Gaussian measure on the dual of $\mathrm{U}(N)$, random partitions, and topological expansion of the partition function
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909574485770240 |
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| author | Lemoine, Thibaut Maïda, Mylène |
| author_facet | Lemoine, Thibaut Maïda, Mylène |
| contents | We study a Gaussian measure with parameter $q\in(0,1)$ on the dual of the unitary group of size $N$: we prove that a random highest weight under this measure is the coupling of two independent $q$-uniform random partitions $α,β$ and a random highest weight of $\mathrm{U}(1)$. We prove deviation inequalities for the $q$-uniform measure, and use them to show that the coupling of random partitions under the Gaussian measure vanishes in the limit $N\to\infty$. We also prove that the partition function of this measure admits an asymptotic expansion in powers of $1/N$, and that this expansion is topological, in the sense that its coefficients are related to the enumeration of ramified coverings of elliptic curves. It provides a rigorous proof of the gauge/string duality for the Yang-Mills theory on a 2D torus with gauge group $\mathrm{U}(N),$ advocated by Gross and Taylor \cite{GT,GT2}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08393 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian measure on the dual of $\mathrm{U}(N)$, random partitions, and topological expansion of the partition function Lemoine, Thibaut Maïda, Mylène Mathematical Physics Probability Representation Theory We study a Gaussian measure with parameter $q\in(0,1)$ on the dual of the unitary group of size $N$: we prove that a random highest weight under this measure is the coupling of two independent $q$-uniform random partitions $α,β$ and a random highest weight of $\mathrm{U}(1)$. We prove deviation inequalities for the $q$-uniform measure, and use them to show that the coupling of random partitions under the Gaussian measure vanishes in the limit $N\to\infty$. We also prove that the partition function of this measure admits an asymptotic expansion in powers of $1/N$, and that this expansion is topological, in the sense that its coefficients are related to the enumeration of ramified coverings of elliptic curves. It provides a rigorous proof of the gauge/string duality for the Yang-Mills theory on a 2D torus with gauge group $\mathrm{U}(N),$ advocated by Gross and Taylor \cite{GT,GT2}. |
| title | Gaussian measure on the dual of $\mathrm{U}(N)$, random partitions, and topological expansion of the partition function |
| topic | Mathematical Physics Probability Representation Theory |
| url | https://arxiv.org/abs/2405.08393 |