[Paper Review] On the moments of roots of Laguerre-polynomials and the Marchenko-Pastur law

Miklós Kornyik, György Michaletzky|arXiv (Cornell University)|Feb 16, 2016

Random Matrices and Applications4 citations

TL;DR

This paper establishes a direct link between the moments of the roots of Laguerre polynomials $ L_p^{(eta)} $ and the Marchenko-Pastur distribution by proving that the leading-order term in the $ k $-th power sum of the roots matches the $ k $-th moment of the Marchenko-Pastur law when $ \alpha_p/p \to c > -1 $. The proof relies on recursive properties of Laguerre polynomials and combinatorial path counting, confirming the weak convergence of the empirical root distribution to the Marchenko-Pastur law in the large-$ p $ limit.

ABSTRACT

In this paper we compute the leading terms in the sum of the $k^{th}$ power of the roots of $L_{p}^{(α)}$, the Laguerre-polynomial of degree $p$ with parameter $α$. The connection between the Laguerre-polynomials and the Marchenko-Pastur distribution is expressed by the fact, among others, that the limiting distribution of the empirical distribution of the normalized roots of the Laguerre-polynomials is given by the Marchenko-Pastur distribution. We give a direct proof of this statement based on the recursion satisfied by the Laguerre-polynomials. At the same time, our main result gives that the leading term in $p$ and $(α+p)$ of the sum of the $k^{th}$ power of the roots of $L_{p}^{(α)}$ coincides with the $k^{th}$ moment of the Marchenko-Pastur law. We also mention the fact that the expectation of the characteristic polynomial of a $XX^T$ type random covariance matrix, where $X$ is a $p imes n$ random matrix with iid elements, is $\ell^{(n-p)}_p$, i.e. the monic version of the $p^{th}$ Laguerre polynomial with parameter $n-p$.

Motivation & Objective

• To establish a direct analytical connection between the moments of roots of Laguerre polynomials and the Marchenko-Pastur distribution.
• To provide a non-complex-analytic proof of the weak convergence of the empirical distribution of normalized roots to the Marchenko-Pastur law.
• To compute the leading-order asymptotic behavior of the $ k $-th power sum of the roots of $ L_p^{(eta)} $ in terms of $ p $ and $ \alpha + p $.
• To clarify the role of Laguerre polynomials as generating functions for the characteristic polynomial of $ XX^T $ random covariance matrices.

Proposed method

• Uses the recursion satisfied by Laguerre polynomials to derive the asymptotic expansion of the $ k $-th power sum of their roots.
• Introduces a combinatorial path-counting model to identify coefficients in the asymptotic expansion, modeling root power sums as weighted sums over lattice paths.
• Defines a generating function $ \mathcal{M}_p^{(\alpha)}(z) $ based on the moments of normalized roots, linking it to the logarithmic derivative of the conjugate polynomial.
• Applies a fixed-point argument to the generating function in the limit $ p \to \infty $, showing convergence to the generating function of the Marchenko-Pastur law.
• Uses the fact that the expectation of the characteristic polynomial of $ XX^T $, where $ X $ is $ p \times n $ i.i.d. with mean 0 and variance 1, is the monic Laguerre polynomial $ \ell_p^{(n-p)} $.
• Employs a cut-and-glue transformation on lattice paths to count valid paths ending at $ (k,1) $, showing that the number of such paths with $ j $ upward steps is $ \frac{1}{k} \binom{k}{j} \binom{k}{j-1} $.

Experimental results

Research questions

• RQ1 Does the leading-order term in the $ k $-th power sum of the roots of $ L_p^{(\alpha)} $ coincide with the $ k $-th moment of the Marchenko-Pastur distribution?
• RQ2 Can the weak convergence of the empirical distribution of normalized roots to the Marchenko-Pastur law be proven using only recursion relations of Laguerre polynomials?
• RQ3 What is the exact asymptotic form of $ M_p^{(\alpha)}(k) = \sum_{i=1}^p (\xi_{p,i}^{(\alpha)})^k $ in terms of $ p $ and $ \alpha + p $?
• RQ4 How does the combinatorial structure of lattice paths relate to the coefficients in the moment expansion of Laguerre polynomial roots?

Key findings

• The leading-order term in $ M_p^{(\alpha)}(k) $ is $ \sum_{j=1}^k \frac{1}{k} \binom{k}{j} \binom{k}{j-1} p^j (\alpha + p)^{k-j+1} $, which matches the $ k $-th moment of the Marchenko-Pastur law.
• When $ \alpha_p / p \to c > -1 $, the empirical distribution of the normalized roots of $ L_p^{(\alpha_p)} $ converges weakly to the Marchenko-Pastur distribution $ \mu_c $.
• For $ c \geq 0 $, the limit measure $ \mu_c $ is absolutely continuous with density $ \frac{\sqrt{(x_+-x)(x-x_-)}}{2\pi x} \mathbf{1}_{[x_-,x_+]}(x) $, where $ x_\pm = (\sqrt{c+1} \pm 1)^2 $.
• Even when $ \alpha < 0 $ is not an integer (so roots may be complex), the $ k $-th moment of the empirical root distribution still converges to the $ k $-th moment of $ \mu_c $, though the limit measure is not supported on $ \mathbb{R}_+ $ in this case.
• The number of legal lattice paths ending at $ (k,1) $ with $ j $ upward edges is $ \frac{1}{k} \binom{k}{j} \binom{k}{j-1} $, which matches the coefficient of $ p^j (\alpha + p)^{k-j+1} $ in the asymptotic expansion.