We prove an asymptotic formula for the partition function of a 2d Coulomb gas at inverse temperature beta > 0, confined to lie on a Jordan curve. The partition function can include a linear statistic. The asymptotic formula involves a Fredholm determinant related to the Loewner energy of the curve, and also an expression involving the sampling function, the exterior conformal map for the curve and the Grunsky operator. The asymptotic formula also gives a central limit theorem for linear statistics of the particles in the gas.

For any integer m < n, where m can depend on n, we study the rate of convergence (Formula Presented) to its limiting Gaussian as n → ∞ for orthogonal, unitary and symplectic Haar distributed random matrices U of size n. In the unitary case, we prove that the total variation distance is less than (Formula Presented) times a constant. This result interpolates between the super-exponential bound obtained for fixed m and the 1/n bound coming from the Berry–Esseen theorem applicable when m ≥ n by a result of Rains. We obtain analogous results for the orthogonal and symplectic groups. In these cases, our total variation upper bound takes the form (Formula Presented) times a constant and the result holds provided n > 2m. For m = 1, we obtain complementary lower bounds and precise asymptotics for the L2-distances as n → ∞, which show how sharp our results are.

We show that the distance in total variation between (Tr U, √12 Tr U2, . . ., √m Tr Um) and a real Gaussian vector, where 1 U is a Haar distributed orthogonal or symplectic matrix of size 2n or 2n + 1, is bounded by 「 (2 mn + 1)− 12 times a correction. The correction term is explicit and holds for all n ≥ m4, for m sufficiently large. For n ≥ m3 we obtain the bound (mn)−c √ mn with an explicit constant c. Our method of proof is based on an identity of Toeplitz + Hankel determinants due to Basor and Ehrhardt, see (Oper. Matrices 3 (2009) 167–86), which is also used to compute the joint moments of the traces.

For any integer $m<n$, where $m$ can depend on $n$, we study the rate of convergence of $\frac{1}{\sqrt{m}}\tr \mathbf{U}^m$ to its limiting Gaussian as $n\to\infty$ for orthogonal, unitary and symplectic Haar distributed random matrices $\mathbf{U}$ of size $n$. In the unitary case, we prove that the total variation distance is less than $\Gamma(\floor{n/m}+2)^{-1}m^{-\floor{n/m}}\floor{n/m}^{1/4}\sqrt{\log n}$ times a constant. This result interpolates between the super-exponential bound obtained for fixed $m$ and the $1/n$ bound coming from the Berry-Esseen theorem applicable when $m\ge n$ by a result of Rains. We obtain analogous results for the orthogonal and symplectic groups. In these cases, our total variation upper bound takes the form $\Gamma(2\floor{n/m}+1)^{-1/2}m^{-\floor{n/m}+1}(\log n)^{1/4}$ times a constant and the result holds provided $n \geq 2m$. For $m=1$, we obtain complementary lower bounds and precise asymptotics for the $L^2$-distances as $n\to\infty$, which show how sharp our results are. 

We show that the distance in total variation between $(\tr U, \frac{1}{\sqrt{2}}\tr U^2, \cdots, \frac{1}{\sqrt{m}}\tr U^m)$ and a real Gaussian vector, where $U$ is a Haar distributed orthogonal or symplectic matrix of size $2n$ or $2n+1$, is bounded by $\Gamma(2\frac{n}{m}+1)^{-\frac{1}{2}}$ times a correction. The correction term is explicit and holds for all $n\geq m^4$, for $m$ sufficiently large. For $n\geq m^3$ we obtain the bound $(\frac{n}{m})^{-c\sqrt{\frac{n}{m}}}$ with an explicit constant $c$. Our method of proof is based on an identity of Toeplitz+Hankel determinants due to Basor and Ehrhardt, see \cite{BE}, which is also used to compute the joint moments of the traces.

We prove an asymptotic formula for the partition function of a 2d Coulomb gas at inverse temperature $\beta>0$ confined to lie on a Jordan curve. This also gives a central limit theorem for a linear statistic of the particles in the gas.  We obtain different expressions for the asymptotic mean and variance which involve either the exterior conformal mapping of the curve or the Grunsky operator.