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The evolution of a complex multi-state system is often interpreted as acontinuous-time Markovian process. To model the relaxation dynamics of suchsystems, we introduce an ensemble of random sparse matrices which can be usedas generators of Markovian evolution. The sparsity is controlled by a parameter$\varphi$, which is the number of non-zero elements per row and column in thegenerator matrix. Thus, a member of the ensemble is characterized by theLaplacian of a directed regular graph with $D$ vertices (number of systemstates) and $2 \varphi D$ edges with randomly distributed weights. We study theeffects of sparsity on the spectrum of the generator. Sparsity is shown toclose the large spectral gap that is characteristic of non-sparse randomgenerators. We show that the first moment of the eigenvalue distribution scalesas $\sim \varphi$, while its variance is $\sim \sqrt{\varphi}$. By usingextreme value theory, we demonstrate how the shape of the spectral edges isdetermined by the tails of the corresponding weight distributions, and clarifythe behavior of the spectral gap as a function of $D$. Finally, we analyzecomplex spacing ratio statistics of ultra-sparse generators, $\varphi =\mathrm{const}$, and find that starting already at $\varphi \geqslant 2$,spectra of the generators exhibit universal properties typical of Ginibre'sOrthogonal Ensemble.

Random sparse generators of Markovian evolution and their spectral properties