Electronic Journal of Probability

First passage time of the frog model has a sublinear variance

Van Hao Can and Shuta Nakajima

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Abstract

In this paper, we show that the first passage time in the frog model on $\mathbb{Z} ^{d}$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not hold at least with the standard diffusive scaling. The proof is based on the method introduced in [4, 11] combined with a control of the maximal weight of paths in a locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 76, 27 pp.

Dates
Received: 1 October 2018
Accepted: 15 June 2019
First available in Project Euclid: 5 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1562292237

Digital Object Identifier
doi:10.1214/19-EJP334

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
frog model first passage time sublinear variance

Rights
Creative Commons Attribution 4.0 International License.

Citation

Can, Van Hao; Nakajima, Shuta. First passage time of the frog model has a sublinear variance. Electron. J. Probab. 24 (2019), paper no. 76, 27 pp. doi:10.1214/19-EJP334. https://projecteuclid.org/euclid.ejp/1562292237


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References

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