## Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 21, Number 6 (2011), 2447-2482.

### Asymptotic behavior of Aldous’ gossip process

Shirshendu Chatterjee and Rick Durrett

#### Abstract

Aldous [(2007) Preprint] defined a gossip process in which space is a discrete *N* × *N* torus, and the state of the process at time *t* is the set of individuals who know the information. Information spreads from a site to its nearest neighbors at rate 1/4 each and at rate *N*^{−α} to a site chosen at random from the torus. We will be interested in the case in which *α* < 3, where the long range transmission significantly accelerates the time at which everyone knows the information. We prove three results that precisely describe the spread of information in a slightly simplified model on the real torus. The time until everyone knows the information is asymptotically *T* = (2 − 2 *α*/3)*N*^{α/3}log *N*. If *ρ*_{s} is the fraction of the population who know the information at time *s* and *ε* is small then, for large *N*, the time until *ρ*_{s} reaches *ε* is *T*(*ε*)* ≈ T* + *N*^{α/3}log(3*ε*/*M*), where *M* is a random variable determined by the early spread of the information. The value of *ρ*_{s} at time *s* = *T*(1/3) + *tN*^{α/3} is almost a deterministic function *h*(*t*) which satisfies an odd looking integro-differential equation. The last result confirms a heuristic calculation of Aldous.

#### Article information

**Source**

Ann. Appl. Probab., Volume 21, Number 6 (2011), 2447-2482.

**Dates**

First available in Project Euclid: 23 November 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1322057327

**Digital Object Identifier**

doi:10.1214/10-AAP750

**Mathematical Reviews number (MathSciNet)**

MR2895421

**Zentralblatt MATH identifier**

1246.60117

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

**Keywords**

Gossip branching process first-passage percolation integro-differential equation

#### Citation

Chatterjee, Shirshendu; Durrett, Rick. Asymptotic behavior of Aldous’ gossip process. Ann. Appl. Probab. 21 (2011), no. 6, 2447--2482. doi:10.1214/10-AAP750. https://projecteuclid.org/euclid.aoap/1322057327