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Project C6: Random matrices and iterated function systems

Principal investigators

Participating scientists


The topic of this project is to study certain aspects of sequences of random functions with a focus on the following two types.
  • The spectrum of (products of) random matrices of increasing dimension with stochastically dependent entries.
  • The convergence and stationary regime of iterated function systems (IFS) which are compositions of random self-mappings of a given complete metric space (typically Rd). This includes products of random matrices as a special case.
The study of high dimensional random matrices originates in statistics, where random matrices occur as sample covariance matrices, and in theoretical physics. However, it is also of major inner-mathematical interest, e.g. via its connections to free probability and hence to von Neumann algebras, or to number theory. Our project is triggered by the observation that many limit theorems of classical probability theory that hold for sequences of independent and identically distributed (i.i.d., for short) random variables are (under suitable conditions) also true for stationary, ergodic sequences of random variables. Hence, the central problem, we want to analyze, is, whether the limit theorems for random matrices and their products due to Wigner, Voiculescu, or Tracy and Widom (among others) carry over to the case of dependent entries and if so, to what extent. On the one hand, it can be expected that for a quickly decaying dependency structure such as Markov random fields, one obtains the same or similar limit laws. On the other hand, it is not true that all symmetric random matrices with weakly correlated entries obey a Wigner semi-circle law, as can be seen, for example, from the fact that the spectral distribution of Wishart matrices converge to a Pastur-Marchenko law. The key challenge will thus be to provide appropriate conditions for random matrices with dependent entries that ensure that the limit laws for the case of independent entries are preserved. One of our main goals is to formulate such conditions in terms of stationary and ergodic random fields. We are partially motivated for this study by similarities between random matrix theory and number theory. It is well known that the distribution of the imaginary parts of the zeroes of the Riemann zeta function can be modeled by random matrix theory. This is the content of the Montgomery-Odlyzko law. However, the Riemann zeta function is not a random object and hence these relations, while fascinating, are unlikely to lead to proofs of properties of the Riemann zeta function. This may be different, if we can prove limit theorems for matrices with stationary entries, because interesting ergodic systems abound in number theory.
As for IFS, the goal is to understand their asymptotic behavior under appropriate smoothness and (global or local) contraction conditions on the chosen functions. The crucial assumption is that the random mechanism by which the functions are drawn is supposed to carry some kind of regenerative structure like i.i.d. or Markov modulated of finite order with a recurrent modulating (driving) Markov chain. A regenerative structure of such kind allows the use of renewal theoretic methods which are to be combined with new ones so as to deal with the nonlinear fluctuations of the system. Primary questions of interest include the validity of a power law behavior for the limiting distribution, as shown by Kesten for products of independent, identically distributed positive matrices, the rate of convergence towards this stationary regime, and sample path behavior. These are also the typical questions of interest in applications of IFS like Markov chain Monte Carlo simulation (Metropolis-Hastings algorithm, Gibbs sampler, Propp-Wilson algorithm) or the evolution of nonlinear time series used to describe population dynamics or financial data.
The project is therefore subdivided into the following subprojects.
  • The spectrum of large random matrices
  • Iterated function systems and renewal theory
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