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Orthogonal polynomials and Jacobi matrices

There are deep and beautiful links between orthogonal polynomials, symmetric tridiagonal matrices, one dimensional Schrödinger operators, and the Hamburger moments problem, see e.g. Van Assche, Szeg Ho, Chihara, Dette and Studden. The relationships with birth-and-death processes on \( {\mathbb{Z}_+} \) are considered in Karlin and Mc Gregor, Karlin and Mc Gregor, Van Assche, Dette and Studden.

Fix a prescribed bounded sequence \( {(\gamma_n)_{n\geq0}} \) of positive reals and define the symmetric tridiagonal matrix

\[ \Gamma= \begin{pmatrix} 0 & \gamma_1 \\ \gamma_1 & 0 & \gamma_2 \\ & \gamma_2 & 0 & \gamma_3 \\ && \ddots & \ddots & \ddots \\ &&& \gamma_{n-2} & 0 & \gamma_{n-1} \\ &&&& \gamma_{n-1} & 0 \end{pmatrix} \ \ \ \ \ (1) \]

with eigenvalues \( {(\lambda_{n,k})_{1\leq k\leq n}} \) and empirical spectral distribution

\[ \mu_{\Gamma}=\frac{1}{n}\sum_{k=1}^n\delta_{\lambda_{n,k}}. \]

The characteristic polynomial \( {P_n} \) of \( {\Gamma} \) is the last term of the second order recursion

\[ P_{k+1}(x)=xP_k(x)-\gamma_k^2P_{k-1}(x) \]

for every \( {0\leq k\leq n-1} \), with \( {P_{-1}\equiv0} \) and \( {P_0\equiv1} \). From the classical theory, \( {(P_n)_{n\geq0}} \) is the sequence of orthogonal polynomials for a compactly supported probability measure \( {\nu} \) on \( {\mathbb{R}} \). Namely, for every \( {i,j\geq0} \),

\[ \int_{\mathbb{R}}\!P_i(x)P_j(x)\,\nu(dx)=\delta_{i,j}. \]

The law \( {\nu} \) is unique and solves a Hamburger moments problem. The matrix \( {\Gamma} \) and the law \( {\nu} \) are known respectively as the Jacobi matrix and the spectral measure of \( {(P_n)_{n\geq0}} \). The roots \( {(\lambda_{n,k})_{1\leq k\leq n}} \) of \( {P_n} \) are real, distinct, belong to the support of \( {\nu} \), and there exists one and only one root of \( {P_{n+1}} \) between two roots of \( {P_n} \) (interlacement). The Cauchy-Stieltjes transform of \( {\nu} \) is given by the continuous fraction

\[ \int_{\mathbb{R}}\!\frac{1}{z-x}\,\nu(dx) =\cfrac{1}{z-\cfrac{\gamma_1^2}{z-\cfrac{\gamma_2^2}{z-\ddots}}} \]

for all \( {z\in\mathbb{C}_+.} \) The \( {n^\text{th}} \) approximant of this continuous fraction is the Cauchy-Stieltjes transform of the Gauss-Christoffel quadrature \( {\nu_n} \) of \( {\nu} \). In other words, \( {\nu_n} \) is the discrete law on \( {\mathbb{R}} \) with atoms \( {(\lambda_{n,k})_{1\leq k\leq n}} \) which is equal to \( {\nu} \) as a linear form on all polynomials of degree \( {\leq n} \). If \( {(v_{n,k})_{1\leq k\leq n}} \) are the orthogonal eigenvectors of the symmetric matrix \( {\Gamma} \) associated to the eigenvalues \( {(\lambda_{n,k})_{1\leq k\leq n}} \), then from Proposition 1.1.9 in Hiai and Petz,

\[ \nu_n=\frac{1}{n}\sum_{k=1}^n\left<v_{n,k},e_1\right>^2\delta_{\lambda_{n,k}} \]

and

\[ \nu_n\underset{n\rightarrow\infty}{\longrightarrow}\nu \]

where the convergence is the weak convergence for continuous bounded functions and all polynomials (Wasserstein topology). It turns out that \( {\nu_n} \) is the so called distribution measure at vector \( {e_1} \) of the self–adjoint operator \( {\Gamma} \), while \( {\nu} \) is the distribution measure at vector \( {e_1} \) of the self–adjoint operator

\[ \overline{\Gamma}:\ell^2(\mathbb{N})\rightarrow\ell^2(\mathbb{N}) \]

where \( {\mathbb{N}=\{1,2,\ldots\}} \), see e.g. Example 1.1.1 and Proposition 1.1.9 in Hiai and Petz. Here \( {\overline{\Gamma}} \) is the operator which coincides with \( {\Gamma} \) on \( {\ell^2(\{1,\ldots,n\})} \) for every \( {n\geq1} \). In other words, for every \( {k\geq0} \),

\[ \left<\overline{\Gamma}^k e_1,e_1\right>_{\ell^2(\mathbb{N})} =\lim_{n\rightarrow\infty}\left<\Gamma^k e_1,e_1\right>_{\ell^2(\{1,\ldots,n\})} =\lim_{n\rightarrow\infty}\int_{\mathbb{R}}\!x^k\,\nu_n(dx) =\int_{\mathbb{R}}\!x^k\,\nu(dx). \]

One should not confuse \( {\nu_n} \) with \( {\mu_\Gamma} \). In general, \( {\mu_\Gamma} \) does not converge weakly to \( {\nu} \). Actually, a famous theorem by Erdös & Turán states that if \( {\nu} \) is supported on \( {[-1,+1]} \) with almost-everywhere positive density, then \( {\mu_\Gamma} \) tends weakly as \( {n\rightarrow\infty} \) to the arc–sine distribution on \( {[-1,+1]} \), see e.g. pages 24–25 in Van Assche. In fact the convergence holds for all moments.

In the special case where \( {\gamma_n=1} \) for every \( {n\geq1} \), then \( {\nu} \) turns out to be a Wigner semi–circle law on \( {[-1,+1]} \) and \( {(P_n)_{n\geq0}} \) are the Jacobi polynomials with parameters \( {\alpha=\beta=1/2} \), also known as the Chebyshev polynomials of the second kind. From the Erdős-Turán theorem, we get then a simple example for which \( {\nu} \) (Wigner semi–circle law on \( {[-1,+1]} \)) is not the weak limit \( {\mu} \) of \( {\mu_\Gamma} \) (arc–sine law on \( {[-1,+1]} \)). On the other hand, it is well known that the arc–sine law on \( {[-1,+1]} \) is a fixed point of the map \( {\nu\mapsto\mu} \) and \( {(P_n)_{n\geq0}} \) are in this case the Chebyshev polynomials of the first kind.

Recall that the recursive relations of the Chebyshev polynomials of the first and second kind differ only on the first coefficients. It is quite natural to consider the empirical measure of the roots \( {\mu_{Q_n}} \) associated to the sequence of random orthogonal polynomials \( {(Q_n)_{n\geq0}} \) defined by the recursive relation

\[ Q_{n+1}(x)=xQ_n(x)-W_nQ_{n-1}(x) \]

for every \( {n\geq1} \) with initial conditions \( {Q_0\equiv1} \) and \( {Q_1(x)=x} \), where this time \( {(W_n)_{n\geq1}} \) are i.i.d. positive random variables. For every \( {n\geq1} \), the random polynomial \( {Q_n} \) is the characteristic polynomial of the random matrix

\[ \begin{pmatrix} 0 & \sqrt{W_1} \\ \sqrt{W_1} & 0 & \sqrt{W_2} \\ &\sqrt{W_2} & 0 & \sqrt{W_3} \\ &&\ddots & \ddots &\ddots \\ &&&\sqrt{W_{n-2}}&0&\sqrt{W_{n-1}} \\ &&&&\sqrt{W_{n-1}}&0 \end{pmatrix}. \]

This matrix is the so called Jacobi matrix of the sequence \( {(Q_n)_{n\geq0}} \), and its empirical spectral distribution is \( {\mu_{Q_n}} \). Since this matrix is symmetric tridiagonal with i.i.d. entries, the method of moments allows to show that a.s. \( {\mu_{Q_n}} \) tends as \( {n\rightarrow\infty} \) to a non-random probability measure which depends on the common law of the \( {W_i} \)’s via its moments. Here again, the limit of \( {\mu_{Q_n}} \) as \( {n\rightarrow\infty} \) does not coincide in general with the (random) law for which \( {(Q_n)_{n\geq0}} \) are orthogonal.

Note: the content of this post is taken from arXiv 0811.1097v1, but was removed from the lightweight published version arXiv 0811.1097v2.

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