 64, 15 pp.

Dates
Accepted: 30 August 2016
First available in Project Euclid: 14 September 2016

http://projecteuclid.org/euclid.ecp/1473854580

Digital Object Identifier
doi:10.1214/16-ECP20

Zentralblatt MATH identifier
1346.60006

#### Citation

Sen, Sanchayan. On the largest component in the subcritical regime of the Bohman-Frieze process. Electron. Commun. Probab. 21 (2016), paper no. 64, 15 pp. doi:10.1214/16-ECP20. http://projecteuclid.org/euclid.ecp/1473854580.

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