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

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}.