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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | https://arxiv.org/abs/2601.01304 |
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| _version_ | 1866914232967102464 |
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| author | Sinclair, Christopher D. |
| author_facet | Sinclair, Christopher D. |
| contents | Random matrix ensembles with Dyson index $β=L^{2}$ describe systems of $M$ charge-$L$ particles interacting logarithmically in the presence of an external potential, yet exact formulas for their physical observables have remained elusive for $L\neq 1,2$. We show that, for $L$ even, $β=L^{2}$ ensembles are governed by the KP hierarchy at finite particle number--paralleling the KP solvability of classical $β=1,2,4$ ensembles. The partition function is a hyperpfaffian $τ$-function satisfying the Hirota bilinear identity, and correlation functions are generated by finite-order differential operators acting on this $τ$-function. The key mechanism is an emergent quantized momentum that stratifies the system into discrete sectors, enforcing momentum conservation as a selection rule. This produces a dramatic dimensional reduction from ${LM\choose L}$ to $O(L^{2}M)$, enabling explicit computation of physical observables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01304 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exact Solvability via the KP Hierarchy for $β=L^2$ Random Matrix Ensembles Sinclair, Christopher D. Mathematical Physics 15B52, 37K10 Random matrix ensembles with Dyson index $β=L^{2}$ describe systems of $M$ charge-$L$ particles interacting logarithmically in the presence of an external potential, yet exact formulas for their physical observables have remained elusive for $L\neq 1,2$. We show that, for $L$ even, $β=L^{2}$ ensembles are governed by the KP hierarchy at finite particle number--paralleling the KP solvability of classical $β=1,2,4$ ensembles. The partition function is a hyperpfaffian $τ$-function satisfying the Hirota bilinear identity, and correlation functions are generated by finite-order differential operators acting on this $τ$-function. The key mechanism is an emergent quantized momentum that stratifies the system into discrete sectors, enforcing momentum conservation as a selection rule. This produces a dramatic dimensional reduction from ${LM\choose L}$ to $O(L^{2}M)$, enabling explicit computation of physical observables. |
| title | Exact Solvability via the KP Hierarchy for $β=L^2$ Random Matrix Ensembles |
| topic | Mathematical Physics 15B52, 37K10 |
| url | https://arxiv.org/abs/2601.01304 |