Eigenvalues Of Random Hermitian Matrices Research Articles

Tracy–Widom method for Jánossy density and joint distribution of extremal eigenvalues of random matrices

Abstract The Jánossy density for a determinantal point process is the probability density that an interval $I$ contains exactly $p$ points except for those at $k$ designated loci. The Jánossy density associated with an integrable kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ is shown to be expressed as a Fredholm determinant $\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel $\tilde{\mathbf{K}}\doteq (\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$. We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom’s criteria if $\mathbf{K}$ does, because of the structure that the map $(\varphi, \psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$ is a meromorphic $\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant sections. This observation enables application of the Tracy–Widom method [7] to Jánossy densities, expressed in terms of a solution to a system of differential equations in the endpoints of the interval. Our approach does not explicitly refer to isomonodromic systems associated with Painlevé equations employed in the preceding works. As illustrative examples we compute Jánossy densities with $k=1, p=0$ for Airy and Bessel kernels, related to the joint distributions of the two largest eigenvalues of random Hermitian matrices and of the two smallest singular values of random complex matrices.

Gaussian fluctuations of eigenvalues of random Hermitian matrices associated with fixed and varying weights

We investigate fluctuations of eigenvalues in the bulk for random Hermitian matrices associated with (i) a fixed weight on a compact interval, (ii) an exponential weight on a possibly unbounded interval and (iii) varying exponential weights. In particular, we show that the normalized difference between the [Formula: see text]th eigenvalue in the bulk, and the [Formula: see text]th zero of an appropriate orthogonal polynomial, is normally distributed in the bulk. This generalizes earlier work of Gustavsson and Zhang.

Local universality of repulsive particle systems and random matrices

We study local correlations of certain interacting particle systems on the real line which show repulsion similar to eigenvalues of random Hermitian matrices. Although the new particle system does not seem to have a natural spectral or determinantal representation, the local correlations in the bulk coincide in the limit of infinitely many particles with those known from random Hermitian matrices; in particular they can be expressed as determinants of the so-called sine kernel. These results may provide an explanation for the appearance of sine kernel correlation statistics in a number of situations which do not have an obvious interpretation in terms of random matrices.

Random matrix central limit theorems for nonintersecting random walks

We consider nonintersecting random walks satisfying the condition that the increments have a finite moment generating function. We prove that in a certain limiting regime where the number of walks and the number of time steps grow to infinity, several limiting distributions of the walks at the mid-time behave as the eigenvalues of random Hermitian matrices as the dimension of the matrices grows to infinity.

Random numbers and random matrices: Quantum chaos meets number theory

The statistical analysis of the eigenvalues of quantum systems has become an important tool in understanding the connections between classical and quantum physics. The statistical properties of the eigenvalues of a quantum system whose classical counterpart is integrable match those of random numbers. The eigenvalues of a chaotic classical system have statistical properties like those of the eigenvalues of random Hermitian matrices. The statistical properties of random numbers and eigenvalues of random Hermitian matrices are examined and the connection between these properties and the statistics of eigenvalues of quantum systems is illustrated, using the quantum standard map as an example. The relevance of these ideas to some problems in the theory of prime numbers is explored.

On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices

A stronger result on the limiting distribution of the eigenvalues of random Hermitian matrices of the form A + XTX*, originally studied in Marčenko and Pastur, is presented. Here, X( N × n), T( n × n), and A( N × N) are independent, with X containing i.i.d. entries having finite second moments, T is diagonal with real (diagonal) entries, A is Hermitian, and n/ N → c > 0 as N → ∞. Under additional assumptions on the eigenvalues of A and T, almost sure convergence of the empirical distribution function of the eigenvalues of A + XTX* is proven with the aid of Stieltjes transforms, taking a more direct approach than previous methods.

On the Distribution of Spacings Between Zeros of the Zeta Function

A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first ${10^5}$ zeros and for zeros number ${10^{12}} + 1$ to ${10^{12}} + {10^5}$ that are accurate to within $\pm {10^{ - 8}}$, and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave like eigenvalues of random Hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by relating them to the primes.

On the distribution of spacings between zeros of the zeta function

A numerical study of the distribution of spacings between zeros of the Riemann zeta function is presented. It is based on values for the first105{10^5}zeros and for zeros number1012+1{10^{12}} + 1to1012+105{10^{12}} + {10^5}that are accurate to within±10−8\pm {10^{ - 8}}, and which were calculated on the Cray-1 and Cray X-MP computers. This study tests the Montgomery pair correlation conjecture as well as some further conjectures that predict that the zeros of the zeta function behave like eigenvalues of random Hermitian matrices. Matrices of this type are used in modeling energy levels in physics, and many statistical properties of their eigenvalues are known. The agreement between actual statistics for zeros of the zeta function and conjectured results is generally good, and improves at larger heights. Several initially unexpected phenomena were found in the data and some were explained by relating them to the primes.