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.
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 study eigenvalues of unitary invariant random matrices and other de-terminantal point processes. Paper A investigates some generalizations ofthe Gaussian Unitary Ensemble which are motivated by the physics of freefermions. We show that these processes exhibit a transition from Poisson tosine statistics at mesoscopic scales and that, at the critical scale, fluctuationsare not Gaussian but are governed by complicated limit laws. In papers Band C, we prove limit theorems which cover the different regimes of randommatrix theory. In particular, this establishes universality of the fluctuations ofinvariant Hermitian random matrices in great generality. The techniques arebased on generalizations of the orthogonal polynomial method and the cumu-lant method developed by Soshnikov. In particular, the results rely on certaincombinatorial identities originating in the theory of random walks and on theasymptotics for Orthogonal polynomials coming from the Riemann-Hilbertsteepest descent introduced by Deift et al.
We study the fluctuations of certain biorthogonal ensembles for which the underlying family {P, Q} satisfies a finite term recurrence relation of the form xP(x) = JP(x). For polynomial linear statistics of such ensembles, we reformulate the cumulant method introduced in [53] in terms of counting certain lattice paths on the adjacency graph of the recurrence matrix J. In the spirit of [12], we show that the asymptotic fluctuations are described by the right-limits of the matrix J. Moreover, whenever the right-limit is a Laurent matrix, we show that the CLT is equivalent to Soshnikov's main combinatorial lemma. We discuss several applications to unitary invariant Hermitian random matrices. In particular, we provide a general central limit theorem (CLT) and a law of large numbers in the one-cut regime. We also prove a CLT for the square singular values of the product of independent complex rectangular Ginibre matrices, as well as for the Laguerre and Jacobi biorthogonal ensembles introduced in [7], and we explain how to recover the equilibrium measure from the asymptotics of the recurrence coefficients. Finally, we discuss the connection with the Strong Szegto limit theorem where this combinatorial method originates.
Considering a determinantal point process on the real line, we establish a connection between the sine-kernel asymptotics for the correlation kernel and the CLT for mesoscopic linear statistics. This implies universality of mesoscopic fluctuations for a large class of unitary invariant Hermitian ensembles. In particular, this shows that the support of the equilibrium measure need not be connected in order to see Gaussian fluctuations at mesoscopic scales. Our proof is based on the cumulants computations introduced in [45] for the CUE and the sine process and the asymptotic formulae derived by Deift et al. [13]. For varying weights e(-N) (Tr) (V) ((H)), in the one-cut regime, we also provide estimates for the variance of linear statistics Tr f (H) which are valid for a rather general function f. In particular, this implies that the logarithm of the absolute value of the characteristic polynomials of such Hermitian random matrices converges in a suitable regime to a regularized fractional Brownian motion with logarithmic correlations introduced in [17]. For the GUE and Jacobi ensembles, we also discuss how to obtain the necessary sine-kernel asymptotics at mesoscopic scale by elementary means.