Scaling limits of complex and symplectic non-Hermitian Wishart ensembles

Abstract

Non-Hermitian Wishart matrices were introduced in the context of quantum chromodynamics with a baryon chemical potential. These provide chiral extensions of the elliptic Ginibre ensembles as well as non-Hermitian extensions of the classical Wishart/Laguerre ensembles. In this work, we investigate eigenvalues of non-Hermitian Wishart matrices in the symmetry classes of complex and symplectic Ginibre ensembles. We introduce a generalised Christoffel-Darboux formula in the form of a certain second-order differential equation, offering a unified and robust method for analyzing correlation functions across all scaling regimes in the model. By employing this method, we derive universal bulk and edge scaling limits for eigenvalue correlations at both strong and weak non-Hermiticity.

Figures (3)

• Figure 1: 100 realisations of eigenvalues of the complex and symplectic non-Hermitian Wishart matrices, where $\tau=0.5$, $\nu=1$ and $N=200$. Here, the full line represents the ellipse given in \ref{['S droplet']}. In both (A) and (B), the accumulation of eigenvalues near the origin is evident, cf. \ref{['MeasMu']}. Additionally, in (B), one can observe repulsion along the real axis.
• Figure 2: In the figure, the closed curve represents the ellipse \ref{['S droplet']}, and the red dots indicate its two foci, where $\tau=1/2$. The figure illustrates three different regions to which different forms of the strong asymptotics of the planar Laguerre polynomial $p_n$ in \ref{['LaguerrePolynomial']} apply: the critical regime near two foci $\{0,4\tau\}$ (inside the red circles); the oscillatory regime near the line segment connecting two foci (blue full line); and the exponential regime, which covers the rest of the complex plane.
• Figure 3: Graph of the function $z \mapsto \Omega(z)$. Here the dashed line indicates the ellipse \ref{['S droplet']}.

Theorems & Definitions (44)

• Theorem 1.1: Differential equations for correlation kernels of planar Wishart ensembles
• Theorem 1.2: Bulk/edge scaling limits of planar Wishart ensembles at strong non-Hermiticity
• Theorem 1.3: Bulk scaling limits of planar Wishart ensembles at weak non-Hermiticity
• Theorem 1.4: Edge scaling limits of planar Wishart ensembles at weak non-Hermiticity
• Remark 1.5: Scaling limits of the elliptic Ginibre ensembles
• Remark 1.6: Real orthogonal ensemble
• Proposition 2.1: Determinantal/Pfaffian point processes
• Lemma 2.2: Exponential regime
• Lemma 2.3: Oscillatory regime
• Lemma 2.4: Critical regime