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The article is devoted to stochastic processes with values in finite- and infinite-dimensional vector spaces over infinite fields $\bf K$ of zero characteristics with non-trivial non-archimedean norms. For different types of stochastic…
We show that for a family of quantum walk models with electric fields, the spectrum is the unit circle for any irrational field. The result also holds for the associated CMV matrices defined by skew-shifts. Generalizations to CMV matrices…
Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial…
We will develop some elements in stochastic analysis in the Wasserstein space $\mathbb{P}_2(M)$ over a compact Riemannian manifold $M$, such as intrinsic It$\^o$ formulae, stochastic regular curves and parallel translations along them. We…
The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schr\"odinger operators. It may be used to exclude eigenvalues for such operators.…
In this article the statistical properties of symmetrical random matrices whose elements are drawn from a q-parameterized non-extensive statistics power-law distribution are investigated. In the limit as q->1 the well known Gaussian…
In this paper we find spectral properties in the large $N$ limit of Dirac operators that come from random finite noncommutative geometries. In particular for a Gaussian potential the limiting eigenvalue spectrum is shown to be universal…
Statistical models on infinite graphs may exhibit inhomogeneous thermodynamic behaviour at macroscopic scales. This phenomenon is of geometrical origin and may be properly described in terms of spectral partitions into subgraphs with well…
A general analytical approach to the statistical description of quantum graph spectra based on the exact periodic orbit expansions of quantum levels is discussed. The exact and approximate expressions obtained in \cite{Anima} for the…
The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation…
We identify three semiclassical parameters in the QCD Dirac operator. Mutual coupling of the different types of degrees of freedom (translational, colour and spin) depends on how the semiclassical limit is taken. We discuss various…
We introduce a new model for investigating spectral properties of quantum graphs, a quantum circulant graph. Circulant graphs are the Cayley graphs of cyclic groups. Quantum circulant graphs with standard vertex conditions maintain…
Non-ergodicity impacts statistical inference in a diverse range of disciplines inside and outside of physics. However the concept of ergodicity is used inconsistently, and may refer to several nonequivalent notions. To help address this, we…
We introduce new natural generalizations of the classical descent and inversion statistics for permutations, called width-$k$ descents and width-$k$ inversions. These variations induce generalizations of the excedance and major statistics,…
We describe the behaviour of semiclassical electrodynamics under gauge transformations. For this purpose we observe the structure of Schr\"odinger equation and matricial elements under these transformations. We conclude this theory is not…
We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic"…
We previously introduced a family of symplectic maps of the torus whose quantization exhibits scarring on invariant co-isotropic submanifolds. The purpose of this note is to show that in contrast to other examples, where failure of Quantum…
Using longer spectra we re-analyze spectral properties of the two-body random ensemble studied thirty years ago. At the center of the spectra the old results are largely confirmed, and we show that the non-ergodicity is essentially due to…
We present results for the equilibrium statistics and dynamic evolution of moderately large ($n = {\mathcal{O}}(10^2 - 10^3)$) numbers of interacting point vortices on the unit sphere under the constraint of zero mean angular momentum. We…
We investigate the quantum properties of a non-random Hamiltonian with a step-like singularity. It is shown that the eigenfunctions are multifractals and, in a certain range of parameters, the level statistics is described exactly by…