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

Thibaut Lemoine, Mylène Maïda

TL;DR

This work constructs a Gaussian-type measure on the dual of the unitary group $U(N)$, showing that a random highest weight is the coupling of two independent $q$-uniform partitions and a $U(1)$ highest weight. It proves asymptotic decoupling as $N\to\infty$ and provides an explicit $1/N$ expansion for the partition function $Z_N(q)$, whose coefficients relate to the enumeration of ramified coverings of elliptic curves, thereby giving a rigorous version of the Gross–Taylor gauge/string duality for 2D Yang–Mills on the torus. The authors also develop the $q$-uniform measure on partitions, derive deviation inequalities, and connect random partitions to Frobenius–Hurwitz measures and Hurwitz numbers, establishing a rich link between random representation theory and enumerative geometry. The topological expansion interpretation is made precise in the torus setting, with a detailed treatment of the chiral sector and potential extensions to higher genus via recursion techniques. Overall, the paper provides a rigorous bridge between probabilistic models on representations, geometric enumeration, and gauge–string duality in low-dimensional quantum field theory.

Abstract

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}.

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

TL;DR

This work constructs a Gaussian-type measure on the dual of the unitary group , showing that a random highest weight is the coupling of two independent -uniform partitions and a highest weight. It proves asymptotic decoupling as and provides an explicit expansion for the partition function , whose coefficients relate to the enumeration of ramified coverings of elliptic curves, thereby giving a rigorous version of the Gross–Taylor gauge/string duality for 2D Yang–Mills on the torus. The authors also develop the -uniform measure on partitions, derive deviation inequalities, and connect random partitions to Frobenius–Hurwitz measures and Hurwitz numbers, establishing a rich link between random representation theory and enumerative geometry. The topological expansion interpretation is made precise in the torus setting, with a detailed treatment of the chiral sector and potential extensions to higher genus via recursion techniques. Overall, the paper provides a rigorous bridge between probabilistic models on representations, geometric enumeration, and gauge–string duality in low-dimensional quantum field theory.

Abstract

We study a Gaussian measure with parameter on the dual of the unitary group of size : we prove that a random highest weight under this measure is the coupling of two independent -uniform random partitions and a random highest weight of . We prove deviation inequalities for the -uniform measure, and use them to show that the coupling of random partitions under the Gaussian measure vanishes in the limit . We also prove that the partition function of this measure admits an asymptotic expansion in powers of , 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 advocated by Gross and Taylor \cite{GT,GT2}.
Paper Structure (20 sections, 23 theorems, 138 equations, 2 figures)

This paper contains 20 sections, 23 theorems, 138 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Coupling of random partitions
  3. Asymptotic expansion of the partition function
  4. Two-dimensional Yang--Mills theory
  5. On the notion of topological expansion
  6. Organisation of the paper
  7. $q$-Uniform measure on partitions
  8. Definitions and main properties
  9. Deviation inequalities
  10. The Gaussian measure on $\widehat{{\mathrm U}}(N)$
  11. Definition of $\mathscr{G}_N(q)$
  12. The case $N=1$
  13. The general case: from highest weights to partitions and back
  14. Proofs of the main theorems
  15. The large $N$ limit of $\mathscr{G}_N(q)$
  16. ...and 5 more sections

Key Result

Theorem 1.1

For any $q\in(0,1)$, we have the following convergence in distribution:

Figures (2)

  • Figure 1: Construction of the highest weight $\lambda_N(\alpha,\beta,n)$ for $N=8$, $\alpha=(2,1,1)$, $\beta=(4,4,3,1)$ and $n=2$.
  • Figure 2: The Young diagram of the partition $(5,4,1)$, in the English convention (on the left), and the French convention (on the right).

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Lem
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 34 more