IST Austria Thesis

The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations

IST Austria Thesis

In the first part of this thesis we consider large random matrices with arbitrary expectation and a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic control of a multivariate cumulant expansion. In the second part we consider Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are uni- versal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta universality conjecture for the last remaining universality type. Our analysis holds not only for exact cusps, but approximate cusps as well, where an ex- tended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow- nian motion to the cusp regime. In the third and final part we explore the entrywise linear statistics of Wigner ma- trices and identify the fluctuations for a large class of test functions with little regularity. This enables us to study the rectangular Young diagram obtained from the interlacing eigenvalues of the random matrix and its minor, and we find that, despite having the same limit, the fluctuations differ from those of the algebraic Young tableaux equipped with the Plancharel measure

Modelling elastic structures with strong nonlinearities with application to stick-slip friction

An exact transformation method is introduced that reduces the governing equations of a continuum structure coupled to strong nonlinearities to a low dimensional equation with memory. The method is general and well suited to problems with point discontinuities such as friction and impact at point contact. It is assumed that the structure is composed of two parts: a continuum but linear structure and finitely many discrete but strong nonlinearites acting at various contact points of the elastic structure. The localised nonlinearities include discontinuities, e.g., the Coulomb friction law. Despite the discontinuities in the model, we demonstrate that contact forces are Lipschitz continuous in time at the onset of sticking for certain classes of structures. The general formalism is illustrated for a continuum elastic body coupled to a Coulomb-like friction model

A New Strategy for Exact Determination of the Joint Spectral Radius

Computing the joint spectral radius of a finite matrix family is, though interesting for many applications, a difficult problem. This work proposes a method for determining the exact value which is based on graph-theoretical ideas. In contrast to some other algorithms in the literature, the purpose of the approach is not to find an extremal norm for the matrix family. To validate that the finiteness property (FP) is satisfied for a certain matrix product, a tree is to be analyzed whose nodes code sets of matrix products. A sufficient, and in certain situations also necessary, criterion is given by existence of a finite tree with special properties, and an algorithm for searching such a tree is proposed. The suggested method applies in case of several FP-products as well and is not limited to asymptotically simple matrix families. In the smoothness analysis of subdivision schemes, joint spectral radius determination is crucial to detect Hölder regularity. The palindromic symmetry of matrices, which results from symmetric binary subdivision, is considered in the context of set-valued trees. Several illustrating examples explore the capabilities of the approach, consolidated by examples from subdivision

The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Amp\ue8re equations

We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Amp\ue8re type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited

Examples of compact Einstein four-manifolds with negative curvature

We give new examples of compact, negatively curved Einstein manifolds of dimension $4$. These are seemingly the first such examples which are not locally homogeneous. Our metrics are carried by a sequence of 4-manifolds $(X_k)$ previously considered by Gromov and Thurston. The construction begins with a certain sequence $(M_k)$ of hyperbolic 4-manifolds, each containing a totally geodesic surface $\Sigma_k$ which is nullhomologous and whose normal injectivity radius tends to infinity with $k$. For a fixed choice of natural number $l$, we consider the $l$-fold cover $X_k \to M_k$ branched along $\Sigma_k$. We prove that for any choice of $l$ and all large enough $k$ (depending on $l$), $X_k$ carries an Einstein metric of negative sectional curvature. The first step in the proof is to find an approximate Einstein metric on $X_k$, which is done by interpolating between a model Einstein metric near the branch locus and the pull-back of the hyperbolic metric from $M_k$. The second step in the proof is to perturb this to a genuine solution to Einstein's equations, by a parameter dependent version of the inverse function theorem. The analysis relies on a delicate bootstrap procedure based on $L^2$ coercivity estimates.Comment: 53 pages. v2 small modifications to exposition. v3 typos corrected. Same text as published version, to appear in the Journal of the American Mathematical Societ

From Monge-Ampere equations to envelopes and geodesic rays in the zero temperature limit

Let X be a compact complex manifold equipped with a smooth (but not necessarily positive) closed form theta of one-one type. By a well-known envelope construction this data determines a canonical theta-psh function u which is not two times differentiable, in general. We introduce a family of regularizations of u, parametrized by a positive number beta, defined as the smooth solutions of complex Monge-Ampere equations of Aubin-Yau type. It is shown that, as beta tends to infinity, the regularizations converge to the envelope u in the strongest possible Holder sense. A generalization of this result to the case of a nef and big cohomology class is also obtained. As a consequence new PDE proofs are obtained for the regularity results for envelopes in [14] (which, however, are weaker than the results in [14] in the case of a non-nef big class). Applications to the regularization problem for quasi-psh functions and geodesic rays in the closure of the space of Kahler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of the regularizations as transcendental Bergman metrics.Comment: 28 pages. Version 2: 29 pages. Improved exposition, references updated. Version 3: 31 pages. A direct proof of the bound on the Monge-Amp\`ere mass of the envelope for a general big class has been included and Theorem 2.2 has been generalized to measures satisfying a Bernstein-Markov propert

Geometric Flows of Diffeomorphisms

The idea of this thesis is to apply the methodology of geometric heat flows to the study of spaces of diffeomorphisms. We start by describing the general form that a geometrically natural flow must take and the implications this has for the evolution equations of associated geometric quantities. We discuss the difficulties involved in finding appropriate flows for the general case, and quickly restrict ourselves to the case of surfaces. In particular the main result is a global existence, regularity and convergence result for a geometrically defined quasilinear flow of maps u between flat surfaces, producing a strong deformation retract of the space of diffeomorphisms onto a finite-dimensional submanifold. Partial extensions of this result are then presented in several directions. For general Riemannian surfaces we obtain a full local regularity estimate under the hypothesis of bounds above and below on the singular values of the first derivative. We achieve these gradient bounds in the flat case using a tensor maximum principle, but in general the terms contributed by curvature are not easy to control. We also study an initial-boundary-value problem for which we can attain the necessary gradient bounds using barriers, but the delicate nature of the higher regularity estimate is not well-adapted for obtaining uniform estimates up to the boundary. To conclude, we show how appropriate use of the maximum principle can provide a proof of well-posedness in the smooth category under the assumption of estimates for all derivatives