On the Outlying Eigenvalues of a Polynomial in Large Independent Random Matrices

Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent Hermitian random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu$ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu ,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\eta \ (\textrm{sometimes denoted}\ P^\square(\mu,\nu))$ and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu ,\nu ,P$, and the spikes of $A_N$. We establish a similar result when $B_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

Free probability theory

This article focuses on free probability theory, which is useful for dealing with asymptotic eigenvalue distributions in situations involving several matrices. In particular, it considers some of the basic ideas and results of free probability theory, mostly from the random matrix perspective. After providing a brief background on free probability theory, the article discusses the moment method for several random matrices and the concept of freeness. It then gives some of the main probabilistic notions used in free probability and introduces the combinatorial theory of freeness. In this theory, freeness is described in terms of free cumulants in relation to the planar approximations in random matrix theory (RMT). The article also examines free harmonic analysis, second-order freeness, operator-valued free probability theory, further free-probabilistic aspects of random matrices, and operator algebraic aspects of free probability.

Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem

We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Fréchet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson’s selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: Let {X_{1}^{(N)},\dots,X_{n}^{(N)}} be selfadjoint {N\times N} random matrices which are, for {N\to\infty} , asymptotically free. Consider a selfadjoint polynomial p in n non-commuting variables and let {P^{(N)}} be the element {P^{(N)}=p(X_{1}^{(N)},\dots,X_{n}^{(N)})} . How can we calculate the asymptotic eigenvalue distribution of {P^{(N)}} out of the asymptotic eigenvalue distributions of {X_{1}^{(N)},\dots,X_{n}^{(N)}} ?

Analysis of fast boundary-integral approximations for modeling electrostatic contributions of molecular binding

We analyze and suggest improvements to a recently developed approximate continuum-electrostatic model for proteins. The model, called BIBEE/I (boundary-integral based electrostatics estimation with interpolation), was able to estimate electrostatic solvation free energies to within a mean unsigned error of 4% on a test set of more than 600 proteins¶a significant improvement over previous BIBEE models. In this work, we tested the BIBEE/I model for its capability to predict residue-by-residue interactions in protein–protein binding, using the widely studied model system of trypsin and bovine pancreatic trypsin inhibitor (BPTI). Finding that the BIBEE/I model performs surprisingly less well in this task than simpler BIBEE models, we seek to explain this behavior in terms of the models’ differing spectral approximations of the exact boundary-integral operator. Calculations of analytically solvable systems (spheres and tri-axial ellipsoids) suggest two possibilities for improvement. The first is a modified BIBEE/I approach that captures the asymptotic eigenvalue limit correctly, and the second involves the dipole and quadrupole modes for ellipsoidal approximations of protein geometries. Our analysis suggests that fast, rigorous approximate models derived from reduced-basis approximation of boundaryintegral equations might reach unprecedented accuracy, if the dipole and quadrupole modes can be captured quickly for general shapes.