arXiv subjects

Patrick Lopatto

Publications and source records attributed to Patrick Lopatto.

41 records · Page 3Linked to original sources

arXiv2019

GOE Statistics for Levy Matrices

In this paper we establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, $N \times N$ random matrices $\textbf{H}$ whose upper triangular entries are independent, identically distributed $α$-stable laws. First, if $α\in (1, 2)$ and $E \in \mathbb{R}$ is any energy bounded away from $0$, we show that every eigenvector of $\textbf{H}$ corresponding to an eigenvalue near $E$ is completely delocalized and that the local spectral statistics of $\textbf{H}$ around $E$ converge to those of the Gaussian Orthogonal Ensemble (GOE) as $N$ tends to $\infty$. Second, we show for almost all $α\in (0, 2)$, there exists a constant $c(α) > 0$ such that the same statements hold if $|E| < c (α)$.

math.PR↗

arXiv2019

Statistical thresholds for Tensor PCA

We study the statistical limits of testing and estimation for a rank one deformation of a Gaussian random tensor. We compute the sharp thresholds for hypothesis testing and estimation by maximum likelihood and show that they are the same. Furthermore, we find that the maximum likelihood estimator achieves the maximal correlation with the planted vector among measurable estimators above the estimation threshold. In this setting, the maximum likelihood estimator exhibits a discontinuous BBP-type transition: below the critical threshold the estimator is orthogonal to the planted vector, but above the critical threshold, it achieves positive correlation which is uniformly bounded away from zero.

math.PR↗

arXiv2019

Universality of the least singular value for sparse random matrices

We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution with parameter $p$. These matrices represent the adjacency matrices of random Erdős--Rényi digraphs and are sparse when $p\ll 1$. We prove that in the regime $pN\gg 1$, the distribution of the least singular value is universal in the sense that it is independent of $p$ and equal to the distribution of the least singular value of a Gaussian matrix ensemble. We also prove the universality of the joint distribution of multiple small singular values. Our methods extend to matrix ensembles whose entries are chosen from arbitrary distributions that may be correlated, complex valued, and have unequal variances.

math.PR↗

arXiv2015

Simplicial Abel-Jacobi maps and reciprocity laws

We describe an explicit morphism of complexes that induces the cycle-class maps from (simplicially described) higher Chow groups to rational Deligne cohomology. The reciprocity laws satisfied by the currents we introduce for this purpose are shown to provide a clarifying perspective on functional equations satisfied by complex-valued di- and trilogarithms.

math.AG↗

arXiv2014

Schatten-class Truncated Toeplitz Operators

We investigate truncated Toeplitz operators belonging to the Schatten ideals. We completely characterize such operators when they have an analytic symbol or belong to the ideal of Hilbert-Schmidt operators. We also study model spaces generated by Blaschke products associated with thin sequences, model spaces generated by certain types of singular inner functions, and operators associated with a class of very smooth symbols.

math.CV↗