arXiv subjects

Aron Wennman

Publications and source records attributed to Aron Wennman.

21 records · Page 2Linked to original sources

arXiv2016

A Central Limit Theorem for Fluctuations in Polyanalytic Ginibre Ensembles

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 $q$-analytic kernels corresponding to different Landau levels. The fluctuations with respect to these pure $q$-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.

math-ph↗

arXiv2016

Discrepancy densities for planar and hyperbolic Zero Packing

We study the problem of geometric zero packing, recently introduced by Hedenmalm. There are two natural densities associated to this problem: the discrepancy density $ρ_{\mathbb{H}}$, given by $$ ρ_{\mathbb{H}} = \liminf_{r\to 1^-} \inf_{f} \frac{\int_{\mathbb{D}(0,r)} \left((1-\lvert z\rvert^2) \lvert f(z)\rvert-1\right)^2 \frac{dA(z)}{1-\lvert z\rvert^2}} {\int_{\mathbb{D}(0,r)} \frac{dA(z)}{1-\lvert z\rvert^2}} $$ which measures the discrepancy in optimal approximation of $(1-\lvert z\rvert^2)^{-1}$ with the modulus of polynomials $f$, and it's relative, the tight discrepancy density $ρ_{\mathbb{H}}^*$, which will trivially satisfy $ρ_{\mathbb{H}}\leqρ_{\mathbb{H}}^*$. These densities have deep connections to the boundary behaviour of conformal mappings with $k$-quasiconformal extensions, which can be seen from the Hedenmalm's result that the universal asymptotic variance $Σ^2$ is related to $ρ_{\mathbb{H}}^*$ by $Σ^2=1-ρ_{\mathbb{H}}^*$. Here we prove that in fact $ρ_{\mathbb{H}}=ρ_{\mathbb{H}}^*$, resolving a conjecture by Hedenmalm in the positive. The natural planar analogues $ρ_{\mathbb{C}}$ and $ρ_{\mathbb{C}}^*$ to these densities make contact with work of Abrikosov on Bose-Einstein condensates. As a second result we prove that also $ρ_{\mathbb{C}}=ρ_{\mathbb{C}}^*$. The methods are based on Ameur, Hedenmalm and Makarov's Hörmander-type $\bar\partial$-estimates with polynomial growth control. As a consequence we obtain sufficiency results on the degrees of approximately optimal polynomials.

math.CV↗

arXiv2014

Carleman-Sobolev classes for small exponents

This paper is devoted to the study of a generalization of Sobolev spaces for small $L^{p}$ exponents, i.e. $0<p<1$. We consider spaces defined as abstract completions of certain classes of smooth functions with respect to weighted quasi-norms, simultaneously inspired by Carleman classes and classical Sobolev spaces. If the class is restricted with a growth condition on the supremum norms of the derivatives, we prove that there exists a condition which guarantees that the resulting space can be embedded into $C^{\infty}(\mathbb{R})$. This is sharp up to some regularity conditions on the weight sequence. We also show that the growth condition is necessary, in the sense that if we drop it entirely we can naturally embed $L^p$ into the resulting completion.

math.CA↗