arXiv subjects
Publications and source records attributed to Vladislav Kargin.
It is shown that the normalized fluctuations of Riemann's zeta zeros around their predicted locations follow the Gaussian law. It is also shown that fluctuations of two zeros, $γ_{k}$ and $γ_{k+x},$ with $x\sim(\log k)^β$, $β>0$, for large $k$ follow the two-variate Gaussian distribution with correlation $(1-β)_{+}$.
The paper provides a new integral formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternion-valued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large and compare the Lyapunov exponents in models with a spike and no spikes.
The paper reviews existing results about the statistical distribution of zeros for the three main types of zeta functions: number-theoretical, geometrical, and dynamical. It provides necessary background and some details about the proofs of main results.
We study Pfaffian random point fields by using the Moore-Dyson quaternion determinants. First, we give sufficient conditions that ensure that a self-dual quaternion kernel defines a valid random point field, and then we prove a CLT for Pfaffian point fields. The proofs are based on a new quaternion extension of the Cauchy-Binet determinantal identity. In addition, we derive the Fredholm determinantal formulas for the Pfaffian point fields which use the quaternion determinant.
Brief lecture notes for a course about random matrices given at the University of Cambridge.
We study the behavior of eigenvalues of matrix P_N + Q_N where P_N and Q_N are two N -by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P_N + Q_N is not universal in the usual sense.
Let H=A+UBU* where A and B are two N-by-N Hermitian matrices and U is a Haar-distributed random unitary matrix, and let μ_H, μ_A, and μ_B be empirical measures of eigenvalues of matrices H, A, and B, respectively. Then, it is known (see, for example, Pastur-Vasilchuk, CMP, 2000, v.214, pp.249-286) that for large N, measure μ_H is close to the free convolution of measures μ_A and μ_B, where the free convolution is a non-linear operation on probability measures. The large deviations of the cumulative distribution function of μ_H from its expectation have been studied by Chatterjee in in JFA, 2007, v. 245, pp.379-389. In this paper we improve Chatterjee's concentration inequality and show that it holds with the rate which is quadratic in N. In addition, we prove a local law for eigenvalues of H, by showing that the normalized number of eigenvalues in an interval converges to the density of the free convolution of μ_A and μ_B provided that the interval has width (log N)^{-1/2}.
In this note we consider the Glauber dynamics for the mean-field Ising model, when all couplings are equal and the external field is uniform. It is proved that the relaxation time of the dynamics is monotonically decreasing in temperature.
Suppose that $X_1,\...,X_n,\...$ are i.i.d. rotationally invariant $N$-by-$N$ matrices. Let $Π_n=X_n\... X_1$. It is known that $n^{-1}\log |Π_n|$ converges to a nonrandom limit. We prove that under certain additional assumptions on matrices $X_i$ the speed of convergence to this limit does not decrease when the size of matrices, $N$, grows.
This paper is concerned with the continuous-time quantum walk on Z, Z^d, and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.
Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov, Ambainis, Kempe and Vazirani (2001) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speed-up in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e., walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.
Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the concept of the extended Fuglede-Kadison determinant. We establish the existence of Lyapunov exponents, derive formulas for their calculation, and show that Lyapunov exponents of free variables are additive with respect to operator product. We illustrate these results using an example of free operators whose singular values are distributed by the Marchenko-Pastur law, and relate this example to C. M. Newman's "triangle" law for the distribution of Lyapunov exponents of large random matrices with independent Gaussian entries. As an interesting by-product of our results, we derive a relation between the extended Fuglede-Kadison determinant and Voiculescu's S-transform.
This paper establishes necessary and sufficient conditions for the products of freely independent unitary operators to converge in distribution to the uniform law on the unit circle.
This paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence.
Let $S_N$ be the sum of vector-valued functions defined on a finite Markov chain. An analogue of the Bernstein--Hoeffding inequality is derived for the probability of large deviations of $S_N$ and relates the probability to the spectral gap of the Markov chain. Examples suggest that this inequality is better than alternative inequalities if the chain has a sufficiently large spectral gap and the function is high-dimensional.
A Central Limit Theorem for non-commutative random variables is proved using the Lindeberg method. The theorem is a generalization of the Central Limit Theorem for free random variables proved by Voiculescu. The Central Limit Theorem in this paper relies on an assumption which is weaker than freeness.
Let $X_i$ denote free identically-distributed random variables. This paper investigates how the norm of products $Π_n=X_1 X_2 ... X_n$ behaves as $n$ approaches infinity. In addition, for positive $X_i$ it studies the asymptotic behavior of the norm of $Y_n=X_1 \circ X_2 \circ ...\circ X_n$, where $\circ$ denotes the symmetric product of two positive operators: $A \circ B=:A^{1/2}BA^{1/2}$. It is proved that if the expectation of $X_i$ is 1, then the norm of the symmetric product $Y_{n}$ is between $c_1 n^{1/2}$ and $c_2 n$ for certain constant $c_1$ and $c_2$. That is, the growth in the norm is at most linear. For the norm of the usual product $Pi_n$, it is proved that the limit of $n^{-1}\log Norm(Pi_n)$ exists and equals $\log \sqrt{E(X_i^{\ast}X_{i})}.$ In other words, the growth in the norm of the product is exponential and the rate equals the logarithm of the Hilbert-Schmidt norm of operator X. Finally, if $π$ is a cyclic representation of the algebra generated by $X_i$, and if $ξ$ is a cyclic vector, then $n^{-1}\log Norm(π(Π_{n}) ξ)=\log \sqrt{E(X_{i}^{\ast}X_{i})}$ for all $n.$ In other words, the growth in the length of the cyclic vector is exponential and the rate coincides with the rate in the growth of the norm of the product. These results are significantly different from analogous results for commuting random variables and generalize results for random matrices derived by Kesten and Furstenberg.
An analogue of the Berry-Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n^{-1/2}, the same as in the classical case. An example with binomial measures shows that this estimate cannot be improved without imposing further restrictions on convolved measures.