Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the determinantal structure of conditional overlaps for the complex Ginibre ensemble
84
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
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.
We prove rates of convergence for the circular law for the complex Ginibre ensemble. Specifically, we bound the expected $L_p$-Wasserstein distance between the empirical spectral measure of the normalized complex Ginibre ensemble and the uniform measure on the unit disc, both in expectation and almost surely. For $1 le p le 2$, the bounds are of the order $n^{-1/4}$, up to logarithmic factors.
We introduce constellation ensembles, in which charged particles on a line (or circle) are linked with charged particles on parallel lines (or concentric circles). We present formulas for the partition functions of these ensembles in terms of either the Hyperpfaffian or the Berezin integral of an appropriate alternating tensor. Adjusting the distances between these lines (or circles) gives an interpolation between a pair of limiting ensembles, such as one-dimensional $beta$-ensembles with $beta=K$ and $beta=K^2$.
We provide a simple extension of Bolthausens Morita type proof cite{Bolt2} of the replica symmetric formula for the Sherrington-Kirkpatrick (SK) model and prove the replica symmetry for all $(beta,h)$ that satisfy $beta^2 E, text{sech}^2(betasqrt{q}Z+h) leq 1$, where $q = Etanh^2(betasqrt{q}Z+h)$. Compared to cite{Bolt2}, the key of the argument is to apply the conditional second moment method to a suitably reduced partition function.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
