Three non-Hermitian random matrix universality classes of complex edge statistics: Spacing ratios and distributions

Abstract

The conjectured three generic local bulk statistics amongst all non-Hermitian random matrix symmetry classes have recently been extended to three generic local edge statistics. We study analytically and numerically complex spacing ratios and nearest-neighbour (NN) spacing distributions that characterise such local statistics. We choose the three simplest representatives of these universality classes, given by the Gaussian ensembles of complex Ginibre, complex symmetric and complex self-dual matrices, denoted by class A, AI$^†$ and AII$^†$. In the first part, we analytically study the complex spacing ratio in class A, at finite matrix size $N$. Introducing a conditional point process, we simplify existing expressions and show why an uncontrolled approximation introduced earlier converges well in the large-$N$ limit in the bulk. When specifying to the elliptic Ginibre ensemble, we present a parameter-dependent $N=3$ surmise for the complex spacing ratio, interpolating to that of the Gaussian unitary ensemble (GUE), where such a surmise is very accurate. In the second numerical part, we compare complex spacing ratios, its moments, and NN spacing distributions for all three ensembles with that of uncorrelated points, the two-dimensional (2D) Poisson process, both in the bulk and at the edge. The varying degree of repulsion within these different edge universality classes can be well understood in terms of an effective 2D Coulomb gas description, at different values of inverse temperature $β$. We find indications that the complex spacing ratio does not fully unfold the local statistics at the edge. Finally we verify that for small argument, in all three symmetry classes the NN spacing distributions in the bulk and at the edge are consistent with the universal cubic repulsion.

Figures (13)

• Figure 1: The unconditional $N=3$ surmise for the eGinUE $\varrho_{\rm A}^{(N=3)}(x+\mathrm{i} y;\tau)$ from \ref{['eq.rho3final']} for the complex spacing ratio (left), its radial (middle) and angular marginal distribution centred at $\theta=0$, with $\theta\in(-\pi,\pi]$ (right), at $\tau=0,0.5,0.99$ (top, middle bottom). Note that the colour coding of the 2D density in the left column strongly depends on $\tau$, as the eigenvalues concentrate more around the $x$-axis with increasing $\tau\nearrow 1$.
• Figure 2: The same plots as in Fig. \ref{['fig:rho3_uncond']} in the conditional eGinUE, $\varrho_{\rm A,C}^{(3)}(x+\mathrm{i} y;\tau)$ from \ref{['eq.rho3Cfinal']}, with $\theta\in[0,2\pi)$. Notice that here the angular marginal distribution is centred at $\theta=\pi$ (right column).
• Figure 3: Left: normal NN and NNN ordering of consecutive eigenvalues on the real line, with labels $k_0=k+1$, and $k_1=k-1$ in \ref{['def.k0k1']}, right: NN and NNN both to the right with labels $k_0=k+1$, $k_1=k+2$.
• Figure 4: The radial spectral density of class A \ref{['R1A']} (blue full curve: analytical; purple triangle: numerical), class AI$^\dagger$\ref{['R1AI+']} (red dashed curve: analytical; orange circle: numerical), both at $N=1024$, and of class AII$^\dagger$ (green diamond: numerical only) with $2N=1024$, with $720\,000$ samples for all ensembles. All 3 ensembles share a constant density in the bulk and show differences at the edge. For our value of $N$ we define the bulk at radius $r_b=0.8$ (orange dash-dotted vertical line, left). The right picture shows details at the edge, with the region to the right of the solid vertical line at radius $r_-=1-1/\sqrt{1024} = 0.96875$ defined as edge, and the dashed vertical line at radius $r_-^\prime=1-2/\sqrt{1024} = 0.9375$ as extended edge.
• Figure 5: The radial spectral density of the 2D Poisson process for $N=1024$, analytically (black full line) and from $720\,000$ ensembles (orange circles). Notice the excellent agreement between the two overlapping curves. The vertical lines show our choice for the bulk $r_b=0.8$ (orange dash-dotted), the edge $r_-=1-1/\sqrt{N}$ (black full line) and the extended edge $r_-^\prime=1-2/\sqrt{N}$ (black dashed).