Stochastic higher spin six vertex model and Macdonald measures

We prove an identity that relates the q-Laplace transform of the height function of a (higher spin inhomogeneous) stochastic six vertex model in a quadrant on one side and a multiplicative functional of a Macdonald measure on the other. The identity is used to prove the GUE Tracy-Widom asymptotics for two instances of the stochastic six vertex model via asymptotic analysis of the corresponding Schur measures.National Science Foundation (U.S.) (Grant DMS-1056390)National Science Foundation (U.S.) (Grant DMS-1607901

On a conjecture of Widom

We prove a conjecture of H.Widom stated in [W] (math/0108008) about the reality of eigenvalues of certain infinite matrices arising in asymptotic analysis of large Toeplitz determinants. As a byproduct we obtain a new proof of A.Okounkov's formula for the (determinantal) correlation functions of the Schur measures on partitions.Comment: 9 page

Factorizations of Rational Matrix Functions with Application to Discrete Isomonodromic Transformations and Difference Painlev\'e Equations

We study factorizations of rational matrix functions with simple poles on the Riemann sphere. For the quadratic case (two poles) we show, using multiplicative representations of such matrix functions, that a good coordinate system on this space is given by a mix of residue eigenvectors of the matrix and its inverse. Our approach is motivated by the theory of discrete isomonodromic transformations and their relationship with difference Painlev\'e equations. In particular, in these coordinates, basic isomonodromic transformations take the form of the discrete Euler-Lagrange equations. Secondly we show that dPV equations, previously obtained in this context by D. Arinkin and A. Borodin, can be understood as simple relationships between the residues of such matrices and their inverses.Comment: 9 pages; minor typos fixed, journal reference adde

Airy processes and variational problems

We review the Airy processes; their formulation and how they are conjectured to govern the large time, large distance spatial fluctuations of one dimensional random growth models. We also describe formulas which express the probabilities that they lie below a given curve as Fredholm determinants of certain boundary value operators, and the several applications of these formulas to variational problems involving Airy processes that arise in physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI Proceedings: Topics in percolative and disordered systems

Evaluating the effectiveness of companies using the DEA method

The paper assesses the effectiveness of Russian oil and gas companies. It is hypothesized that the DEA method for evaluating the effectiveness of companies gives the most accurate results compared to the method based on profitability coefficients and allows you to determine the individual and industry-wide effectiveness of companie

From interacting particle systems to random matrices

In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on the class on initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses. Some of the fluctuation laws were first discovered in random matrix models. Moreover, the limit process for curved limit shape turned out to show up in a dynamical version of hermitian random matrices, but this analogy does not extend to the case of symmetric matrices. Therefore the connections between growth models and random matrices is only partial.Comment: 18 pages, 8 figures; Contribution to StatPhys24 special issue; minor corrections in scaling of section 2.

The impact of the publication of non-financial statements on the financial performance of companies with the identification of interpectoral features

The purpose of this article is to study the impact of the publication of non-financial statements on the financial performance of Russian companies with the identification of intersectoral feature

Sociocultural Component in the Content of Teaching Foreign Languages to Pre-school Children

The paper is aimed at studying efficiency of learning languages by pre-school children. Different types of memory that are at high level of its development at this age promote children's abilities to acquire languages. Also psychologically, pre-school children are fit to develop speech and communication skills by using adequate materials and tasks. A set of criteria are evaluated to introduce socio-cultural content for promoting effective language learning