Nikolai Makarov

Fluctuations of eigenvalues of random normal matrices

In this article, we consider a fairly general potential in the plane and the corresponding Boltzmann-Gibbs distribution of eigenvalues of random normal matrices. As the order of the matrices tends to infinity, the eigenvalues condensate on a certain compact subset of the plane—the “droplet.” We prove that fluctuations of linear statistics of eigenvalues of random normal matrices converge on compact subsets of the interior of the droplet to a Gaussian field, and we discuss various ramifications of this result.

Operators with singular continuous spectrum. II. Rank one operators

For an operator,A, with cyclic vector ϕ, we studyA+λP, whereP is the rank one projection onto multiples of ϕ. If [α,β] ⊂ spec (A) andA has no a.c. spectrum, we prove thatA+λP has purely singular continuous spectrum on (α,β) for a denseGδ of λs.

Meromorphic Inner Functions, Toeplitz Kernels and the Uncertainty Principle

This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. Their common feature is the close relation to the theory of complex Fourier transform of compactly supported measures or, more generally, Fourier–Weyl–Titchmarsh transforms associated with selfadjoint differential operators with compact resolvent.

RANDOM NORMAL MATRICES AND WARD IDENTITIES

We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman’s solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

Singular continuous spectrum is generic

In a variety of contexts, we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators, those with singular spectrum are a dense G delta.

On Thermodynamics of Rational Maps. II: Non-Recurrent Maps

The pressure function p(t )o fa non-recurrent map is real analytic on some interval (0 ,t ∗ )w it ht∗ strictly greater than the dimension of the Julia set. The proof is an adaptation of the well known tower techniques to the complex dynamics situation. In general, p(t )n eed not be analytic on the whole positive axis. In this paper we study analyticity properties of the pressure function of non-recurrent maps. Our approach is based on the well known tower techniques adapted to the complex dynamics situation. The pressure function p(t), which is defined in terms of the Poincar´ es erie s( see (1.4)), carries essential information about ergodic and dimensional properties of the maximal measure. In particular, it characterizes the dimension spectrum of harmonic measure on the Julia set in the case of a polynomial dynamics. According to the classical theory of Sinai, Ruelle and Bowen, p(t )i s real analytic if the dynamics is hyperbolic ,t hat is, expanding on the Julia set. This fact is closely related to the so called ‘spectral gap’ phenomenon, which also implies other important features of hyperbolic dynamics such as the existence of equilibrium states, exponential decay of correlations, etc. The problem of extending (some parts of) the classical theory to the non-hyperbolic case has become one of the central themes in the ergodic theory of conformal dynamics. In the first part of this work [8], we provided a detailed analysis of the negative part t 0 is substantially more complicated (and more important). The main difficulty arises from the presence of singularities (critical points) on the Julia set. To circumvent this difficulty, we propose to use a tower construction which forces the dynamics to be expanding on some auxiliary space. The tower method has been widely used in the general theory of dynamical systems with some degree of hyperbolicity (see especially [12]), and in particular in 1-dimensional real dynamics, where the construction is known as Hof bauer’s tower .T oa pply this method in the complex case, it is natural to use some basic elements of the Yoccoz jigsaw puzzle structure (see [9]). We will discuss only the simplest type of non-hyperbolic behavior: the case of nonrecurrent dynamics (every critical point in the Julia set is non-recurrent, that is, is away from its iterates) without parabolic cycles (see [2 ]f orvarious characterizations

Laplacian path models

In this paper, we study two growth models in the complex plane--the needle and the geodesic η-models, defined below.

Off-critical lattice models and massive SLEs

We suggest how versions of Schramm’s SLE can be used to describe the scaling limit of some off-critical 2D lattice models. Many open questions remain.

Harmonic measure and polynomial Julia sets

There is a natural conjecture that the universal bounds for the dimension spectrum of harmonic measure are the same for simply connected and for non-simply connected domains in the plane. Because of the close relation to conformal mapping theory, the simply connected case is much better understood, and proving the above statement would give new results concerning the properties of harmonic measure in the general case. We establish the conjecture in the category of domains bounded by polynomial Julia sets. The idea is to consider the coefficients of the dynamical zeta-function as subharmonic functions on a slice of Teichmuller’s space of the polynomial, and then to apply the maximum principle. 1. Dimension spectrum of harmonic measure In this paper we discuss some properties of harmonic measure in the complex plane. For a domain Ω ⊂ Ĉ and a point a ∈ Ω, let ω = ωa denote the harmonic measure of Ω evaluated at a. The measure ωa can be defined, for instance, as the hitting distribution of a Brownian motion started at a: if e ⊂ ∂Ω, then ωa(e) is the probability that a random Brownian path first hits the boundary at a point of e. Much work has been devoted to describing dimensional properties of ω when the domain is as general as possible. In particular, Jones and Wolff [7] proved that no matter what the domain Ω is, harmonic measure is concentrated on a Borel set of Hausdorff dimension at most one; in other words, dimω ≤ 1 for all plane domains. (1.1) We are interested in finding similar (but stronger) universal estimates involving the dimension spectrum of ω. 1.1. Universal spectrum. For every positive α, we denote f ω (α) = dim{αω(z) ≤ α}, where αω(z) is the lower pointwise dimension of ω: αω(z) = lim inf δ→0 logωB(z, δ) log δ . B(z, δ) is a general notation for the disc with center z and radius δ. The universal dimension spectrum is the function Φ(α) = sup ω f ω (α), (1.2) where the supremum is taken over harmonic measures of all planar domains. The first author is supported by N.S.F. Grant DMS-9970283. The second author is supported by N.S.F. Grant DMS-9800714.

Topology of quadrature domains

We address the problem of topology of quadrature domains, namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity.