The Annals of Probability

Stochastic networks with multiple stable points

Nelson Antunes, Christine Fricker, Philippe Robert, and Danielle Tibi

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This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated Markov jump processes are analyzed under a thermodynamic limit regime, that is, when the networks have some symmetry properties and when the number of nodes goes to infinity. An intriguing stability property is proved: under some conditions on the parameters, it is shown that, in the limit, several stable equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Networks with a unique equilibrium point are also presented.

Article information

Ann. Probab., Volume 36, Number 1 (2008), 255-278.

First available in Project Euclid: 28 November 2007

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Energy function fixed point equations stable equilibrium points metastability mean field limit


Antunes, Nelson; Fricker, Christine; Robert, Philippe; Tibi, Danielle. Stochastic networks with multiple stable points. Ann. Probab. 36 (2008), no. 1, 255--278. doi:10.1214/009117907000000105.

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