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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2024

Gaussian measure on the dual of $\mathrm{U}(N)$, random partitions, and topological expansion of the partition function

Résumé

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 $\alpha,\beta$ 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}.
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hal-04572877 , version 1 (13-05-2024)

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  • HAL Id : hal-04572877 , version 1

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Thibaut Lemoine, Mylène Maïda. Gaussian measure on the dual of $\mathrm{U}(N)$, random partitions, and topological expansion of the partition function. 2024. ⟨hal-04572877⟩
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