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Universal scaling limits at the spectral singularity of structured random matrices

Markus Ebke, Torben Krüger

TL;DR

The paper analyzes local spectral statistics at a spectral singularity (origin) for Hermitian block random matrices with a Gaussian variance profile $S$ across $K$ blocks. By deriving a finite-$N$ resolvent integral via the SUSY method and superbosonization, it reduces the problem to a saddle-point analysis in the origin limit $z o 0$, and identifies two nontrivial block configurations: $K=2$ with $S=egin{pmatrix}s_{11}&s_{12}\ s_{12}&0\\end{pmatrix}$ and $K=3$ with $S=egin{pmatrix}s_{11}&s_{12}&s_{13}\ s_{12}&s_{22}&0\ s_{13}&0&0\\end{pmatrix}$. For these patterns, the microscopic scaling limits of the resolvent (hence the density of states) exist on scales $oldsymbol{ u}_N$ and are universal, depending only on the zero-pattern of $S$. The $K=2$ limit is expressed through Meijer $G$-functions and ${}_0F_2$, while the $K=3$ limit is described by modified Bessel functions with a pronounced logarithmic singularity at the origin. The work also explores a weak non-chirality regime where $S$ vanishes with $N$, revealing interpolating limits between the new universality classes and the chiral GUE, and establishing the methodological framework to obtain these results via a higher-dimensional saddle-point expansion. Overall, the paper provides a rigorous, functionally explicit description of new local universality classes at spectral singularities in structured random matrices with block variance profiles.

Abstract

The empirical spectral distribution of Hermitian $K \times K$-block random matrices converges to a deterministic density on the real line with a potential atom at the origin as the dimension of the blocks tends to infinity. In this model the variances of the entries depends on the block and the limiting density is determined by these variances. In the absence of an atom the density is either bounded or has a power law singularity at the origin. We determine all scaling limits of the spectral density on the eigenvalue spacing scale at this singularity for Gaussian matrices with block numbers $K \leq 3$. The appropriate scaling for the universal limit is correctly predicted by the global eigenvalue density. For $K=3$ the local one-point function exhibits an additional logarithmic singularity. The scaling limits depend only on the zero pattern within the variance profile, but not on the values of its positive entries.

Universal scaling limits at the spectral singularity of structured random matrices

TL;DR

The paper analyzes local spectral statistics at a spectral singularity (origin) for Hermitian block random matrices with a Gaussian variance profile across blocks. By deriving a finite- resolvent integral via the SUSY method and superbosonization, it reduces the problem to a saddle-point analysis in the origin limit , and identifies two nontrivial block configurations: with and with . For these patterns, the microscopic scaling limits of the resolvent (hence the density of states) exist on scales and are universal, depending only on the zero-pattern of . The limit is expressed through Meijer -functions and , while the limit is described by modified Bessel functions with a pronounced logarithmic singularity at the origin. The work also explores a weak non-chirality regime where vanishes with , revealing interpolating limits between the new universality classes and the chiral GUE, and establishing the methodological framework to obtain these results via a higher-dimensional saddle-point expansion. Overall, the paper provides a rigorous, functionally explicit description of new local universality classes at spectral singularities in structured random matrices with block variance profiles.

Abstract

The empirical spectral distribution of Hermitian -block random matrices converges to a deterministic density on the real line with a potential atom at the origin as the dimension of the blocks tends to infinity. In this model the variances of the entries depends on the block and the limiting density is determined by these variances. In the absence of an atom the density is either bounded or has a power law singularity at the origin. We determine all scaling limits of the spectral density on the eigenvalue spacing scale at this singularity for Gaussian matrices with block numbers . The appropriate scaling for the universal limit is correctly predicted by the global eigenvalue density. For the local one-point function exhibits an additional logarithmic singularity. The scaling limits depend only on the zero pattern within the variance profile, but not on the values of its positive entries.

Paper Structure

This paper contains 30 sections, 4 theorems, 142 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Main Results
  4. Integral Representation for Finite N
  5. SUSY Representation
  6. Evaluation of the Expectation
  7. Superbosonization
  8. Anti-Commuting Differentiation
  9. Large N Limit for z -> 0
  10. 2x2 case
  11. Transformation of the integrals
  12. Approximation of the u-integral
  13. Computation of the v- and w-integrals
  14. 3x3 case
  15. Transformation of the integrals
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $H$ be given by eq: matrix model definition. Then for all $z \in \mathbb{C}$ with $\mathop{\mathrm{Im}}\nolimits(z) > 0$ and all $N \in \mathbb{N}$ the expectation of the resolvent has the integral representation where $\mathrm{d} a$ is the $K$-dimensional Lebesgue-measure and the $b$-integral is $\oint^K \mathrm{d} b := \oint \mathrm{d} b_1 \dots \oint \mathrm{d} b_K$ where each $\oint \math

Figures (3)

  • Figure 2.1: Plots of the eigenvalue density of the random matrix $H$. The finite $N$ densities (colored lines) are histograms obtained from numerical simulations of \ref{['eq: matrix model definition']}. The asymptotic density $\rho_\infty$ (black line) is derived from \ref{['eq: Asymptotic density from saddle points']}. For comparison the gray dashed line is the semicircle law that would hold if all entries of $S$ were 1.
  • Figure 2.2: Plots of $\xi \mapsto K N \eta_N \rho_N(\eta_N \xi)$, the microscopic scaling of the one-point function at the origin. The graphs for finite $N$ (coloured lines) are histograms obtained from numerical simulations of \ref{['eq: matrix model definition']}. The limit $\lim_{N \to \infty} K N \eta_N \rho_N(\eta_N \xi)$ (black line) is derived from \ref{['eq: Stieltjes inversion']} and Theorem \ref{['thm: microscopic resolvent expectation']}.
  • Figure 2.3: Plots of $\xi \mapsto \lim_{N \to \infty} K N \eta_N \rho_N(\eta_N \xi)$ in the weak non-chirality limit derived from Theorem \ref{['thm: microscopic resolvent expectation weak non-chirality']}.

Theorems & Definitions (8)

  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • Remark 2.8