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## Inﬁnite Products of Random Isotropically Distributed Matrices

Journal of Statistical Physics. 2017. Vol. 166. P. 24-38.

Statisticalpropertiesofinﬁniteproductsofrandomisotropicallydistributedmatrices are investigated. Both for continuous processes with ﬁnite correlation time and discrete sequencesofindependentmatrices,aformalismthatallowstocalculateeasilytheLyapunov spectrum and generalized Lyapunov exponents is developed. This problem is of interest to probability theory, statistical characteristics of matrix T-exponentials are also needed for turbulent transport problems, dynamical chaos and other parts of statistical physics.

A.V. Kopiev, Zybin K., Journal of Turbulence 2018 Vol. 19 No. 9 P. 717-730

On the grounds of Kolmogorov's 4/5 law analytical relations for triple two-point correlations of velocity and velocity gradients in homogeneous isotropic incompressible turbulence are derived. The corresponding correlation tensor can be expressed in terms of dissipation ε, the second-order longitudinal velocity structure function and one more scalar function of distance between the points. However, some ...

Added: February 21, 2019

Ilyin A., Zybin K., Валерия Александровна Сирота, Physica Scripta 2019 Vol. 94 P. 064001

Statistical properties of a statistically homogeneous random magnetic ﬁeld in a viscous diffusive ﬂuid are derived from the evolution of a single blob of the magnetic ﬁeld. It is shown that, although the magnetic ﬁeld of a single blob decreases in time, the volume occupied by the magnetic ﬁeld and its energy increase; this is ...

Added: October 30, 2019

Zybin K., Ilyin A., Валерия Александровна Сирота et al., EPL 2020 Vol. 132 P. 24001

Evolution of a stochastically homogeneous magnetic field advected by an incompressible turbulent flow with large magnetic Prandtl numbers is considered at scales smaller than the Kolmogorov viscous scale. It is shown that, despite the unlimited growth of the magnetic field, its feedback on the fluid's dynamics remains negligibly small. ...

Added: December 23, 2020

Orlov A. Y., , in: Geometric Methods in Physics XXXVI. Workshop and Summer School, Białowieża, Poland, 2017. Contains papers. .: Springer, 2019.. P. 355-373.

I show that Hurwitz numbers may be generated by certain correlation functions which appear in quantum chaos. ...

Added: June 28, 2019

Ilyin A., Zybin K., Валерия Александровна Сирота, EPL 2018 Vol. 121 P. 34002-p1-34002-p6

Statistical moments of magnetic ﬁeld in a viscous range of turbulence are calculated for arbitrary initial conditions. It is shown that the evolution of magnetic ﬁeld in the case of ﬁnite initial distribution in a linear velocity ﬁeld consists of two or three consecutive regimes: exponential growth is followed by exponential decay. This solves the ...

Added: October 30, 2019

Алексей Копьев, Александр Киселев, Ilyin A. et al., Astrophysical Journal 2022 Vol. 927 No. 2 Article 172

We consider a natural generalization of the Kazantsev–Kraichnan model for a small-scale turbulent dynamo. This generalization takes into account the statistical time asymmetry of a turbulent flow and thus allows one to describe velocity fields with energy cascade. For three-dimensional velocity fields, a generalized Kazantsev equation is derived, and the evolution of the second-order magnetic ...

Added: March 15, 2022

Ilyin A., A. V. Kopyev, V.A. Sirota et al., Physics of Fluids 2021 Vol. 33 No. 7 P. 075105-1-075105-10

Kinematic dynamo in incompressible isotropic turbulent flows with high magnetic Prandtl number is considered. The approach interpreting an arbitrary magnetic field distribution as a superposition of localized perturbations (blobs) is developed. We derive a general relation between stochastic properties of an isolated blob and a stochastically homogenous distribution of magnetic field advected by the same ...

Added: July 20, 2021

Friedrich Goetze, Naumov A.A., Tikhomirov A., Bernoulli: a journal of mathematical statistics and probability 2017 Vol. 23 No. 4B P. 3067-3113

In this paper we consider the product of two independent random matrices X^(1) and X^(2). Assume that X_{jk}^{(q)},1\le j,k \le n,q=1,2,, are i.i.d. random variables with \EX_{jk}^{q}=0, VarX_{jk}^{(q)}=1/ Denote by s_1(W),…,s_n(W) the singular values of W:=n^{-1}X^(1)X^(2). We prove the central limit theorem for linear statistics of the squared singular values s_1^2(W),…,s_n^2(W) showing that the limiting variance depends on \kappa_4:=\E(X_{11}^{(1)})^4−3. ...

Added: April 28, 2018

Alexei Doludenko, Svetlana Fortova, Kolokolov I. et al., Physics of Fluids 2021 Vol. 33 No. 011704 P. 1-6

We investigate the coherent vortex produced by two-dimensional turbulence excited in a finite box. We establish analytically the mean velocity
profile of the vortex for the case where the bottom friction is negligible and express its characteristics via the parameters of pumping. Our
theoretical predictions are verified and confirmed by direct numerical simulations in the framework of ...

Added: January 20, 2021

Алексей Копьев, Zybin K., Валерия Александровна Сирота et al., Physics of Fluids 2021 Vol. 33 No. 3 P. 031703-031704

The paper [Djenidi et al., Phys. Fluids 33(3), 031703 (2021)] considers a classical issue of an anomalous scaling of velocity structure functions in a high-Reynolds number turbulent flow. The paper offers a mathematical proof of the ground-breaking result: the intermittency is an artifact of the Reynolds number finiteness. However, the proof contains a technical error that makes ...

Added: November 5, 2021

Фридрих Гётце, Naumov A., Tikhomirov A., Random Matrices-Theory and Applications 2020 Vol. 9 No. 4 P. 2150004

We consider products of independent \(n \times n\) non-Hermitian random matrices \(\X^{(1)}, \ldots, \X^{(m)}\). Assume that their entries, \(X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, \ldots, m\), are independent identically distributed random variables with zero mean, unit variance. G\"otze -- Tikhomirov~\cite{GotTikh2011} and O'Rourke--Sochnikov~\cite{Soshnikov2011} proved that under these assumptions the empirical spectral distribution (ESD) ...

Added: September 13, 2019

Ilyin A., A.V. Kopyev, V.A. Sirota et al., Physics of Fluids 2020 Vol. 32 No. 12 P. 125114

We consider fluctuations of magnetic field excited by external force and advected by isotropic turbulent flow. It appears that non-Gaussian velocity gradient statistics and finite region of pumping force provide the existence of stationary solution. The mean-square magnetic field is calculated for arbitrary velocity gradient statistics. An estimate for possible feedback of magnetic field on ...

Added: December 16, 2020

Ilyin A., Zybin K., Валерия Александровна Сирота, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2017 Vol. 96 P. 013117-1-013117-8

WeanalyzepassivescalaradvectionbyaturbulentﬂowintheBatchelorregime.Norestrictionsonthevelocity statistics of the ﬂow are assumed. The properties of the scalar are derived from the statistical properties of velocity; analytic expressions for the moments of scalar density are obtained. We show that the scalar statistics can differ signiﬁcantly from that obtained in the frames of the Kraichnan model. ...

Added: October 30, 2019

Ilyin A., Zybin K., Валерия Александровна Сирота, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2019 Vol. 99 No. 5 P. 052220-1-052220-6

The impact of turbulent advection in reaction-diffusion systems is investigated for the viscous range of scales. We show that the population size can increase exponentially even in systems with density saturation, at the expense of exponential propagation of the reaction front. Exact expressions for scaling exponents of the density and population size are calculated in ...

Added: October 30, 2019

A.S. Il'yn, V.A. Sirota, Zybin K., Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2017 Vol. 96 No. 1 P. 013117-1-013117-8

We analyze passive scalar advection by a turbulent flow in the Batchelor regime. No restrictions on the velocity statistics of the flow are assumed. The properties of the scalar are derived from the statistical properties of velocity; analytic expressions for the moments of scalar density are obtained. We show that the scalar statistics can differ ...

Added: February 21, 2019

Ilyin A., V. Sirota, Zybin K., Journal of Statistical Physics 2016 Vol. 163 No. 4 P. 765-783

A functional method for calculating averages of the time-ordered exponential of a continuous isotropic random N×NN×N matrix process is presented. The process is not assumed to be Gaussian. In particular, the Lyapunov exponents and higher correlation functions of the T-exponent are derived from the statistical properties of the process. The approach may be of use ...

Added: October 30, 2019