  The nearest-neighbor level-spacing distributions are a fundamental quantity
of disordered systems and universal. It is well-known that extended and
localized states of random Hermitian systems follow the Wigner-Dyson and the
Poison distributions, respectively, while the Ginibre distributions describe
random non-Hermitian systems with complex eigenvalues. However, the level
distribution of systems of neither Hermitian nor non-Hermitian with full
complex eigenvalues is still unknown. Here we show a new class of universal
level distributions in the vicinity of exceptional points of non-Hermitian
Hamiltonians. Two universal distribution functions, $P_{\text{SP}}(s)$ for the
symmetry-preserved phase and $P_{\text{SB}}(s)$ for the symmetry-broken phase,
are needed to describe the nearest-neighbor level-spacing distributions near
exceptional points. Surprisingly, both $P_{\text{SP}}(s)$ and
$P_{\text{SB}}(s)$ are proportional to $s$ for small $s$, or a linear level
repulsion, in contrast to the cubic level repulsions of the Ginibre ensembles.
For non-Hermitian disordered tight-binding Hamiltonians, $P_{\text{SP}}(s)$ and
$P_{\text{SB}}(s)$ can be well described by
$P_{\text{SP(SB)}}(s)=\tilde{c}_1s\exp[-\tilde{c}_2s^{\tilde{\alpha}}]$ in the
thermodynamic limit (of infinite systems) with a constant $\tilde{\alpha}$ that
depends on the localization nature of states at exceptional points.

Disordered Systems and Neural Networks