Related papers: Full expectation value statistics for randomly sam…
These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero…
The asymptotic normality of the maximum likelihood estimator (MLE) under regularity conditions is a cornerstone of statistical theory. In this paper, we give explicit upper bounds on the distributional distance between the distribution of…
The filtering distribution is a time-evolving probability distribution on the state of a dynamical system, given noisy observations. We study the large-time asymptotics of this probability distribution for discrete-time, randomly…
We consider the quantum dynamical evolution of a fully-connected quantum system subjected to random projective measurements and study the first detection time of an extended subspace of the Hilbert space. Exact analytical expressions are…
We study the optimal way to estimate the quantum expectation value of a physical observable when a finite number of copies of a quantum pure state are presented. The optimal estimation is determined by minimizing the squared error averaged…
A quantum statistical random system with energy dissipation is studied. Its statistics is governed by random complex-valued non-Hermitean Hamiltonians belonging to complex Ginibre ensemble of random matrices. The eigenenergies of…
We discuss the time development of Gaussian wave packet solutions of the quantum bouncer' (a quantum mechanical particle subject to a uniform downward force, above an impermeable flat surface). We focus on the evaluation and visualization…
In statistical physics lately a specific kind of average, called the q-expectation value, has been extensively used in the context of q-generalized statistics dealing with distributions following power-laws. In this context q-expectation…
As the vacuum state of a quantum field is not an eigenstate of the Hamiltonian density, the vacuum energy density can be represented as a random variable. We present an analytical calculation of the probability distribution of the vacuum…
The purpose of this paper is to analyze the distribution distance between random vectors derived from the magnitude of the analytic wavelet transform of the squared envelopes of Gaussian processes and their large-scale limits. When the…
This paper focuses on investigating Stein's invariant shrinkage estimators for large sample covariance matrices and precision matrices in high-dimensional settings. We consider models that have nearly arbitrary population covariance…
In standard quantum mechanics (QM), a state vector $| \psi \rangle$ may belong to infinitely many different orthogonal bases, as soon as the dimension $N$ of the Hilbert space is at least three. On the other hand, a complete physical…
We have calculated the joint probability distribution function for random reverse-cyclic matrices and shown that it is related to an N-body exactly solvable model. We refer to this well-known model potential as a screened harmonic…
A recent conjecture regarding the average of the minimum eigenvalue of the reduced density matrix of a random complex state is proved. In fact, the full distribution of the minimum eigenvalue is derived exactly for both the cases of a…
Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random. If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these…
We study random Morse functions on a Riemann manifold $(M^m,g)$ defined as a random Gaussian weighted superpositions of eigenfunctions of the Laplacian of the metric $g$. The randomness is determined by a fixed Schwartz function $w$ and a…
Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${\mathbb H}$ with covariance operator $\Sigma:={\mathbb E}(X\otimes X).$ Let $\Sigma=\sum_{r\geq 1}\mu_r P_r$ be the spectral decomposition of $\Sigma$ with…
In this article we derive a useful expectation identity using the language of quantum statistical mechanics, where density matrices represent the state of knowledge about the system. This identity allows to establish relations between…
We consider a measure $\psi$ k of dispersion which extends the notion of Wilk's generalised variance, or entropy, for a d-dimensional distribution, and is based on the mean squared volume of simplices of dimension k $\le$ d formed by k + 1…
We construct the generalized entropy optimized by a given arbitrary statistical distribution with a finite linear expectation value of a random quantity of interest. This offers, via the maximum entropy principle, a unified basis for a…