Scaling limit of soliton lengths in a multicolor box-ball system
The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this paper, we consider a multicolor box-ball system with two types of random initial con...
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000744/type/journal_article |
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| author | Joel Lewis Hanbaek Lyu Pavlo Pylyavskyy Arnab Sen |
| author_facet | Joel Lewis Hanbaek Lyu Pavlo Pylyavskyy Arnab Sen |
| author_sort | Joel Lewis |
| collection | DOAJ |
| description | The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this paper, we consider a multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain a sharp scaling limit of soliton lengths that turns out to be more delicate than that in the single color case established in [LLP20]. A large part of our analysis is devoted to studying the associated carrier process, which is a multidimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes. |
| format | Article |
| id | doaj-art-93813c257a16481c91b2a2c15da55b2d |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-93813c257a16481c91b2a2c15da55b2d2024-12-10T06:02:49ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.74Scaling limit of soliton lengths in a multicolor box-ball systemJoel Lewis0Hanbaek Lyu1https://orcid.org/0000-0002-6323-5240Pavlo Pylyavskyy2https://orcid.org/0000-0001-7211-6115Arnab Sen3Department of Mathematics, The George Washington University, 801 22nd St. NW, Washington, DC, 20052, United States; E-mail:Department of Mathematics, University of Wisconsin, 480 Lincoln Dr, Madison, Wisconsin, 53706, United States;Department of Mathematics, University of Minnesota, 127 Vincent Hall 206 Church St. SE Minneapolis, 55455, United States; E-mail:Department of Mathematics, University of Minnesota, 127 Vincent Hall 206 Church St. SE Minneapolis, 55455, United States; E-mail:The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this paper, we consider a multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain a sharp scaling limit of soliton lengths that turns out to be more delicate than that in the single color case established in [LLP20]. A large part of our analysis is devoted to studying the associated carrier process, which is a multidimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.https://www.cambridge.org/core/product/identifier/S2050509424000744/type/journal_article37B1560K3582C2237K40 |
| spellingShingle | Joel Lewis Hanbaek Lyu Pavlo Pylyavskyy Arnab Sen Scaling limit of soliton lengths in a multicolor box-ball system Forum of Mathematics, Sigma 37B15 60K35 82C22 37K40 |
| title | Scaling limit of soliton lengths in a multicolor box-ball system |
| title_full | Scaling limit of soliton lengths in a multicolor box-ball system |
| title_fullStr | Scaling limit of soliton lengths in a multicolor box-ball system |
| title_full_unstemmed | Scaling limit of soliton lengths in a multicolor box-ball system |
| title_short | Scaling limit of soliton lengths in a multicolor box-ball system |
| title_sort | scaling limit of soliton lengths in a multicolor box ball system |
| topic | 37B15 60K35 82C22 37K40 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000744/type/journal_article |
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