Dynamics of the ultra-discrete Toda lattice via Pitman's transformation
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866915935359598592 |
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| author | Croydon, David A. Sasada, Makiko Tsujimoto, Satoshi |
| author_facet | Croydon, David A. Sasada, Makiko Tsujimoto, Satoshi |
| contents | By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between $-1$ and $1$, we show that the dynamics of the system can be described in terms of a shifted version of Pitman's transformation (that is, reflection in the past maximum of the path encoding). This characterisation of the dynamics applies to finite configurations in both the non-periodic and periodic cases, and also admits an extension to infinite configurations. The latter point is important in the study of invariant measures for the ultra-discrete Toda lattice, which is pursued in a parallel work. We also describe a generalisation of the result to a continuous version of the box-ball system, whose states are described by continuous functions whose gradient may take values other than $\pm1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1904_13185 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Dynamics of the ultra-discrete Toda lattice via Pitman's transformation Croydon, David A. Sasada, Makiko Tsujimoto, Satoshi Mathematical Physics 37B15 (primary), 82B99 (secondary) By encoding configurations of the ultra-discrete Toda lattice by piecewise linear paths whose gradient alternates between $-1$ and $1$, we show that the dynamics of the system can be described in terms of a shifted version of Pitman's transformation (that is, reflection in the past maximum of the path encoding). This characterisation of the dynamics applies to finite configurations in both the non-periodic and periodic cases, and also admits an extension to infinite configurations. The latter point is important in the study of invariant measures for the ultra-discrete Toda lattice, which is pursued in a parallel work. We also describe a generalisation of the result to a continuous version of the box-ball system, whose states are described by continuous functions whose gradient may take values other than $\pm1$. |
| title | Dynamics of the ultra-discrete Toda lattice via Pitman's transformation |
| topic | Mathematical Physics 37B15 (primary), 82B99 (secondary) |
| url | https://arxiv.org/abs/1904.13185 |