Universality in spectral statistics of open quantum graphs

The quantum evolution maps of closed chaotic quantum graphs are unitary and known to have universal spectral correlations matching predictions of random matrix theory. In chaotic graphs with absorption the quantum maps become nonunitary. We show that their spectral statistics exhibit universality at the soft edges of the spectrum. The same spectral behavior is observed in many classical nonunitary ensembles of random matrices with rotationally invariant measures.

Figure 1

Shown on the left is the spectral density of 2-regular connectivity graphs ($\mu =1$) vs the universal function (1) [solid (blue) line]; the two dashed lines are analytic results from [18] for RMEs of truncated unitary matrices of the same dimensions. On the right are the spectral form factors for the same family of graphs plotted as dashed lines vs (2) [solid (blue) line].

Figure 2

The (nonrescaled) spectral densities of quantum maps $S^{\left(N\right)}{\mathrm{\Lambda }}_{\mathbf{\varphi }}$ vs asymptotics (1) (solid blue lines) for (a) doubly stochastic 10-regular graph with $N=1000$, where each vertex matrix ${\sigma }^v$ is fixed by a random choice of $10\times 10$ unitary matrix $u^v$, and (b) a damped de Brujin graph with $N=2^7$, $D_{i,j}={\delta }_{i,j}f(2j\pi /N)$, and $f\left(x\right)=3.2+sin\left(x\right)+sin\left(2x\right)+sin\left(3x\right)$. The parameters are $\mu =0.1542$ and ${\lambda }_1=12.6578$ for the outer edge and ${\mu }^{\prime }=0.3133$ and ${\lambda }_1^{\prime }=0.1952$ for the inner edge. The inner $\sqrt{1/{\lambda }_1^{\prime }}$ and outer $\sqrt{{\lambda }_1}$ radii are depicted by vertical solid (red) left and right lines. The (red) line in the middle shows the mean value of $logf\left(x\right)$, the point, where $\rho$ clusters at $N\to \infty$ (see [23]).

Figure 3

Diagram of periodic orbits with two 2-encounters $e_1,e_2$ and four loops $a,b,c,d$ ($m=1$) contributing to (3). The two orbits of the same length are represented by the sequences $\left[a{e}_{1}c{e}_{2}b{e}_{1}d{e}_2\right]$ and $\left[a{e}_{1}d{e}_{2}b{e}_{1}c{e}_2\right]$.