Cusp Universality for Correlated Random Matrices

Abstract

For correlated real symmetric or complex Hermitian random matrices, we prove that the local eigenvalue statistics at any cusp singularity are universal. Since the density of states typically exhibits only square root edge or cubic root cusp singularities, our result completes the proof of the Wigner-Dyson-Mehta universality conjecture in all spectral regimes for a very general class of random matrices. Previously only the bulk and the edge universality were established in this generality [arXiv:1804.07744], while cusp universality was proven only for Wigner-type matrices with independent entries [arXiv:1809.03971, arXiv:1811.04055]. As our main technical input, we prove an optimal local law at the cusp using the Zigzag strategy, a recursive tandem of the characteristic flow method and a Green function comparison argument. Moreover, our proof of the optimal local law holds uniformly in the spectrum, thus also re-establishing universality of the local eigenvalue statistics in the previously studied bulk [arXiv:1705.10661] and edge [arXiv:1804.07744] regimes.

Figures (4)

• Figure 1: The left panel depicts several trajectories of the flow \ref{['eq:z_flow']} that terminate at the scale curve$\rho_T(z)N\Im z = c$ (solid black line), while the the graph of scDOS $\rho_T$ is superimposed in light blue. The right panel depicts trajectories up to an intermediate time $t\in(0,T)$ with their continuations beyond $t$ shown as thin dotted lines. The pre-image of the scale curve at the time $t$ is depicted as a solid black line, and the scale curve itself is depicted as a dashed black line. The graph of scDOS $\rho_t$ is superimposed in light blue. In both panels, the black markers along the trajectories of \ref{['eq:z_flow']} are evenly spaced in time.
• Figure 2: The time-dependent domain $\mathcal{D}^\mathrm{abv}_{t}$, defined in \ref{['eq:abvD_t']}, is illustrated in blue at three distinct times: the initial time $t = 0$ (left), an intermediate time $0 < t < T$ (center), and the terminal time $t=T$ (right). The graph of the scDOS $\rho_t$ is superimposed in black on each panel (not to scale).
• Figure 3: Schematic representation of the Zigzag induction. The random matrices $H_k, H^k$, as defined in \ref{['eq:H_k_defs']}, are situated within an abstract coordinate system. The horizontal axis represents the size of the Gaussian component, while the vertical axis indicates the lower bound on $\rho(z)^{-1}\eta$ in the domains, c.f. \ref{['eq:abvD_t']}, where we prove the local laws \ref{['eq:ll_template']}. Solid arrows denote applications of Proposition \ref{['prop:zig']} (referred to as Zig steps), and dashed arrows indicate applications of Proposition \ref{['prop:zag']} (Zag steps).
• Figure 4: Shaded in blue is the illustration of the time-dependent domain $\mathcal{D}^\mathrm{sub}_{t}$, defined in \ref{['eq:below_Dt']}, at three distinct times: the initial time $t = 0$ (left), an intermediate time $0 < t < T$ (center), and the terminal time $t=T$ (right). The domain $\mathcal{D}^\mathrm{abv}_{t}$ at the corresponding time $t$ is indicated with crosshatching in the zoomed-in insert, with its boundary indicated by a dashed line in the main plot. The zoomed-in insert also depicts the distance $f(t)$, defined in \ref{['eq:front_func']}, between the edge of the support of $\rho_t$ and the corresponding horizontal cut-off of the domain $\mathcal{D}^\mathrm{sub}_{t}$. The graph of the scDOS $\rho_t$ is superimposed in black on each panel (not to scale).

Theorems & Definitions (53)

• Example 2.6: Polynomially Decaying Metric Correlation Structure
• Definition 2.7: Local fluctuation scale
• Theorem 2.8: Optimal Local Laws
• Theorem 2.9: No eigenvalues outside the support of the scDOS
• Corollary 2.10: Eigenvector delocalization
• Corollary 2.11: Band rigidity and eigenvalue rigidity
• Remark 2.12: Integer mass
• Theorem 2.13: Cusp universality for correlated random matrices
• Remark 2.14: On $p_{k, \alpha}^{\rm GUE/GOE}$
• Definition 3.1: Local Laws