Limit shapes and fluctuations for $(GL_n, GL_k)$ skew Howe duality
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913475593240576 |
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| author | Betea, Dan Nazarov, Anton Nikitin, Pavel Scrimshaw, Travis |
| author_facet | Betea, Dan Nazarov, Anton Nikitin, Pavel Scrimshaw, Travis |
| contents | We consider the probability measures on Young diagrams in the $n \times k$ rectangle obtained by piecewise-continuously differentiable specializations of Schur polynomials in the dual Cauchy identity. We use a free fermionic representation of the correlation kernel to study its asymptotic behavior and derive the uniform convergence to a limit shape of Young diagrams in the limit $n,k \to \infty$. More specifically, we show the bulk is the discrete sine kernel with boundary fluctuations generically given by the Tracy-Widom distribution with the Airy kernel. When our limit shape touches the boundary corner of the rectangle, the fluctuations with a second order correction are given by the discrete Hermite kernel, and we recover the discrete distribution of Gravner-Tracy-Widom (2001) [arXiv:math/0005133] restricting to the leading order. Finally, we demonstrate our limit shapes can have sections with no or full density of particles, where the Pearcey kernel appears when such a section is infinitely small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11419 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit shapes and fluctuations for $(GL_n, GL_k)$ skew Howe duality Betea, Dan Nazarov, Anton Nikitin, Pavel Scrimshaw, Travis Probability Mathematical Physics Combinatorics Representation Theory 05A19, 60C05, 60G55 We consider the probability measures on Young diagrams in the $n \times k$ rectangle obtained by piecewise-continuously differentiable specializations of Schur polynomials in the dual Cauchy identity. We use a free fermionic representation of the correlation kernel to study its asymptotic behavior and derive the uniform convergence to a limit shape of Young diagrams in the limit $n,k \to \infty$. More specifically, we show the bulk is the discrete sine kernel with boundary fluctuations generically given by the Tracy-Widom distribution with the Airy kernel. When our limit shape touches the boundary corner of the rectangle, the fluctuations with a second order correction are given by the discrete Hermite kernel, and we recover the discrete distribution of Gravner-Tracy-Widom (2001) [arXiv:math/0005133] restricting to the leading order. Finally, we demonstrate our limit shapes can have sections with no or full density of particles, where the Pearcey kernel appears when such a section is infinitely small. |
| title | Limit shapes and fluctuations for $(GL_n, GL_k)$ skew Howe duality |
| topic | Probability Mathematical Physics Combinatorics Representation Theory 05A19, 60C05, 60G55 |
| url | https://arxiv.org/abs/2408.11419 |