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We prove a local central limit theorem (LCLT) for the number of points $N(J)$ in a region $J$ in $mathbb R^d$ specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of $N(J)$ tends to infinity as $|J| to infty$. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for ${E(k;J)}$ --- the probabilities of there being exactly $k$ points in $J$ --- all lie on the negative real $z$-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the $k$-th largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors in the bulk.
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We study the many body quantum evolution of bosonic systems in the mean field limit. The dynamics is known to be well approximated by the Hartree equation. So far, the available results have the form of a law of large numbers. In this paper we go one step further and we show that the fluctuations around the Hartree evolution satisfy a central limit theorem. Interestingly, the variance of the limiting Gaussian distribution is determined by a time-dependent Bogoliubov transformation describing the dynamics of initial coherent states in a Fock space representation of the system.
We continue the study of joint statistics of eigenvectors and eigenvalues initiated in the seminal papers of Chalker and Mehlig. The principal object of our investigation is the expectation of the matrix of overlaps between the left and the right eigenvectors for the complex $Ntimes N$ Ginibre ensemble, conditional on an arbitrary number $k=1,2,ldots$ of complex eigenvalues.These objects provide the simplest generalisation of the expectations of the diagonal overlap ($k=1$) and the off-diagonal overlap ($k=2$) considered originally by Chalker and Mehlig. They also appear naturally in the problem of joint evolution of eigenvectors and eigenvalues for Brownian motions with values in complex matrices studied by the Krakow school. We find that these expectations possess a determinantal structure, where the relevant kernels can be expressed in terms of certain orthogonal polynomials in the complex plane. Moreover, the kernels admit a rather tractable expression for all $N geq 2$. This result enables a fairly straightforward calculation of the conditional expectation of the overlap matrix in the local bulk and edge scaling limits as well as the proof of the exact algebraic decay and asymptotic factorisation of these expectations in the bulk.
We show that the central limit theorem for linear statistics over determinantal point processes with $J$-Hermitian kernels holds under fairly general conditions. In particular, We establish Gaussian limit for linear statistics over determinantal point processes on union of two copies of $mathbb{R}^d$ when the correlation kernels are $J$-Hermitian translation-invariant.
In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of $k$-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on $kgeq2$ complex eigenvalues. As a new result we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.
Determinantal point processes (DPPs) are popular probabilistic models of diversity. In this paper, we investigate DPPs from a new perspective: property testing of distributions. Given sample access to an unknown distribution $q$ over the subsets of a ground set, we aim to distinguish whether $q$ is a DPP distribution, or $epsilon$-far from all DPP distributions in $ell_1$-distance. In this work, we propose the first algorithm for testing DPPs. Furthermore, we establish a matching lower bound on the sample complexity of DPP testing. This lower bound also extends to showing a new hardness result for the problem of testing the more general class of log-submodular distributions.
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