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Dynamics of the Box-Ball System with Random Initial Conditions via Pitman’s Transformation
About this Title
David A. Croydon, Tsuyoshi Kato, Makiko Sasada and Satoshi Tsujimoto
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 283, Number 1398
ISBNs: 978-1-4704-5633-7 (print); 978-1-4704-7399-0 (online)
DOI: https://doi.org/10.1090/memo/1398
Published electronically: January 19, 2023
Keywords: Box-ball system,
integrated current,
Pitman’s transformation,
simple random walk,
solitons,
tagged particle
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
- 2. Path encodings of the BBS
- 3. Random initial configurations
- 4. Connections with Pitman’s theorem and exclusion processes
- 5. BBS on $\mathbb {R}$
- 6. Open questions
Abstract
The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than $\frac {1}{2}$, and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by ‘reflection in the past maximum’, which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible (i.e. can be inverted) for all times. The techniques developed to understand the deterministic dynamics are subsequently applied to study the evolution of the BBS from a random initial configuration. Specifically, we give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter $p\uparrow \frac 12$ (which may be considered the ‘high density’ regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.- Márton Balázs and Ross Bowen, Product blocking measures and a particle system proof of the Jacobi triple product, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 1, 514–528 (English, with English and French summaries). MR 3765898, DOI 10.1214/16-AIHP813
- Philippe Biane, From Pitman’s theorem to crystals, Noncommutativity and singularities, Adv. Stud. Pure Math., vol. 55, Math. Soc. Japan, Tokyo, 2009, pp. 1–13. MR 2463487, DOI 10.2969/aspm/05510001
- Philippe Biane, Philippe Bougerol, and Neil O’Connell, Continuous crystal and Duistermaat-Heckman measure for Coxeter groups, Adv. Math. 221 (2009), no. 5, 1522–1583. MR 2522427, DOI 10.1016/j.aim.2009.02.016
- David A. Croydon and Makiko Sasada, Invariant measures for the box-ball system based on stationary Markov chains and periodic Gibbs measures, J. Math. Phys. 60 (2019), no. 8, 083301, 25. MR 3987815, DOI 10.1063/1.5095622
- David A. Croydon and Makiko Sasada, Duality between box-ball systems of finite box and/or carrier capacity, Stochastic analysis on large scale interacting systems, RIMS Kôkyûroku Bessatsu, B79, Res. Inst. Math. Sci. (RIMS), Kyoto, 2020, pp. 63–107. MR 4279012
- David A. Croydon, Makiko Sasada, and Satoshi Tsujimoto, Dynamics of the ultra-discrete Toda lattice via Pitman’s transformation, Mathematical structures of integrable systems and their applications, RIMS Kôkyûroku Bessatsu, B78, Res. Inst. Math. Sci. (RIMS), Kyoto, 2020, pp. 235–250. MR 4279006
- Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. MR 2571413, DOI 10.1007/978-3-642-03311-7
- Lester E. Dubins and Meir Smorodinsky, The modified, discrete, Lévy-transformation is Bernoulli, Séminaire de Probabilités, XXVI, Lecture Notes in Math., vol. 1526, Springer, Berlin, 1992, pp. 157–161. MR 1231991, DOI 10.1007/BFb0084318
- P. A. Ferrari and L. R. G. Fontes, Current fluctuations for the asymmetric simple exclusion process, Ann. Probab. 22 (1994), no. 2, 820–832. MR 1288133
- Pablo A. Ferrari and Davide Gabrielli, BBS invariant measures with independent soliton components, Electron. J. Probab. 25 (2020), Paper No. 78, 26. MR 4125783, DOI 10.1214/20-ejp475
- Pablo A. Ferrari and Davide Gabrielli, Box-ball system: soliton and tree decomposition of excursions, XIII Symposium on Probability and Stochastic Processes, Progr. Probab., vol. 75, Birkhäuser/Springer, Cham, [2020] ©2020, pp. 107–152. MR 4181381, DOI 10.1007/978-3-030-57513-7_{5}
- Pablo A. Ferrari, Chi Nguyen, Leonardo T. Rolla, and Minmin Wang, Soliton decomposition of the box-ball system, Forum Math. Sigma 9 (2021), Paper No. e60, 37. MR 4308823, DOI 10.1017/fms.2021.49
- Patrik L. Ferrari and Herbert Spohn, Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process, Comm. Math. Phys. 265 (2006), no. 1, 1–44. MR 2217295, DOI 10.1007/s00220-006-1549-0
- Peter W. Glynn and Dirk Ormoneit, Hoeffding’s inequality for uniformly ergodic Markov chains, Statist. Probab. Lett. 56 (2002), no. 2, 143–146. MR 1881167, DOI 10.1016/S0167-7152(01)00158-4
- Peter W. Glynn and Hermann Thorisson, Two-sided taboo limits for Markov processes and associated perfect simulation, Stochastic Process. Appl. 91 (2001), no. 1, 1–20. MR 1807359, DOI 10.1016/S0304-4149(00)00050-8
- P. Gonçalves, Equilibrium fluctuations for totally asymmetric particle systems, VDM Verlag Dr. Müller e.K., 2010.
- B. M. Hambly, J. B. Martin, and N. O’Connell, Pitman’s $2M-X$ theorem for skip-free random walks with Markovian increments, Electron. Comm. Probab. 6 (2001), 73–77. MR 1855343, DOI 10.1214/ECP.v6-1036
- J. M. Harrison and R. J. Williams, On the quasireversibility of a multiclass Brownian service station, Ann. Probab. 18 (1990), no. 3, 1249–1268. MR 1062068
- Ryogo Hirota, Nonlinear partial difference equations. II. Discrete-time Toda equation, J. Phys. Soc. Japan 43 (1977), no. 6, 2074–2078. MR 460935, DOI 10.1143/JPSJ.43.2074
- Remco van der Hofstad and Michael Keane, An elementary proof of the hitting time theorem, Amer. Math. Monthly 115 (2008), no. 8, 753–756. MR 2456097, DOI 10.1080/00029890.2008.11920588
- Rei Inoue, Atsuo Kuniba, and Taichiro Takagi, Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry, J. Phys. A 45 (2012), no. 7, 073001, 64. MR 2892302, DOI 10.1088/1751-8113/45/7/073001
- Saburo Kakei, Jonathan J. C. Nimmo, Satoshi Tsujimoto, and Ralph Willox, Linearization of the box-ball system: an elementary approach, J. Integrable Syst. 3 (2018), no. 1, xyy002, 32. MR 3800067, DOI 10.1093/integr/xyy002
- Tsuyoshi Kato, Dynamical scale transform in tropical geometry, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. MR 3561545
- Tsuyoshi Kato, Satoshi Tsujimoto, and Andrzej Zuk, Spectral analysis of transition operators, automata groups and translation in BBS, Comm. Math. Phys. 350 (2017), no. 1, 205–229. MR 3606474, DOI 10.1007/s00220-016-2702-z
- Michael Keane and Neil O’Connell, The $M/M/1$ queue is Bernoulli, Colloq. Math. 110 (2008), no. 1, 205–210. MR 2353906, DOI 10.4064/cm110-1-9
- John Kent, Time-reversible diffusions, Adv. in Appl. Probab. 10 (1978), no. 4, 819–835. MR 509218, DOI 10.2307/1426661
- Tomasz Komorowski, Claudio Landim, and Stefano Olla, Fluctuations in Markov processes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 345, Springer, Heidelberg, 2012. Time symmetry and martingale approximation. MR 2952852, DOI 10.1007/978-3-642-29880-6
- Takis Konstantopoulos and Michael Zazanis, A discrete-time proof of Neveu’s exchange formula, J. Appl. Probab. 32 (1995), no. 4, 917–921. MR 1363333, DOI 10.1017/s0021900200103389
- D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. (5) 39 (1895), no. 240, 422–443. MR 3363408, DOI 10.1080/14786449508620739
- Atsuo Kuniba, Masato Okado, Reiho Sakamoto, Taichiro Takagi, and Yasuhiko Yamada, Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection, Nuclear Phys. B 740 (2006), no. 3, 299–327. MR 2214663, DOI 10.1016/j.nuclphysb.2006.02.005
- David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2009. With a chapter by James G. Propp and David B. Wilson. MR 2466937, DOI 10.1090/mbk/058
- Lionel Levine, Hanbaek Lyu, and John Pike, Double jump phase transition in a soliton cellular automaton, Int. Math. Res. Not. IMRN 1 (2022), 665–727. MR 4366029, DOI 10.1093/imrn/rnaa166
- Paul Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars & Cie, Paris, 1965 (French). Suivi d’une note de M. Loève; Deuxième édition revue et augmentée. MR 190953
- Thomas M. Liggett, Interacting particle systems, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 276, Springer-Verlag, New York, 1985. MR 776231, DOI 10.1007/978-1-4613-8542-4
- Grigori L. Litvinov and Victor P. Maslov, The correspondence principle for idempotent calculus and some computer applications, Idempotency (Bristol, 1994) Publ. Newton Inst., vol. 11, Cambridge Univ. Press, Cambridge, 1998, pp. 420–443. MR 1608383, DOI 10.1017/CBO9780511662508.026
- Jun Mada and Tetsuji Tokihiro, Correlation functions for a periodic box-ball system, J. Phys. A 43 (2010), no. 13, 135205, 15. MR 2602102, DOI 10.1088/1751-8113/43/13/135205
- Kazuki Maeda and Satoshi Tsujimoto, Box-ball systems related to the nonautonomous ultradiscrete Toda equation on the finite lattice, JSIAM Lett. 2 (2010), 95–98. MR 3009389, DOI 10.14495/jsiaml.2.95
- S. Moriwaki, A. Nagai, J. Satsuma, T. Tokihiro, M. Torii, D. Takahashi, and J. Matsukidaira, $(2+1)$-dimensional soliton cellular automaton, Symmetries and integrability of difference equations (Canterbury, 1996) London Math. Soc. Lecture Note Ser., vol. 255, Cambridge Univ. Press, Cambridge, 1999, pp. 334–342. MR 1705240, DOI 10.1017/CBO9780511569432.027
- Neil O’Connell, Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), no. 2, 437–458. MR 2952082, DOI 10.1214/10-AOP632
- N. O’Connell, From Pitman’s $2M-X$ theorem to random polymers and integrable systems, slides from Stochastic Processes and their Applications, Boulder, Colorado, 2013. Available at: www.maths.ucd.ie/~noconnell/doob.pdf.
- Neil O’Connell and Marc Yor, Brownian analogues of Burke’s theorem, Stochastic Process. Appl. 96 (2001), no. 2, 285–304. MR 1865759, DOI 10.1016/S0304-4149(01)00119-3
- Richard Otter, The multiplicative process, Ann. Math. Statistics 20 (1949), 206–224. MR 30716, DOI 10.1214/aoms/1177730031
- Goran Peskir, On reflecting Brownian motion with drift, Proceedings of the 37th ISCIE International Symposium on Stochastic Systems Theory and its Applications, Inst. Syst. Control Inform. Engrs. (ISCIE), Kyoto, 2006, pp. 1–5. MR 2410622
- J. W. Pitman, One-dimensional Brownian motion and the three-dimensional Bessel process, Advances in Appl. Probability 7 (1975), no. 3, 511–526. MR 375485, DOI 10.2307/1426125
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- L. C. G. Rogers and J. W. Pitman, Markov functions, Ann. Probab. 9 (1981), no. 4, 573–582. MR 624684
- D. Takahashi, On a fully discrete soliton system, Nonlinear evolution equations and dynamical systems (Baia Verde, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 245–249. MR 1192582
- Daisuke Takahashi and Junta Matsukidaira, Box and ball system with a carrier and ultradiscrete modified KdV equation, J. Phys. A 30 (1997), no. 21, L733–L739. MR 1603434, DOI 10.1088/0305-4470/30/21/005
- Daisuke Takahashi and Junkichi Satsuma, A soliton cellular automaton, J. Phys. Soc. Japan 59 (1990), no. 10, 3514–3519. MR 1082435, DOI 10.1143/JPSJ.59.3514
- D. Takahashi and J. Satsuma, On cellular automata as a simple soliton system, Transactions of the Japan Society for Industrial and Applied Mathematics 1 (1991), no. 1, 41–60.
- Tetsuji Tokihiro, Ultradiscrete systems (cellular automata), Discrete integrable systems, Lecture Notes in Phys., vol. 644, Springer, Berlin, 2004, pp. 383–424. MR 2087745, DOI 10.1007/978-3-540-40357-9_{9}
- T. Tokihiro, The mathematics of box-ball systems, Asakura Shoten, 2010.
- Daisuke Takahashi and Junta Matsukidaira, On discrete soliton equations related to cellular automata, Phys. Lett. A 209 (1995), no. 3-4, 184–188. MR 1363168, DOI 10.1016/0375-9601(95)00780-8
- Satoshi Tsujimoto and Ryogo Hirota, Ultradiscrete KdV equation, J. Phys. Soc. Japan 67 (1998), no. 6, 1809–1810. MR 1632405, DOI 10.1143/JPSJ.67.1809
- V. S. Varadarajan, Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191–220. MR 159923, DOI 10.1090/S0002-9947-1963-0159923-5
- Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
- Daisuke Yoshihara, Fumitaka Yura, and Tetsuji Tokihiro, Fundamental cycle of a periodic box-ball system, J. Phys. A 36 (2003), no. 1, 99–121. MR 1958135, DOI 10.1088/0305-4470/36/1/307
- Fumitaka Yura and Tetsuji Tokihiro, On a periodic soliton cellular automaton, J. Phys. A 35 (2002), no. 16, 3787–3801. MR 1913801, DOI 10.1088/0305-4470/35/16/317