Random sparse generators of Markovian evolution and their spectral properties.

Area of Science:

• Complex systems dynamics
• Statistical physics
• Random matrix theory

Background:

• Complex multistate systems often follow continuous-time Markovian processes.
• Modeling relaxation dynamics requires generators of Markovian evolution.
• Nonsparse random generators exhibit large spectral gaps.

Purpose of the Study:

• To introduce an ensemble of random sparse matrices for modeling Markovian evolution.
• To investigate the impact of sparsity on the spectrum of generator matrices.
• To analyze the spectral gap and eigenvalue distributions as a function of sparsity.

Main Methods:

• Ensemble of random sparse matrices with controlled sparsity (parameter φ).
• Characterization via Laplacian of directed regular graphs with random weights.
• Application of extreme value theory to analyze spectral edges.
• Analysis of complex spacing ratio statistics for ultrasparse generators.

Main Results:

• Sparsity closes the large spectral gap characteristic of nonsparse generators.
• The first moment of the eigenvalue distribution scales as ∼φ; variance scales as ∼sqrt[φ].
• Spectral edge shapes depend on weight distribution tails; spectral gap behavior clarified as a function of D.
• Ultrasparse generators (φ⩾2) exhibit universal spectral properties akin to the Ginibre orthogonal ensemble.

Conclusions:

• Sparsity is a crucial factor in determining the spectral properties of Markovian evolution generators.
• The introduced ensemble provides a tunable model for studying relaxation dynamics in complex systems.
• The findings offer insights into the transition to universal spectral behavior in sparse random matrices.