Time-inhomogeneous random Markov chains

Abstract

We consider Markov chains with random transition probabilities which, moreover, fluctuate randomly with time. We describe such a system by a product of stochastic matrices, U (t) = Mt · · · M1, with the factors Mi drawn independently from an ensemble of random Markov matrices, whose columns are independent Dirichlet random variables. The statistical properties of the columns of U (t), its largest eigenvalue and its spectrum are obtained exactly for N = 2 and numerically investigated for general N . For large t, the columns are Dirichlet-distributed, however the distribution is different from the initial one. As for the spectrum, we find that the eigenvalues converge to zero exponentially fast and investigate the statistics of the largest stability and Lyapunov exponents, which is are approximated by Gamma distributions. We also observe a concentration of the spectrum on the real line for large t.

Singular value distribution of dense random matrices with block Markovian dependence

“…It is shown that the singular values and eigenvalues of √ nP follow the appropriate universal laws. Different models for random transition matrices have been considered in [12,13,15,19,20,24,41,42,46,54,56,79].…”

“…Viewing i as a cycle on G i we let r 1 (i) be the number of edges which are traversed exactly once Figure 5. Visualization of a sequence of integers i for which P (i) has an asymptotically relevant contribution to (42). The corresponding undirected graph G i is the tree found by identifying the doubled edges.…”

On the number of real eigenvalues of a product of truncated orthogonal random matrices

“…Statistics of the real eigenvalues in particular have been shown to arise in many diverse areas, notably in connection to annihilating Brownian motions [43,42,37,16], random dynamical systems [23] and recently to an inverse scattering solution of the Zakharov-Shabat system [6]. Real eigenvalues of products have been applied to physical problems [31,26] and recently appeared in connection to random Markov matrices [27]. The analogue of the results discussed above for products of complex matrices also hold here: products of independent real Ginibre matrices are again described by a Pfaffian point process [28,18].…”

On the number of real eigenvalues of a product of truncated orthogonal random matrices

“…Statistics of the real eigenvalues have been shown to arise in many diverse areas, notably in connection to annihilating Brownian motions [41,40,35,15], random dynamical systems [22] and recently to an inverse scattering solution of the Zakharov-Shabat system [6]. Real eigenvalues of products have been applied to physical problems [30,25] and recently appeared in connection to random Markov matrices [26]. The analogue of the results discussed above for products of complex matrices also hold here: products of independent real Ginibre matrices are again described by a Pfaffian point process [27,17].…”