Spectra of Sparse Non-Hermitian Random Matrices: An Analytical Solution

We present the exact analytical expression for the spectrum of a sparse non-Hermitian random matrix ensemble, generalizing two standard results in random-matrix theory: this analytical expression constitutes a non-Hermitian version of the Kesten-McKay measure as well as a sparse realization of Girko’s elliptic law. Our exact result opens new perspectives in the study of several physical problems modelled on sparse random graphs, which are locally treelike. In this context, we show analytically that the convergence rate of a transport process on a very sparse graph depends in a nonmonotonic way upon the degree of symmetry of the graph edges.

Figure 1

An example of a Bethe lattice and a polarized Bethe lattice, both with degree $k=2$. The matrix elements corresponding with the red (blue) lines are given by $A_{+}(A_{-})$.

Figure 2

Direct diagonalization results of matrices of size $\mathcal{O}(1e+3)$ (symbols) compared with the support (8) (lines) for $k=3$, $p_{+}=1$ and different values of $p_{-}$.

Figure 3

Three cuts of ${\rho }_0(\lambda )$, Eq. (7), along the real direction are compared with direct diagonalization results of matrices of size $\mathcal{O}(1e+3)$, averaged over $\mathcal{O}(1e+4)$ samples. The parameters are $k=3$, $p_{+}=1$, and $p_{-}=0.5$.

Figure 4

Direct diagonalization results of matrices of size $\mathcal{O}(1e+3)$ (symbols) compared with the rotated support of Eq. (8) (solid lines) for $k=3$, $p_{+}=1$, ${\theta }_{+}=0$, $p_{-}=0.2$, and different values of ${\theta }_{-}$. The orange circle denotes the eigenvalues isolated from the bulk, given by $k(A_{+}+A_{-})$. We notice the nontrivial rotation of the bulk spectrum of an angle $\theta$, different from the rotation of the isolated eigenvalue of an angle $2\theta$ around $k$.

Figure 5

Spectral gap as a function of the degree of symmetry $\alpha =p_{-}/p_{+}$ for $p_{+}=1$ and $k$ given. When $k\to \infty$, the curves become straight lines from 0.5 to 1.