Download Random Matrix Universality, Spectral Measures, and Composite Media

Random Matrix Universality, Spectral Measures,
and Composite Media
We consider composite media with a broad range of scales, whose effective properties are
important in materials science, biophysics, and climate modeling. Examples include random
resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean
eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The
analytic continuation method provides a rigorous approach to treating the effective properties of
such systems. At the heart of this method is a random matrix which depends only on the
composite geometry. In this lecture we will discuss computations of the spectral measures of
this operator which yield effective properties, as well as statistical measures of its eigenvalues.
In particular, the effective transport coefficients of these systems often exhibit large changes
associated with transitions in the connectedness or percolation properties of a particular phase.
We will demonstrate that this critical behavior of transport can be characterized by the collapse
of gaps in the spectral measures, underlying Stieltjes integral representations for the effective
transport coefficients. For example, in the case of a random resistor network with a low volume
fraction, there are spectrum free regions at the spectral endpoints. However, as the percolation
transition is approached and the system becomes increasingly connected, these gaps in the
spectrum shrink and then vanish. This leads to the formation of delta-components of the
measures at the spectral endpoints, precisely at the percolation threshold, and to critical
behavior of the integral representation formulas for the effective transport coefficients. We will
discuss how this gap behavior of the measures provides a detailed description of critical
transport transitions, and how we used it to recover all of the classical critical exponent scaling
relations without use of heuristic scaling forms. Instead we used the exact integral representation
formulas for the effective transport coefficients, involving spectral measures of random matrices.
The random matrix theory description of such connectedness transitions in composites provides
a mechanism for the gap behavior of the spectral measures. For example, consider again the
case of a low volume fraction random resistor network. We will demonstrate that the statistics of
eigenvalues of the underlying random matrix are governed by that of weakly correlated Poissonlike spectra, and have weak eigenvalue repulsions typical of such spectra. This allows the
eigenvalues to coalesce in high density configurations, leading to “geometric” resonances and
spectral gaps in the measures. However, as the percolation transition is approached and the
system becomes increasingly connected, the eigenvalues become increasingly correlated with
statistics approaching that of Wigner—Dyson spectra, and they repel one another with large
eigenvalue repulsions typical of such spectra. This forces the eigenvalues to spread out,
resulting in the collapse of the spectral gaps. We will show that this transitional behavior, from
Poisson-like statistics toward Wigner—Dyson statistics of eigenvalues, is exhibited by various
eigenvalue statistics describing the long and short range correlations of the eigenvalues, and
that this behavior is pervasive in a variety of composite systems. Moreover, this transitional
behavior is captured by a one parameter universality class of random matrix ensembles.
Remarkably, this mathematical framework also describes the effective electromagnetic transport
properties of polycrystalline media, as well as the effective thermal transport properties of
advection enhanced diffusion by steady and dynamic fluid velocity fields. Each formulation
yielding Stieltjes integral representations for the effective transport coefficients of the system,
involving spectral measures of random matrices. We will discuss the formulations of these
effective parameter problems as well as computations of the spectral measures and eigenvalue
statistics associated with the underlying random matrices.