Modeling flocks and prices: Jumping particles with an attractive interaction
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, p. 425-454

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.

DOI : https://doi.org/10.1214/12-AIHP512
Classification:  60K35,  60J75
Keywords: competing particles, center of mass, mean field evolution, traveling wave, fluid limit, extreme value statistics
@article{AIHPB_2014__50_2_425_0,
     author = {Bal\'azs, M\'arton and R\'acz, Mikl\'os Z. and T\'oth, B\'alint},
     title = {Modeling flocks and prices: Jumping particles with an attractive interaction},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     pages = {425-454},
     doi = {10.1214/12-AIHP512},
     zbl = {1302.60130},
     mrnumber = {3189078},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_2_425_0}
}
Balázs, Márton; Rácz, Miklós Z.; Tóth, Bálint. Modeling flocks and prices: Jumping particles with an attractive interaction. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 425-454. doi : 10.1214/12-AIHP512. http://www.numdam.org/item/AIHPB_2014__50_2_425_0/

[1] L. P. Arguin. Competing particle systems and the Ghirlanda-Guerra identities. Electron. J. Probab. 13 (2008) 2101-2117. | MR 2461537 | Zbl 1192.60103

[2] L. P. Arguin and M. Aizenman. On the structure of quasi-stationary competing particle systems. Ann. Probab. 37 (2009) 1080-1113. | MR 2537550 | Zbl 1177.60050

[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. | Zbl 1302.60130 | Zbl pre06298322

[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. | MR 2187296 | Zbl 1099.91056

[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. | MR 1700749 | Zbl 0172.21201

[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. | MR 2594349 | Zbl 1188.60049

[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. | MR 2446819 | Zbl 1145.82304

[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. | MR 2738344 | Zbl 1226.60118

[13] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. | MR 838085 | Zbl 1089.60005

[14] J. Feng and T. G. Kurtz. Large Deviations for Stochastic Processes, Mathematical Surveys and Monographs 131. Amer. Math. Soc., Providence, RI, 2006. | MR 2260560 | Zbl 1113.60002

[15] E. R. Fernholz. Stochastic Portfolio Theory. Springer, New York, 2002. | MR 1894767 | Zbl 1049.91067

[16] E. R. Fernholz and I. Karatzas. Stochastic portfolio theory: An overview. In Handbook of Numerical Analysis 15 89-167. Elsevier, Amsterdam, 2009. | Zbl 1180.91267

[17] A. L. Gibbs and F. E. Su. On choosing and bounding probability metrics. International Statistical Review 70 (2002) 419-435. | Zbl 1217.62014

[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. | MR 1403093 | Zbl 0902.60072

[19] A. Greven and F. D. Hollander. Phase transitions for the long-time behaviour of interacting diffusions. Ann. Probab. 35 (2007) 1250-1306. | MR 2330971 | Zbl 1126.60085

[20] I. Grigorescu and M. Kang. Steady state and scaling limit for a traffic congestion model. ESAIM Probab. Stat. 14 (2010) 271-285. | Numdam | MR 2779484 | Zbl 1227.60108

[21] E. J. Gumbel. Statistics of Extremes. Dover, New York, 1958. | MR 96342 | Zbl 1113.62057

[22] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin, 2003. | MR 1943877 | Zbl 0635.60021

[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. | MR 2550362 | Zbl 1189.60123

[24] A. Manita and V. Shcherbakov. Asymptotic analysis of a particle system with mean-field interaction. Markov Process. Related Fields 11 (2005) 489-518. | MR 2175025 | Zbl 1099.60073

[25] S. Pal and J. Pitman. One-dimensional Brownian particle systems with rank dependent drifts. Ann. Appl. Probab. 18 (2008) 2179-2207. | MR 2473654 | Zbl 1166.60061

[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. | Zbl 1020.60075

[27] A. Ruzmaikina and M. Aizenman. Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 (2005) 82-113. | MR 2118860 | Zbl 1096.60042

[28] M. Shkolnikov. Competing particle systems evolving by IID increments. Electron. J. Probab. 14 (2009) 728-751. | MR 2486819 | Zbl 1190.60039

[29] M. Shkolnikov. Competing particle systems evolving by interacting Levy processes. Ann. Appl. Probab. 21 (2011) 1911-1932. | MR 2884054 | Zbl 1238.60113

[30] M. Shkolnikov. Large volatility-stabilized markets. Preprint, 2011. Available at arXiv:1102.3461. | MR 2988116 | Zbl 1288.60092

[31] M. Shkolnikov. Large systems of diffusions interacting through their ranks. Stochastic Process. Appl. 122 (2012) 1730-1747. | MR 2914770 | Zbl 1276.60087

[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. | MR 2048350 | Zbl 1052.54001