We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density B(z,·) for the polynomial Bergman space, where the point z ∈ C is fixed. When z = ∞, the Berezin density is expressed in terms of the squared modulus of the corresponding normalized orthogonal polynomial P. We use an approximate version of this characterization to study the asymptotics of the orthogonal polynomials in the context of exponentially varying weights. This builds on earlier works by Its-Takhtajan and by the first author on a soft Riemann-Hilbert problem for planar orthogonal polynomials, where in place of the Laplacian we have the ∂̄-operator. We adapt the soft Riemann-Hilbert approach to the nonlinear potential problem, where the nonlinearity is due to the appearance of |P|² in place of P. Moreover, we suggest how to adapt the potential theory method to the study of the asymptotics of more general Berezin densities B(z,w) in the off-spectral regime, that is, when z is fixed outside the droplet. This is a first installment in a program to obtain an explicit global expansion formula for the polynomial Bergman kernel, and, in particular, of the one-point function of the associated random normal matrix ensemble.

Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zeros of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero density vanishes. Moreover, we give a description of the class of holes for which the forbidden region is a disk. The connecting link between random zeros and potential theory is supplied by a constrained extremal problem for the Zeitouni-Zelditch functional. To solve this problem, we recast it in terms of a seemingly novel obstacle problem, where the solution is forced to be harmonic inside the hole.

We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after subtracting the mean, the random function becomes stationary. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as electric fields generated by an infinite random distribution of point charges.

Consider a random planar point process whose law is invariant under planar isometries. We think of the process as a random distribution of point charges and consider the electric field generated by the charge distribution. In Part I of this work, we found a condition on the spectral side which characterizes when the field itself is invariant with a well-defined second-order structure. Here, we fix a process with an invariant field, and study the fluctuations of the flux through large arcs and curves in the plane. Under suitable conditions on the process and on the curve, denoted Γ , we show that the asymptotic variance of the flux through RΓ grows like R times the signed length of Γ . As a corollary, we find that the charge fluctuations in a dilated Jordan domain is asymptotic with the perimeter, provided only that the boundary is rectifiable. The proof is based on the asymptotic analysis of a closely related quantity (the complex electric action of the field along a curve). A decisive role in the analysis is played by a signed version of the classical Ahlfors regularity condition.

The asymptotic analysis of Bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention. Such interfaces typically occur when the associated limiting Bergman density function vanishes on a portion of the plane, the off-spectral region. This type of behavior is observed when the metric is negatively curved somewhere, or when we study partial Bergman kernels in the context of positively curved metrics. In this work, we cover these two situations in a unified way, for exponentially varying weights on the complex plane. We obtain a uniform asymptotic expansion of the coherent state of depthn rooted at an off-spectral point, which we also refer to as the root function at the point in question. The expansion is valid in the entire off-spectral component containing the root point, and protrudes into the spectrum as well. This allows us to obtain error function transition behavior of the density of states along the smooth interface. Previous work on asymptotic expansions of Bergman kernels is typically local, and valid only in the bulk region of the spectrum, which contrasts with our non-local expansions.

In this paper, we explore a sharp phase transition phenomenon which occurs for (Formula presented.)-Carleman classes with exponents (Formula presented.). These classes are defined as for the standard Carleman classes, only the (Formula presented.)-bounds are replaced by corresponding (Formula presented.)-bounds. We study the quasinorms (Formula presented.)for some weight sequence (Formula presented.) of positive real numbers, and consider as the corresponding (Formula presented.)-Carleman space the completion of a given collection of smooth test functions. To mirror the classical definition, we add the feature of dilatation invariance as well, and consider a larger soft-topology space, the (Formula presented.)-Carleman class. A particular degenerate instance is when (Formula presented.) for (Formula presented.) and (Formula presented.) for (Formula presented.). This would give the (Formula presented.)-Sobolev spaces, which were analyzed by Peetre, following an initial insight by Douady. Peetre found that these (Formula presented.)-Sobolev spaces are highly degenerate for (Formula presented.). Indeed, the canonical map (Formula presented.) fails to be injective, and there is even an isomorphism (Formula presented.)corresponding to the canonical map (Formula presented.) acting on the test functions. This means that e.g. the function and its derivative lose contact with each other (they “disconnect”). Here, we analyze this degeneracy for the more general (Formula presented.)-Carleman classes defined by a weight sequence (Formula presented.). If (Formula presented.) has some regularity properties, and if the given collection of test functions is what we call (Formula presented.)-tame, then we find that there is a sharp boundary, defined in terms of the weight (Formula presented.): on the one side, we get Douady–Peetre’s phenomenon of “disconnexion”, while on the other, the completion of the test functions consists of (Formula presented.)-smooth functions and the canonical map (Formula presented.) is correspondingly well-behaved in the completion. We also look at the more standard second phase transition, between non-quasianalyticity and quasianalyticity, in the (Formula presented.) setting, with (Formula presented.).

We study fluctuations of linear statistics in polyanalytic Ginibre ensembles, a family of point processes describing planar free fermions in a uniform magnetic field at higher Landau levels. Our main result is asymptotic normality of fluctuations, extending a result of Rider and Virág. As in the analytic case, the variance is composed of independent terms from the bulk and the boundary. Our methods rely on a structural formula for polyanalytic polynomial Bergman kernels which separates out the different pure q" role="presentation">q-analytic kernels corresponding to different Landau levels. The fluctuations with respect to these pure q" role="presentation">q-analytic Ginibre ensembles are also studied, and a central limit theorem is proved. The results suggest a stabilizing effect on the variance when the different Landau levels are combined together.

We study the problem of geometric zero packing, recently introduced by Hedenmalrn [7]. There are two natural densities associated with this problem: the discrepancy density pa, given by rho(H) = lim (r -> 1-) inf inf(f) integral(D(0,r)) ((1 - vertical bar z vertical bar(2)) vertical bar f(z)vertical bar - 1)(2) dA(z)/1 - vertical bar z vertical bar(2)/ integral(D(0,r)) dA(z)/1 - vertical bar z vertical bar(2) which measures the discrepancy in optimal approximation of (1 - vertical bar z vertical bar(2))(-1) with the modulus of polynomials f, and its relative, the tight discrepancy density rho*(H), which will trivially satisfy pH < per. These densities have deep connections to the boundary behaviour of conformal mappings with k-quasiconformal extensions, which can be seen from Hedenmalm's result that the universal asymptotic variance Sigma(2) is related to rho(H)* by Sigma(2) = 1 - rho(H)* . Here we prove that in fact rho(H) = rho(H)*, resolving a conjecture by Hedenmalm in the positive. The natural planar analogues rho(C) and rho(C)* to these densities make contact with work of Abrikosov on Bose Einstein condensates. As a second result we prove that also rho(C) = rho(C)*. The methods are based on Ameur, Hedenmalm and Makarov's Hormander-type <(partial derivative)over bar>-estimates with polynomial growth control [2]. As a consequence we obtain sufficiency results on the degrees of approximately optimal polynomials.

We prove an operator inequality for the Bergman shift operator on weighted Bergman spaces of analytic functions in the unit disc with weight function controlled by a curvature parameter α assuming nonnegative integer values. This generalizes results by Shimorin, Hedenmalm and Jakobsson concerning the cases α = 0 and α = 1. A naturally derived scale of Hilbert space operator inequalities is studied and shown to be relaxing as the parameter α > -1 increases. Additional examples are provided in the form of weighted shift operators.

We prove an operator identity for the shift operator in the scale of standard weighted Bergman spaces in the unit disc. This operator identity is then applied in the context of functional calculus for the shift operator and a characterization of harmonic symbol Bergman space Toeplitz operators is obtained generalizing an earlier result by Louhichi and Olofsson. Duality arguments lead to operator inequalities and structure formulas for reproducing kernel functions which make contact with work of Richter, Shimorin, and others.