What is a $p$-adic Dyson Brownian motion?
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866914832962289664 |
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| author | Van Peski, Roger |
| author_facet | Van Peski, Roger |
| contents | We consider the singular numbers of a certain explicit continuous-time Markov jump process on $\mathrm{GL}_N(\mathbb{Q}_p)$, which we argue gives the closest $p$-adic analogue of multiplicative Dyson Brownian motion. We do so by explicitly classifying the possible dynamics of singular numbers of processes on $\mathrm{GL}_N(\mathbb{Q}_p)$ satisfying natural properties possessed by Brownian motion on $\mathrm{GL}_N(\mathbb{C})$. Computing the evolution of singular numbers explicitly, we find that the $N$-tuple of singular numbers in decreasing order evolves as a Poisson jump process on $\mathbb{Z}^N$, with ordering enforced by reflection off the walls of the positive type $A$ Weyl chamber. This contrasts with -- and provides a $p$-adic analogue to -- the behavior of classical Dyson Brownian motion, where ordering is enforced by conditioning to avoid the Weyl chamber walls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02865 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | What is a $p$-adic Dyson Brownian motion? Van Peski, Roger Probability Combinatorics Number Theory We consider the singular numbers of a certain explicit continuous-time Markov jump process on $\mathrm{GL}_N(\mathbb{Q}_p)$, which we argue gives the closest $p$-adic analogue of multiplicative Dyson Brownian motion. We do so by explicitly classifying the possible dynamics of singular numbers of processes on $\mathrm{GL}_N(\mathbb{Q}_p)$ satisfying natural properties possessed by Brownian motion on $\mathrm{GL}_N(\mathbb{C})$. Computing the evolution of singular numbers explicitly, we find that the $N$-tuple of singular numbers in decreasing order evolves as a Poisson jump process on $\mathbb{Z}^N$, with ordering enforced by reflection off the walls of the positive type $A$ Weyl chamber. This contrasts with -- and provides a $p$-adic analogue to -- the behavior of classical Dyson Brownian motion, where ordering is enforced by conditioning to avoid the Weyl chamber walls. |
| title | What is a $p$-adic Dyson Brownian motion? |
| topic | Probability Combinatorics Number Theory |
| url | https://arxiv.org/abs/2309.02865 |