A Path-Counting Analysis of Phase Shifts in Box-Ball Systems
In this paper, we perform a detailed analysis of the phase shift phenomenon of the classical soliton cellular automaton known as the box-ball system, ultimately resulting in a statement and proof of a formula describing this phase shift. This phenomenon has been observed since the nineties, when the...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2022 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2022
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211724 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems. Nicholas M. Ercolani and Jonathan Ramalheira-Tsu. SIGMA 18 (2022), 063, 42 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862569074692718592 |
|---|---|
| author | Ercolani, Nicholas M. Ramalheira-Tsu, Jonathan |
| author_facet | Ercolani, Nicholas M. Ramalheira-Tsu, Jonathan |
| citation_txt | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems. Nicholas M. Ercolani and Jonathan Ramalheira-Tsu. SIGMA 18 (2022), 063, 42 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we perform a detailed analysis of the phase shift phenomenon of the classical soliton cellular automaton known as the box-ball system, ultimately resulting in a statement and proof of a formula describing this phase shift. This phenomenon has been observed since the nineties, when the system was first introduced by Takahashi and Satsuma, but no explicit global description was made beyond its observation. By using the Gessel-Viennot-Lindström lemma and path-counting arguments, we present here a novel proof of the classical phase shift formula for the continuous-time Toda lattice, as discovered by Moser, and use this proof to derive a discrete-time Toda lattice analogue of the phase shift phenomenon. By carefully analysing the connection between the box-ball system and the discrete-time Toda lattice, through the mechanism of tropicalisation/dequantisation, we translate this discrete-time Toda lattice phase shift formula into our new formula for the box-ball system phase shift.
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| first_indexed | 2026-03-13T11:24:01Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211724 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T11:24:01Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ercolani, Nicholas M. Ramalheira-Tsu, Jonathan 2026-01-09T12:49:46Z 2022 A Path-Counting Analysis of Phase Shifts in Box-Ball Systems. Nicholas M. Ercolani and Jonathan Ramalheira-Tsu. SIGMA 18 (2022), 063, 42 pages 1815-0659 2020 Mathematics Subject Classification: 17B80; 37J70; 37K10 arXiv:2106.07129 https://nasplib.isofts.kiev.ua/handle/123456789/211724 https://doi.org/10.3842/SIGMA.2022.063 In this paper, we perform a detailed analysis of the phase shift phenomenon of the classical soliton cellular automaton known as the box-ball system, ultimately resulting in a statement and proof of a formula describing this phase shift. This phenomenon has been observed since the nineties, when the system was first introduced by Takahashi and Satsuma, but no explicit global description was made beyond its observation. By using the Gessel-Viennot-Lindström lemma and path-counting arguments, we present here a novel proof of the classical phase shift formula for the continuous-time Toda lattice, as discovered by Moser, and use this proof to derive a discrete-time Toda lattice analogue of the phase shift phenomenon. By carefully analysing the connection between the box-ball system and the discrete-time Toda lattice, through the mechanism of tropicalisation/dequantisation, we translate this discrete-time Toda lattice phase shift formula into our new formula for the box-ball system phase shift. NSF grant DMS-1615921 supported this work. We thank the referees for their very careful reading of the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Path-Counting Analysis of Phase Shifts in Box-Ball Systems Article published earlier |
| spellingShingle | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems Ercolani, Nicholas M. Ramalheira-Tsu, Jonathan |
| title | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems |
| title_full | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems |
| title_fullStr | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems |
| title_full_unstemmed | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems |
| title_short | A Path-Counting Analysis of Phase Shifts in Box-Ball Systems |
| title_sort | path-counting analysis of phase shifts in box-ball systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211724 |
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