Synchronization and functional central limit theorems for interacting reinforced random walks

Irene Crimaldi (1) , Paolo Dai Pra (2) , Pierre-Yves Louis (3) , Ida Germana Minelli (4)

1 IMT Institute for Advanced Studies [Lucca]
2 Dipartimento di Matematica Pura e Applicata [Padova]
3 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 7348]
4 UNIVAQ - University of L'Aquila [Italy]

We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic approximation. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. Under certain conditions, synchronization is faster than convergence.

Mots clés: dynamics on random graphs; functional central limit theorems; synchronization; Interacting random systems; urn models; reinforced processes

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Adaptation et Systèmes auto-organisés [nlin.AO]; Mécanique statistique [cond-mat.stat-mech]

Irene Crimaldi, Paolo Dai Pra, Pierre-Yves Louis, Ida Germana Minelli. Synchronization and functional central limit theorems for interacting reinforced random walks. Stochastic Processes and their Applications, 2019, 129 (1), pp.70-101. ⟨10.1016/j.spa.2018.02.012⟩. ⟨hal-01277974⟩