Random sparse generators of Markovian evolution and their spectral properties

Abstract

The evolution of a complex multi-state system is often interpreted as a continuous-time Markovian process. To model the relaxation dynamics of such systems, we introduce an ensemble of random sparse matrices which can be used as generators of Markovian evolution. The sparsity is controlled by a parameter $\phi$, which is the number of non-zero elements per row and column in the generator matrix. Thus, a member of the ensemble is characterized by the Laplacian of a directed regular graph with $D$ vertices (number of system states) and $2\phi D$ edges with randomly distributed weights. We study the effects of sparsity on the spectrum of the generator. Sparsity is shown to close the large spectral gap that is characteristic of non-sparse random generators. We show that the first moment of the eigenvalue distribution scales as $\sim \phi$, while its variance is $\sim \sqrt{\phi }$. By using extreme value theory, we demonstrate how the shape of the spectral edges is determined by the tails of the corresponding weight distributions, and clarify the behavior of the spectral gap as a function of $D$. Finally, we analyze complex spacing ratio statistics of ultra-sparse generators, $\phi =\mathrm{const}$, and find that starting already at $\phi ⩾2$, spectra of the generators exhibit universal properties typical of Ginibre's Orthogonal Ensemble.

arXiv comment20 pages, 7 figures

References in corpus (58)

• Evolutionary games on graphs
• Localization of interacting fermions at high temperature
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Phase transitions in an exactly soluble one-dimensional exclusion process
• Nonequilibrium Statistical Mechanics of the Zero-Range Process and Related Models
• Non-equilibrium statistical mechanics: From a paradigmatic model to biological transport
• Spectral signatures of many-body localization with interacting photons
• Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices
• Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles
• Spectra of complex networks
• Complex Spacing Ratios: A Signature of Dissipative Quantum Chaos
• The asymmetric simple exclusion process: an integrable model for non-equilibrium statistical mechanics
• Eigenvalue statistics of the real Ginibre ensemble
• Exact Spectral Gaps of the Asymmetric Exclusion Process with Open Boundaries
• Sparse random matrices: the eigenvalue spectrum revisited
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Random matrices: Universality of local spectral statistics of non-Hermitian matrices
• Spectra of Sparse Random Matrices
• Universality classes of non-Hermitian random matrices
• Averaging Fluctuations in Resolvents of Random Band Matrices
• Universal spectra of random Lindblad operators
• Absorbing state phase transitions with quenched disorder
• A single defect approximation for localized states on random lattices
• Generalized Bose-Einstein condensation into multiple states in driven-dissipative systems
• Spectral gaps and mid-gap states in random quantum master equations
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• Statistical Properties of Random Banded Matrices with Strongly Fluctuating Diagonal Elements
• Anderson localization transition and eigenfunction multifractality in ensemble of ultrametric random matrices
• Spectral gap of the totally asymmetric exclusion process at arbitrary filling
• Statistics of Real Eigenvalues in Ginibre's Ensemble of Random Real Matrices
• Random Lindblad Dynamics
• Evolutionary games of condensates in coupled birth-death processes
• Delocalization and Diffusion Profile for Random Band Matrices
• Bethe Ansatz calculation of the spectral gap of the asymmetric exclusion process
• Spectral and Steady-State Properties of Random Liouvillians
• Analytic computation of the Instantaneous Normal Modes spectrum in low density liquids
• Cavity and replica methods for the spectral density of sparse symmetric random matrices
• Slowest relaxation mode of the partially asymmetric exclusion process with open boundaries
• Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
• Truncations of Random Orthogonal Matrices
• General Eigenvalue Correlations for the Real Ginibre Ensemble
• Random generators of Markovian evolution: A quantum-classical transition by superdecoherence
• Random transition-rate matrices for the master equation
• Mechanical Bounds to Transcriptional Noise
• Universal finite-size scaling amplitudes in anisotropic scaling
• Spectrum of Markov generators on sparse random graphs
• Spectrum of the totally asymmetric simple exclusion process on a periodic lattice -- first excited states
• Transient and persistent particle subdiffusion in a disordered chain coupled to bosons
• Random matrices with row constraints and eigenvalue distributions of graph Laplacians
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Relaxation rate of the reverse biased asymmetric exclusion process
• Glassy dynamics on networks: local spectra and return probabilities
• Methods for approximating stochastic evolutionary dynamics on graphs
• Breakdown of random matrix universality in Markov models
• From entropic to energetic barriers in glassy dynamics: The Barrat-Mézard trap model on sparse networks
• Perturbative solution for the spectral gap of the weakly asymmetric exclusion process
• Einstein relation for a driven disordered quantum chain in subdiffusive regime
• Rate-equation approach for multi-level quantum systems

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• Random matrices with row constraints and eigenvalue distributions of graph Laplacians
• Hammerstein equations for sparse random matrices