Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Modeling flocks and prices: Jumping particles with an attractive interaction

Márton Balázs, Miklós Z. Rácz, and Bálint Tóth

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Abstract

We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. We prove that in the fluid limit, as the number of particles goes to infinity, the evolution of the system is described by a mean field equation that exhibits traveling wave solutions. A connection to extreme value statistics is also provided.

Résumé

Nous introduisons et étudions un nouveau modèle comprenant un nombre fini de particules situées sur la droite réelle et pouvant effectuer des sauts vers la droite. Les longueurs des sauts sont indépendantes du reste, mais le taux de saut de chaque particule dépend de la position relative de la particule par rapport au centre de masse du système. Les taux sont plus grands pour celles qui sont en retard, et plus petits pour celles qui sont en avance par rapport au centre de masse; cela crée ainsi une interaction attractive qui favorise la cohésion des particules. Nous montrons qu’à la limite fluide, lorsque le nombre de particules tend vers l’infini, l’évolution du système est décrite par une équation de champ moyen ayant des solutions d’ondes progressives. On présente également un lien avec les statistiques des valeurs extrêmes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 2 (2014), 425-454.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1395856134

Digital Object Identifier
doi:10.1214/12-AIHP512

Mathematical Reviews number (MathSciNet)
MR3189078

Zentralblatt MATH identifier
1302.60130

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J75: Jump processes

Keywords
Competing particles Center of mass Mean field evolution Traveling wave Fluid limit Extreme value statistics

Citation

Balázs, Márton; Rácz, Miklós Z.; Tóth, Bálint. Modeling flocks and prices: Jumping particles with an attractive interaction. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 2, 425--454. doi:10.1214/12-AIHP512. https://projecteuclid.org/euclid.aihp/1395856134


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