Project C6: Random matrices and iterated function systems
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