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Gaussian random fields on finite dimensional smooth manifolds whose variances reach their maximum value at smooth submanifolds are considered. Exact asymptotic behaviors of large excursion probabilities have been evaluated. Vector Gaussian…
It is well known that the joint probability density of the eigenvalues of Gaussian ensembles of random matrices may be interpreted as a Coulomb gas. We review these classical results for hermitian and complex random matrices, with special…
We uncover geometric aspects that underlie the sum of two independent stochastic variables when both are governed by q-Gaussian probability distributions. The pertinent discussion is given in terms of random vectors uniformly distributed on…
In this paper we study the quantization problem for probability measures on Riemannian manifolds. Under a suitable assumption on the growth at infinity of the measure we find asymptotic estimates for the quantization error, generalizing the…
The paper considers the distribution of a general linear combination of central and non-central chi-square random variables by exploring the branch cut regions that appear in the standard Laplace inversion process. Due to the original…
A random vector whose norm and overlap (inner product with an independent copy) concentrates is shown to have random low-dimensional projections that are approximately random Gaussians. Conversely, asymptotically random Gaussian projections…
We prove a pointwise version of the multi-dimensional central limit theorem for convex bodies. Namely, let X be an isotropic random vector in R^n with a log-concave density. For a typical subspace E in R^n of dimension n^c, consider the…
We study the persistence probability of a centered stationary Gaussian process on $\mathbb{Z}$ or $\mathbb{R}$, that is, its probability to remain positive for a long time. We describe the delicate interplay between this probability and the…
Generalized quantum measurements with N distinct outcomes are used for determining the density matrix, of order d, of an ensemble of quantum systems. The resulting probabilities are represented by a point in an N-dimensional space. It is…
Similar to the Berry conjecture of quantum chaos we consider elastic analogue which incorporates longitudinal and transverse elastic displacements with corresponding wave vectors. Based on that we derive the correlation functions for…
How likely is the high level of a continuous Gaussian random field on an Euclidean space to have a "hole" of a certain dimension and depth? Questions of this type are difficult, but in this paper we make progress on questions shedding new…
We prove a number of decoupling inequalities for nonhomogeneous random polynomials with coefficients in Banach space. Degrees of homogeneous components enter into comparison as exponents of multipliers of terms of certain Poincar\'e-type…
We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen's expansion approach…
We study the complexity of Banach space valued integration in the randomized setting. We are concerned with $r$-times continuously differentiable functions on the $d$-dimensional unit cube $Q$, with values in a Banach space $X$, and…
Context: Two-point correlation functions are used throughout cosmology as a measure for the statistics of random fields. When used in Bayesian parameter estimation, their likelihood function is usually replaced by a Gaussian approximation.…
We discuss recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N(>>1)-dimensional Gaussian landscape. The particular attention is paid to the case of landscapes with…
We determine the inner product on the Hilbert space of wavefunctions of the universe by imposing the Hermiticity of the quantum Hamiltonian in the context of the minisuperspace model. The corresponding quantum probability density reproduces…
One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions are a special case where the density matrix is restricted to be diagonal. Density…
There seems to be an upper limit to predicting the outcome of matches in (semi-)professional sports. Recent work has proposed that this is due to chance and attempts have been made to simulate the distribution of win percentages to identify…
This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for…