Andrei G. Pronko

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• D. Sc.
• leading research fellow at St. Petersburg Department of Steklov Institute of Mathematics

Motzkin chain is a model of nearest-neighbor interacting quantum $s=1$ spins with open boundary conditions. It is known that it has a unique ground state which can be viewed as a sum of Motzkin paths. We consider the case of periodic boundary conditions and provide several conjectures about structure of the ground state space and symmetries of the...

We discuss determinant formulas for the partition function of the six-vertex model with domain wall boundary conditions, which are parametrized by an arbitrary basis of polynomials. In this note we show that our recent result on this problem admits a one-parameter extension.

We consider the six-vertex model at its free-fermion point with domain wall boundary conditions, which is equivalent to random domino tilings of the Aztec diamond. We compute the scaling limit of a particular non-local correlation function, essentially equivalent to the partition function for the domino tilings of a pentagon-shaped domain, obtained...

We consider the homogeneous five-vertex model on a rectangle domain of the square lattice with so-called scalar-product boundary conditions. Peculiarity of these boundary conditions is that the configurations of the model are in an one-to-one correspondence with the 3D Young diagrams limited by a box of a given size. We address the thermodynamics o...

We consider the four-vertex model with a special choice of fixed boundary conditions giving rise to limit shape phenomena. More generally, the considered boundary conditions relate vertex models to scalar products of off-shell Bethe states, boxed plane partitions, and fishnet diagrams in quantum field theory. In the scaling limit, the model exhibit...

We consider the four-vertex model on a finite domain of the square lattice with the so-called scalar-product boundary conditions. It can be described in terms of nonintersecting lattice paths which are additionally restricted in their propagation in one of the two spacial directions. We compute the one-point function measuring the probability to ob...

We consider the four-vertex model with a particular choice of fixed boundary conditions, closely related to scalar products of off-shell Bethe states. In the scaling limit, the model exhibits the limit shape phenomenon, with the emergence of an arctic curve separating a central disordered region from six frozen `corners' of ferroelectric or anti-fe...

We consider the problem of construction of determinant formulas for the partition function of the six-vertex model with domain wall boundary conditions that depend on two sets of spectral parameters. In the pioneering works of Korepin and Izergin a determinant formula was proposed and proved using a recursion relation. In later works, equivalent d...

We consider the problem of construction of determinant formulas for the partition function of the six-vertex model with domain wall boundary conditions. In pioneering works of Korepin and Izergin a determinant formula was proposed, and proved using a recursion relation. In later works, another determinant formulas were given by Kostov, for the rati...

We formulate the six-vertex model with domain wall boundary conditions in terms of an integral over Grassmann variables. Relying on this formulation, we propose a method of calculation of correlation functions of the model for the case of the weights satisfying the free-fermion condition. We consider here in details the one-point correlation functi...

We consider solutions of the RLL-relation with the R-matrix related to the five-vertex model. We show that in the case where the quantum space of the L-operator is infinite-dimensional and coincides with the Fock space of quantum oscillator, the solution of the RLL-relation gives a phase model with two external fields. In the case of a two-dimensio...

We consider the six-vertex model with the rational weights on an s by N square lattice with partial domain wall boundary conditions. We study the one-point function at the boundary where the free boundary conditions are imposed. For a finite lattice, it can be computed by the quantum inverse scattering method in terms of determinants. In the large...

The six-vertex model on a finite square lattice with the so-called partial domain wall boundary conditions is considered. For the case of rational Boltzmann weights, the polarization on the free boundary of the lattice is computed. For the finite lattice the result is given in terms of a ratio of determinants. In the limit, where the side of the la...

We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably comb...

We consider the six-vertex model with the rational weights on an $s\times N$ square lattice, $s\leq N$, with partial domain wall boundary conditions. We study the one-point function at the boundary where the free boundary conditions are imposed. For a finite lattice, it can be computed by the quantum inverse scattering method in terms of determinan...

We consider the problem of calculation of correlation functions in the six-vertex model with domain wall boundary conditions. To this aim, we formulate the model as a scalar product of off-shell Bethe states, and, by applying the quantum inverse scattering method, we derive three different integral representations for these states. By suitably comb...

We consider the five-vertex model on a finite square lattice with fixed boundary conditions such that the configurations of the model are in a one-to-one correspondence with the boxed plane partitions (3D Young diagrams which fit into a box of given size). The partition function of an inhomogeneous model is given in terms of a determinant. For the...

We consider the five-vertex model on a finite square lattice with fixed boundary conditions such that the configurations of the model are in a one-to-one correspondence with the boxed plane partitions (3D Young diagrams which fit into a box of given size). The partition function of an inhomogeneous model is given in terms of a determinant. For the...

We study the symmetric six-vertex model on a finite square lattice with partial domain wall boundary conditions. We use the known connection of the model to the off-shell Bethe states of the Heisenberg XXZ spin chain. We obtain various formulas for the partition function, and also discuss the model in the limit of semiinfinite lattice.

We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the...

We study the relationship between various integral formulas for nonlocal correlation functions of the six-vertex model with domain wall boundary conditions. Specifically, we show how the known representation for the emptiness formation probability can be derived from that for the so-called row configuration probability. A crucial ingredient in the...

We study a multi-point correlation function of the six-vertex model on the square lattice with domain wall boundary conditions which is called the generalized emptiness formation probability. This function describes the probability of observing ferroelectric order around all vertices of any Ferrers diagram λ at the top left corner of the lattice. F...

We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of free-fermion vertex weights. We describe how the recently developed ‘Tangent method’ can be used to determine the form of the arctic curve. The obtained result is in agreement with numerics.

We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of free-fermion vertex weights. We describe how the recently developed `Tangent Method' can be used to determine the form of the Arctic curve. The obtained result is in agreement with numerical simulations.

In the six-vertex model with domain wall boundary conditions, the emptiness formation probability is the probability that a rectangular region in the top left corner of the lattice is frozen. We generalize this notion to the case where the frozen region has the shape of a generic Young diagram. We derive here a multiple integral representation for...

We show that the emptiness formation probability of the six-vertex model with domain wall boundary conditions at its free-fermion point is a τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69...

In the six-vertex model with domain wall boundary conditions, the emptiness formation probability is the probability that a rectangular region in the top left corner of the lattice is frozen. We generalize this notion to the case where the frozen region has the shape of a generic Young diagram. We derive here a multiple integral representation for...

We investigate a free one-dimensional spinless Fermi gas, and the Tonks–Girardeau gas interacting with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for the time-dependent correlation function of the impurity particle. This representation is valid for an arbitrary temperature and an arbitrary repulsive or...

We consider the five-vertex model on an M ×2N lattice with fixed boundary conditions of special type. We discuss a determinantal formula for the partition function in application to description of various enumerations of N × N × (M − N) boxed plane partitions. It is shown that at the free-fermion point of the model, this formula reproduces the MacM...

In Spring 2015, the Galileo Galilei Institute for Theoretical Physics hosted an eight-week Workshop on “Statistical Mechanics, Integrability and Combinatorics”. The Workshop addressed a series of questions in the realm of exactly solvable models of statistical mechanics, featuring numerous ties and overlaps with various problems in modern combinato...

We investigate a free one-dimensional spinless Fermi gas, and the Tonks-Girardeau gas interacting with a single impurity particle of equal mass. We obtain a Fredholm determinant representation for the time-dependent correlation function of the impurity particle. This representation is valid for an arbitrary temperature and an arbitrary repulsive or...

We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions. For free-fermion vertex weights the partition function can be expressed in terms of some Hankel determinant, or equivalently as a Coulomb gas with discrete measure and a non-polynomial potential with two hard walls. We use Coulomb gas...

We consider the one-dimensional gas of fermions interacting with $\delta$-function interaction, at finite positive coupling constant. We compute the time-dependent two-point correlation function of a spin down fermion in a gas of fully polarized fermions, all having spin up. For this correlation function a representation in terms of a Fredholm dete...

We consider the domino tilings of an Aztec diamond with a cut-off corner of macroscopic square shape and given size and address the bulk properties of tilings as the size is varied. We observe that the free energy exhibits a third-order phase transition when the cut-off square, increasing in size, reaches the arctic ellipse-the phase separation cur...

We consider the five-vertex model on a square lattice with fixed boundary conditions which corresponds to weighted (with weight q per elementary cube) enumerations of boxed plane partitions. We calculate the one-point correlation function of the model which describes the probability of a given state on an edge (polarization). This generalizes an an...

Various representations are derived for the emptiness formation probability (a nonlocal correlation function describing the probability of ferroelectric order) in the six-vertex model with domain wall boundary conditions in the case of weights satisfying the free-fermion condition. Starting from the known representation in terms of a multiple integ...

The two-dimensional directed sandpile with dissipation is transformed into a (1+1)-dimensional problem with discrete space and continuous “time”. The master equation for the conditional probability that K grains preserve their initial order during an avalanche is solved exactly. Explicit expressions for asymptotic forms of solutions are given for t...

We consider the one-dimensional delta-interacting electron gas in the case of infinite repulsion. We use determinant representations to study the long time, large distance asymptotics of correlation functions of local fields in the gas phase. We derive differential equations which drive the correlation functions. Using a related Riemann–Hilbert pro...

The two dimensional directed sandpile with dissipation is transformed into a (1+1) dimensional problem with discrete space and continuous `time'. The master equation for the conditional probability that K grains preserve their initial order during an avalanche can thereby be solved exactly, and an explicit expression is given for the asymptotic for...

We address the problem of calculating correlation functions in the six-vertex model with domain wall boundary conditions by considering a particular nonlocal correlation function, called the row configuration probability. This correlation function can be used as a building block for computing various (both local and nonlocal) correlation functions...

The arctic curve, i.e. the spatial curve separating ordered (or `frozen') and disordered (or `temperate) regions, of the six-vertex model with domain wall boundary conditions is discussed for the root-of-unity vertex weights. In these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution...

An explicit expression for the spatial curve separating the region of ferroelectric order (`frozen' zone) from the disordered one (`temperate' zone) in the six-vertex model with domain wall boundary conditions in its anti-ferroelectric regime is obtained.

The problem of the form of the ‘arctic’ curve of the six-vertex model with domain wall boundary conditions in its disordered regime is addressed. It is well-known that in the scaling limit the model exhibits phase-separation, with regions of order and disorder sharply separated by a smooth curve, called the arctic curve. To find this curve, we stud...

Boxed plane partitions are considered in terms of the five-vertex model on a finite lattice with fired boundary conditions. Assuming that all weights of the model have the same value, the one-point correlation function. describing the probability of having a given state on an arbitrary horizontal edge of the lattice is calculated. This is equivalen...

The problem of the limit shape of large alternating sign matrices (ASMs) is addressed by studying the emptiness formation probability (EFP) in the domain-wall six-vertex model. Assuming that the limit shape arises in correspondence to the `condensation' of almost all solutions of the saddle-point equations for certain multiple integral representati...

The emptiness formation probability in the six-vertex model with domain wall boundary conditions is considered. This correlation function allows one to address the problem of limit shapes in the model. We apply the quantum inverse scattering method to calculate the emptiness formation probability for the inhomogeneous model. For the homogeneous mod...

The problem of limit shapes in the six-vertex model with domain wall boundary conditions is addressed by considering a specially tailored bulk correlation function, the emptiness formation probability. A closed expression of this correlation function is given, both in terms of certain determinant and multiple integral, which allows for a systematic...

The Hankel determinant representations for the partition function and boundary correlation functions of the six-vertex model with domain wall boundary conditions are investigated by the methods of orthogonal polynomial theory. For specific values of the parameters of the model, corresponding to 1-, 2- and 3-enumerations of Alternating Sign Matrices...

The six-vertex model with domain wall boundary conditions on an N × N square lattice is considered. The two-point correlation function describing the probability of having two vertices in a given state at opposite (top and bottom) boundaries of the lattice is calculated. It is shown that this two-point boundary correlator is expressible in a very s...

The six-vertex model with domain wall boundary conditions (DWBC) on an N x N square lattice is considered. The two-point correlation function describing the probability of having two vertices in a given state at opposite (top and bottom) boundaries of the lattice is calculated. It is shown that this two-point boundary correlator is expressible in a...

The six-vertex model with Domain Wall Boundary Conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how t...

An explicit expression for the numbers A(n,r;3) describing the refined 3-enumeration of alternating sign matrices is given. The derivation is based on the recent results of Stroganov for the corresponding generating function. As a result, A(n,r;3)'s are represented as 1-fold sums which can also be written in terms of terminating series of argument...

The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a...

The partition function of the six-vertex model on the finite lattice with domain wall boundary conditions is considered. Starting from Hankel determinant representation, some alternative representations for the partition function are given. It is argued that one of these representations can be rephrased in the language of the angular quantization m...

The one-dimensional XXZ Heizenberg magnet for = - is considered, and the time-dependent temperature correlation function of the z components of local spin operators is calculated. In the thermodynamic limit, the correlation function is expressed in terms of the Fredholm determinants of linear integral operators. Bibliography: 23 titles.

The XX0 chain in the external magnetic field directed along the z axis is discussed. The Hamiltonian describing the exchange interaction between two and four neighboring sites of the chain is constructed. An integral representation for the equal-time temperature correlation function of the third spin components is given. From the long-distance asym...

Time-dependent temperature correlators of the anisotropic Heisenberg XY chain are calculated by the integration techniques with respect to Grassmann variables. For a chain of length M, the correlators are represented as the determinants of 2M 2M matrices. In the thermodynamic limit, the correlation functions are expressed in terms of the Fredholm d...

For the one-dimensional XXZ Heisenberg chain of spin 1/2 at D=-∞, we calculate the two-time temperature correlation function of the third components of local spins. For the correlation function in the thermodynamic limit, we obtain the expression in terms of Fredholm determinants for linear integral operators.

We consider the six-vertex model on an $N \times N$ square lattice with the domain wall boundary conditions. Boundary one-point correlation functions of the model are expressed as determinants of $N\times N$ matrices, generalizing the known result for the partition function. In the free fermion case the explicit answers are obtained. The introduced...

For the one-dimensional XXZ Heisenberg chain of spin 1/2 at = –, we calculate the two-time temperature correlation function of the third components of local spins. For the correlation function in the thermodynamic limit, we obtain the expression in terms of Fredholm determinants for linear integral operators.

An effective boson Hamiltonian applicable to atomic beam splitters, coupled Bose-Einstein condensates, and optical lattices can be made exactly solvable by including all n-body interactions. The model can include an arbitrary number of boson components. In the strong interaction limit the model becomes a quantum phase model, which also describes a...

Temperature and time dependent correlation functions of the spin-1/2 ladder model with infinitely strong coupling are calculated. Bibliography: 34 titles.

The one-dimensional Hubbard model with infinitely strong repulsion between electrons is considered. Explicit expressions for the two-point correlators of local densities (dependent on time, temperature, the chemical potential, and the external field) are obtained. Bibliography: 12 titles.

We consider the one-dimensional Hubbard model with the infinitely strong repulsion. The two-point dynamical temperature correlation functions are calculated. They are represented as Fredholm determinants of linear integrable integral operators. 1 Introduction The one-dimensional Hubbard model [1] is one of the most interesting and important model o...

The results of taking into account quark masses in effective low-energy theory featuring an extended chiral field that contains scalar-diquark fields and pseudoscalar-meson fields on equal footing have been discussed. It has been found that extended chiral dynamics generates relations between the masses of scalar diquarks and their decay constants.

We introduce extended chiral transformation, which depends both on pseudoscalar and diquark fields as parameters and determine its group structure. Assuming soft symmetry breaking in diquark sector, bosonisation of a quasi-Goldstone $ud$-diquark is performed. In the chiral limit the $ud$-diquark mass is defined by the gluon condensate, $m_{ud}\appr...

We consider the one-dimensional delta-interacting electron gas in the case of infinite repulsion. We use determinant representations to study the long time, large distance asymptotics of correlation functions of local fields in the gas phase. We derive differential equations which drive the correlation functions. Using a related Riemann-Hilbert pro...

We consider the one-dimensional Hubbard model with the infinitely strong repulsion. The two-point dynamical temperature correlation functions are calculated. They are represented as Fredholm determinants of linear integrable integral operators.

The quantum nonrelativistic two-component Bose and Fermi gases with the infinitely strong point-like coupling between particles in one space dimension are considered. Time and temperature dependent correlation functions are represented in the thermodynamic limit as Fredholm determinants of integrable linear integral operators. Comment: 40 pages, La...

Quantum nonrelativistic two-component Bose and Fermi gases with an infinitely strong δ-function interaction between particles are considered. The two-point correlation functions depending on temperature, time, distance, chemical potential and external field are represented as Fredholm determinants of linear integral operators.

We consider bosonisation of the low-energy QCD based on integrating the anomaly of the extended chiral (E$\chi$) transformation which depends both on pseudoscalar meson and scalar diquark fields as parameters. The relationship between extended chiral and usual chiral anomalies and related anomalous actions is studied. The effective action for the e...

We introduce an extended chiral transformation, which depends both on pseudoscalar and diquark fields as parameters, and determine its group structure. Assuming soft symmetry breaking in the diquark sector, the bosonisation of a quasi-Goldstone ud-diquark is performed. In the chiral limit the ud-diquark mass is defined by the gluon condensate, mud...

We introduce extended chiral transformation, which depends both on pseudoscalar and diquark fields as parameters and determine its group structure. Assuming soft symmetry breaking in diquark sector, bosonisation of a quasi-Goldstone $ud$-diquark is performed. In the chiral limit the $ud$-diquark mass is defined by the gluon condensate, $m_{ud}\appr...

A path integral with diquark currents is continued into Euclidean space, and the chiral and conformal anomalies are calculated. The chiral anomaly is integrated to the anomaly action, on the basis of which additional terms of the pion chiral action that depend on the diquark currents are obtained.

The quark path integral with diquark currents is continued into Euclidean space and its chiral and conformal anomalies are calculated. Additional terms in the chiral action for pions due to diquark currents are derived.