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Oskari Ajanki

Publications and source records attributed to Oskari Ajanki.

7 recordsLinked to original sources
arXiv2020

Quadratic vector equations on complex upper half-plane

We consider the nonlinear equation $-\frac{1}{m}=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb{H} $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb{H}$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\mathbb{R}$. In [AEK17a] we qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper we give a comprehensive analysis of these singularities with uniform quantitative controls. We also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the companion paper [AEK16b], we present a complete stability analysis of the equation for any $z\in \mathbb{H}$, including the vicinity of the singularities.

math.PR↗
arXiv2018

Stability of the Matrix Dyson Equation and Random Matrices with Correlations

We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.

math.PR↗
arXiv2017

Universality for general Wigner-type matrices

We consider the local eigenvalue distribution of large self-adjoint $N\times N$ random matrices $\mathbf{H}=\mathbf{H}^*$ with centered independent entries. In contrast to previous works the matrix of variances $s_{ij} = \mathbb{E}\, |h_{ij}|^2 $ is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper [1]. We show that as $N$ grows, the resolvent, $\mathbf{G}(z)=(\mathbf{H}-z)^{-1}$, converges to a diagonal matrix, $ \mathrm{diag}(\mathbf{m}(z)) $, where $\mathbf{m}(z)=(m_1(z),\dots,m_N(z))$ solves the vector equation $ -1/m_i(z) = z + \sum_j s_{ij} m_j(z) $ that has been analyzed in [1]. We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.

math.PR↗
arXiv2017

Singularities of solutions to quadratic vector equations on complex upper half-plane

Let $ S $ be a positivity preserving symmetric linear operator acting on bounded functions. The nonlinear equation $ -\frac{1}{m}=z+Sm $ with a parameter $ z $ in the complex upper half-plane $ \mathbb{H} $ has a unique solution $ m $ with values in $ \mathbb{H} $. We show that the $ z $-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures $ v $ on $ \mathbb{R} $. Under suitable conditions on $ S $, we show that $ v $ has a real analytic density apart from finitely many algebraic singularities of degree at most three. Our motivation comes from large random matrices. The solution $ m $ determines the density of eigenvalues of two prominent matrix ensembles; (i) matrices with centered independent entries whose variances are given by $ S $ and (ii) matrices with correlated entries with a translation invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or a cubic root cusps; no other singularities occur.

math.PR↗
arXiv2013

Local semicircle law with imprimitive variance matrix

We extend the proof of the local semicircle law for generalized Wigner matrices given in [4] to the case when the matrix of variances has an eigenvalue $ -1 $. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices $ \boldsymbol{\mathrm{X}}^\ast \boldsymbol{\mathrm{X}} $, where the variances of the entries of $ \boldsymbol{\mathrm{X}} $ may vary.

math.PR↗
arXiv2005

Connection state overhead in a dynamic linear network

We consider a dynamical linear network where nearest neighbours communicate via links whose states form binary (open/closed) valued independent and identically distributed Markov processes. Our main result is the tight information-theoretic lower bound on the network traffic required by the connection state overhead, or the information required for all nodes to know their connected neighbourhood. These results, and especially their possible generalisations to more realistic network models, could give us valuable understanding of the unavoidable protocol overheads in rapidly changing Ad hoc and sensor networks.

cs.IT↗