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

Full-text: Open access


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.


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

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

First available in Project Euclid: 26 March 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • [1] L. P. Arguin. Competing particle systems and the Ghirlanda–Guerra identities. Electron. J. Probab. 13 (2008) 2101–2117.
  • [2] L. P. Arguin and M. Aizenman. On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 (2009) 1080–1113.
  • [3] M. Balázs, M. Z. Rácz, and B. Tóth. Modeling flocks and prices: Jumping particles with an attractive interaction. Preprint, 2011. Available at arXiv:1107.3289.
  • [4] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl. Acad. Sci. USA 105 (2008) 1232–1237.
  • [5] A. D. Banner, R. Fernholz and I. Karatzas. Atlas models of equity markets. Ann. Appl. Probab. 15 (2005) 2296–2330.
  • [6] D. Ben-Avraham, S. N. Majumdar and S. Redner. A toy model of the rat race. J. Stat. Mech. Theory Exp. 2007 (2007) L04002.
  • [7] E. Bertin. Global fluctuations and Gumbel statistics. Phys. Rev. Lett. 95 (2005) 170601.
  • [8] P. Billingsley. Convergence of Probability Measures, 2nd edition. Wiley Series in Probability and Statistics. Wiley, New York, 1999.
  • [9] S. Chatterjee and S. Pal. A phase transition behavior for Brownian motions interacting through their ranks. Probab. Theory Related Fields 147 (2009) 123–159.
  • [10] M. Clusel and E. Bertin. Global fluctuations in physical systems: A subtle interplay between sum and extreme value statistics. Internat. J. Modern Phys. B 22 (2008) 3311–3368.
  • [11] A. Czirók, A. L. Barabási and T. Vicsek. Collective motion of self-propelled particles: Kinetic phase transition in one dimension. Phys. Rev. Lett. 82 (1999) 209–212.
  • [12] J. Engländer. The center of mass for spatial branching processes and an application for self-interaction. Electron. J. Probab. 15 (2010) 1938–1970.
  • [13] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986.
  • [14] J. Feng and T. G. Kurtz. Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs 131. Amer. Math. Soc., Providence, RI, 2006.
  • [15] E. R. Fernholz. Stochastic Portfolio Theory. Springer, New York, 2002.
  • [16] E. R. Fernholz and I. Karatzas. Stochastic portfolio theory: An overview. In Handbook of Numerical Analysis 15 89–167. Elsevier, Amsterdam, 2009.
  • [17] A. L. Gibbs and F. E. Su. On choosing and bounding probability metrics. International Statistical Review 70 (2002) 419–435.
  • [18] A. G. Greenberg, V. A. Malyshev and S. Y. Popov. Stochastic models of massively parallel computation. Markov Process. Related Fields 1 (1995) 473–490.
  • [19] A. Greven and F. D. Hollander. Phase transitions for the long-time behaviour of interacting diffusions. Ann. Probab. 35 (2007) 1250–1306.
  • [20] I. Grigorescu and M. Kang. Steady state and scaling limit for a traffic congestion model. ESAIM Probab. Stat. 14 (2010) 271–285.
  • [21] E. J. Gumbel. Statistics of Extremes. Dover, New York, 1958.
  • [22] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin, 2003.
  • [23] P. M. Kotelenez and T. G. Kurtz. Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type. Probab. Theory Related Fields 146 (2010) 189–222.
  • [24] A. Manita and V. Shcherbakov. Asymptotic analysis of a particle system with mean-field interaction. Markov Process. Related Fields 11 (2005) 489–518.
  • [25] S. Pal and J. Pitman. One-dimensional Brownian particle systems with rank dependent drifts. Ann. Appl. Probab. 18 (2008) 2179–2207.
  • [26] E. A. Perkins. Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999) 125–324 Lecture. Notes in Math. 1781. Springer, Berlin, 2002.
  • [27] A. Ruzmaikina and M. Aizenman. Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 (2005) 82–113.
  • [28] M. Shkolnikov. Competing particle systems evolving by IID increments. Electron. J. Probab. 14 (2009) 728–751.
  • [29] M. Shkolnikov. Competing particle systems evolving by interacting Levy processes. Ann. Appl. Probab. 21 (2011) 1911–1932.
  • [30] M. Shkolnikov. Large volatility-stabilized markets. Preprint, 2011. Available at arXiv:1102.3461.
  • [31] M. Shkolnikov. Large systems of diffusions interacting through their ranks. Stochastic Process. Appl. 122 (2012) 1730–1747.
  • [32] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75 (1995) 1226–1229.
  • [33] S. Willard. General Topology. Dover, New York, 2004.