## Electronic Journal of Probability

- Electron. J. Probab.
- Volume 24 (2019), paper no. 45, 33 pp.

### Analysis of large urn models with local mean-field interactions

#### Abstract

The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some random amount of time *all the balls* of any urn are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results.

When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of $d$ choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies.

#### Article information

**Source**

Electron. J. Probab., Volume 24 (2019), paper no. 45, 33 pp.

**Dates**

Received: 5 March 2018

Accepted: 9 April 2019

First available in Project Euclid: 10 May 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ejp/1557453644

**Digital Object Identifier**

doi:10.1214/19-EJP304

**Mathematical Reviews number (MathSciNet)**

MR3949270

**Zentralblatt MATH identifier**

1412.60113

**Subjects**

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 60K25: Queueing theory [See also 68M20, 90B22]

Secondary: 68M15: Reliability, testing and fault tolerance [See also 94C12]

**Keywords**

local mean-field interaction nonlinear Markov processes urn models

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Sun, Wen; Philippe, Robert. Analysis of large urn models with local mean-field interactions. Electron. J. Probab. 24 (2019), paper no. 45, 33 pp. doi:10.1214/19-EJP304. https://projecteuclid.org/euclid.ejp/1557453644