David W Farmer, Francesco Mezzadri, Nina C Snaith
We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.
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Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group U(N) (also known as the CUE) and relate the moment to the solution of a Painleve differential equation. We also calculate a simple form for the asymptotic behaviour of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle.
We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szego weight and polynomials orthonormal on R with respect to varying weights and having the same union of intervals as the set of oscillations of asymptotics. In both cases we construct double infinite Jacobi matrices with generically quasiperiodic coefficients and show that each of them is an isospectral deformation of another. Related results on …
The fermionic, bosonic and supersymmetric variants of the colour-flavour transformation are derived for the orthogonal group. These transformations are then used to calculate the ensemble averages of characteristic polynomials of real random matrices.
We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like, equations. These equations are valid for matrices of arbitrary size. Their solutions can be given an integral representation that allows for a simple study of their asymptotic behaviors for a broad range of initial conditions.
This work provides explicit characterizations and formulae for the minimal polynomials of a wide variety of structured $4\times 4$ matrices. These include symmetric, Hamiltonian and orthogonal matrices. Applications such as the complete determination of the Jordan structure of skew-Hamiltonian matrices and the computation of the Cayley transform are given. Some new classes of matrices are uncovered, whose behaviour insofar as minimal polynomials are concerned, is remarkably similar to those of skew-Hamiltonian and Hamiltonian matrices. The main technique is the invocation of the associative al…
Particular class of skew orthogonal polynomials are introduced and investigated, which possess Laurent symmetry. They are also shown to appear as eigenfunctions of symplectic generalized eigenvalue problems. The modification of these polynomials gives some symplectic eigenvalue problem and the corresponding matrix is shown to be equivalent to butterfly matrix, which is a canonical form of symplectic matrices.
We prove that the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian real symmetric band matrices coincides with those for the Gaussian Orthogonal Ensemble (GOE). Here we adapt the approach of T. Shcherbina'14, where the case of 1D Hermitian random band matrices was considered.
Integer moments of the spectral determinant $|\det(zI-W)|^2$ of complex random matrices $W$ are obtained in terms of the characteristic polynomial of the Hermitian matrix $WW^*$ for the class of matrices $W=AU$ where $A$ is a given matrix and $U$ is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results in this context are discussed.
The paper continues previous works which study the behavior of second correlation function of characteristic polynomials of the special case of $n\times n$ one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Applying the transfer matrix approach, we study the case when the bandwidth $W$ is proportional to the threshold $\sqrt{n}$
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct discriminants and the determinant as the discriminant of order $d$, where $d$ is the dimension of the matrix. The characteristic polynomials and the Cayley--Hamilton theorem for higher rank matrices are obtained there from.
Using asymptotics of Toeplitz+Hankel determinants, we establish formulae for the asymptotics of the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices, as the matrix-size tends to infinity. Our results are analogous to those that Fahs obtained for random unitary matrices in [14]. A key feature of the formulae we derive is that the phase transitions in the moments of moments are seen to depend on the symmetry group in question in a significant way.
We consider the asymptotic behavior of the second mixed moment of the characteristic polynomials of the 1D Gaussian band matrices, i.e. of the hermitian matrices $H_n$ with independent Gaussian entries such that $< H_{ij}H_{lk}>=δ_{ik}δ_{jl}J_{ij}$, where $J=(-W^2\triangle+1)^{-1}$. Assuming that $W^2=n^{1+θ}$, $0<θ<1$, we show that this asymptotic behavior (as $n\to\infty$) in the bulk of the spectrum coincides with those for the Gaussian Unitary Ensemble.
We study the special case of $n\times n$ 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by $J=(-W^2\triangle+1)^{-1}$. Assuming that the band width $W\ll \sqrt{n}$, we prove that the limit of the normalized second mixed moment of characteristic polynomials (as $W, n\to \infty$) is equal to one, and so it does not coincides with those for GUE. This complements the previous result of T. Shcherbina and proves the expected crossover for 1D Hermitian random band matrices at $W\sim \sqrt{n}$ on the level of characteristic polynomials.
We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related with the theory of CMV matrices. It contains an arbitrary parameter which leads to a linear operator pencil. We show that the little and big -1 Jacobi polynomials are naturally obtained under this map from the Jacobi polynomials on the unit circle.
The Boubaker polynomials are investigated in this paper. Using Riordan matrices analysis, a sequence of relations outlining the relations with Chebyshev and Fermat polynomials have been obtained. The obtained expressions are a meaningful supply to recent applied physics studies using the Boubaker polynomials expansion scheme (BPES).
We present new and streamlined proofs of various formulae for products and ratios of characteristic polynomials of random Hermitian matrices that have appeared recently in the literature.
In a recent article we have discussed the connections between averages of powers of Riemann's $ζ$-function on the critical line, and averages of characteristic polynomials of random matrices. The result for random matrices was shown to be universal, i.e. independent of the specific probability distribution, and the results were derived for arbitrary moments. This allows one to extend the previous results to logarithmic moments, for which we derive the explicit universal expressions in random matrix theory. We then compare these results to various results and conjectures for $ζ$-functions, and …
One way of constructing explicit expressions of solutions of integrable systems of Partial Differential Equations (PDEs) goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to obtain such matrices in polynomial form. Our method is illustrated by applying it to the classical Massive Thirring Model (MTM), and by combining it with the Dihedral group of symmetries possessed by this model.
Banded totally positive matrices and normality for mixed multiple orthogonal polynomials
本文引入带状全正矩阵，证明其构成半群并含正双对角分解子集，进而通过谱Favard定理确保混合多重正交多项式在阶梯线上的正规性。
In these lecture notes we present some connections between random matrices, the asymmetric exclusion process, random tilings. These three apparently unrelated objects have (sometimes) a similar mathematical structure, an interlacing structure, and the correlation functions are given in terms of a kernel. In the basic examples, the kernel is expressed in terms of orthogonal polynomials.
Our main result asserts that a certain natural non-linear operator on Jacobi matrices built by a hyperbolic polynomial with real Julia set is a contraction in operator norm if the polynomial is sufficiently hyperbolic. This allows us to get for such polynomials the solution of a problem of Bellissard, in other words, to prove the limit periodicity of the limit Jacobi matrix. This fact does not require the iteration of the same fixed polynomial, and therefore it gives a wide class of limit periodic Jacobi matrices with singular continuous spectrum.
Spectral and factorization properties of oscillatory matrices leads to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue-Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given.
We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices.
Several well-known results from the random matrix theory, such as Wigner's law and the Marchenko--Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a role in this approach.
We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the characteristic polynomial of the conjugate matrix. Surprisingly, the results agree with that of the product of $M$ real or complex Ginibre matrices at finite-$N$, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two cha…
Banded Hermitian Matrices, Matrix Orthogonal Polynomials, and the Toda Lattice
研究有限带状厄米矩阵的谱理论，利用矩阵正交多项式从矩阵值测度重构矩阵，并建立测度为谱测度的充要条件，同时分析其与块三对角化算法及Toda格演化的联系。
Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices
本文首次在首阶意义上确定了Wigner矩阵和β-系综特征多项式的最大值，并在Gaussian-divisible情形下证明其紧性普适性，从而回答了谱的最优刚性问题，同时回答了Fyodorov--Hiary--Keating猜想。
Motivated by recent results in random matrix theory we will study the distributions arising from products of complex Gaussian random matrices and truncations of Haar distributed unitary matrices. We introduce an appropriately general class of measures and characterize them by their moments essentially given by specific Jacobi polynomials with varying parameters. Solving this moment problem requires a study of the Riemann surfaces associated to a class of algebraic equations. The connection to random matrix theory is then established using methods from free probability.
Formulae of Plancherel-Rotach type are established for the average characteristic polynomials of certain Hermitian products of rectangular Ginibre random matrices on the region of zeros. These polynomials form a general class of multiple orthogonal hypergeometric polynomials generalizing the classical Laguerre polynomials. The proofs are based on a multivariate version of the complex method of saddle points. After suitable rescaling the asymptotic zero distributions for the polynomials are studied and shown to coincide with the Fuss-Catalan distributions. Moreover, introducing appropriate coor…
Asymptotic freeness through unitaries generated by polynomials of Wigner matrices
本文研究确定性矩阵与独立Wigner矩阵自伴多项式处函数乘积的确定性近似与波动控制，并证明由Wigner矩阵多项式生成的量子演化在长时间下产生渐近自由性。
Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this conjecture is sufficient to explicitly calculate HOMFLY polynomials for all the 3-strand braids in arbitrary (anti)symmetric representations. We list the examples of so obtained polynomials for V=[3] and V=[4], and they are in accordance with the known answers for torus and figur…
Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems", Comm. Math. Phys. 207, pp. 589-620 (1999), we have shown that $m$-periodic sequences of weights lead to "moments", polynomials defined by determinants of matrices involving these moments and $2m+1$-step relations between them, thus leading to $2m+1$-band matrices $L$. Given a Darboux transformations on $L$, which effect does it have on the $m$-periodic sequence of weights and on the associated polynomials ? The…
We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on $\mathbb{Z}_+$ to higher dimension. We illustrate importance of this connection by proving…
We establish formulae for the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices in terms of certain lattice point count problems. This allows us to establish asymptotic formulae when the matrix-size tends to infinity in terms of the volumes of certain regions involving continuous Gelfand-Tsetlin patterns with constraints. The results we find differ from those in the unitary case considered previously
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