The two dimensional directed sandpile with dissipation is transformed into a (1 + 1) dimensional ... more 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 form of the solution for an infinite as well as for a semi-infinite lattice in the horizontal direction. Non-trivial scaling is found in both cases. This conditional probability of the sandpile model is shown to be equal to a K-spin correlation function of the Heisenberg XX spin chain, and the sandpile problem is also shown to be equivalent to the 'random-turns' version of vicious walkers.
The six-vertex model on a finite square lattice with the so-called partial domain wall boundary c... more 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 lattice with free boundary tends to infinity (the limit of a semi-infinite lattice), they are simplified and can be evaluated in a closed form. Bibliography: 19 titles.
In Spring 2015, the Galileo Galilei Institute for Theoretical Physics hosted an eight-week Worksh... more 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 combinatorics, probability theory, and representation theory. Much recent progress in these areas exploits the underlying notion of quantum integrability. We report here on the scientific motivations and background for this activity and on its main outputs.
We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall bo... more We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of freefermion 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 five-vertex model on an M × 2N lattice with fixed boundary conditions of special ... more 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 MacMahon formula for the number of boxed plane partitions, while for generic weights (outside the free-fermion point), it describes enumerations with the weight depending on the cumulative number of jumps along vertical (or horizontal) rows. Various representations for the partition function which describes such enumerations are obtained. Bibliography: 31 titles.
ABSTRACT Quantum nonrelativistic two-component Bose and Fermi gases with an infinitely strong δ-f... more ABSTRACT 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.
The one-dimensional XXZ Heizenberg magnet for ∆ = −∞ is considered, and the time-dependent temper... more 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 Hamilton... more 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 asymptotic of this correlation function, the correlation radius is calculated and its low-temperature behavior is studied. Bibliography: 11 titles.
ABSTRACT For the one-dimensional XXZ Heisenberg chain of spin 1/2 at = –, we calculate the two-ti... more ABSTRACT 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.
ABSTRACT The results of taking into account quark masses in effective low-energy theory featuring... more ABSTRACT 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.
The problem of the limit shape of large alternating sign matrices (ASMs) is addressed by studying... more 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 representation for EFP, the limit shape of large ASMs is found. The case of 3-enumerated ASMs is also considered.
The partition function of the six vertex model on the finite lattice with domain wall boundary co... more 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 method applied to certain fermionic model.
An effective boson Hamiltonian applicable to atomic beam splitters, coupled Bose-Einstein condens... more 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 tight-binding lattice particle. Through exact results for dynamic correlation functions, it is shown how the previous weak interaction dynamics of these systems are extended to strong interactions, now becoming relevant in the experiments. The effect of the number of boson components is also analyzed.
Temperature and time dependent correlation functions of the spin?-1/2 ladder model with infinitel... more Temperature and time dependent correlation functions of the spin?-1/2 ladder model with infinitely strong coupling are calculated. Bibliography: 34 titles.
Journal of Physics A: Mathematical and Theoretical, 2011
The arctic curve, i.e. the spatial curve separating ordered (or 'frozen') and disordered (or 'tem... more 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 rootof-unity vertex weights. In these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve. Some interesting examples are discussed in detail. The upper bound on the maximal degree of the equation in a generic root-of-unity case is obtained.
The two dimensional directed sandpile with dissipation is transformed into a (1 + 1) dimensional ... more 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 form of the solution for an infinite as well as for a semi-infinite lattice in the horizontal direction. Non-trivial scaling is found in both cases. This conditional probability of the sandpile model is shown to be equal to a K-spin correlation function of the Heisenberg XX spin chain, and the sandpile problem is also shown to be equivalent to the 'random-turns' version of vicious walkers.
The six-vertex model on a finite square lattice with the so-called partial domain wall boundary c... more 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 lattice with free boundary tends to infinity (the limit of a semi-infinite lattice), they are simplified and can be evaluated in a closed form. Bibliography: 19 titles.
In Spring 2015, the Galileo Galilei Institute for Theoretical Physics hosted an eight-week Worksh... more 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 combinatorics, probability theory, and representation theory. Much recent progress in these areas exploits the underlying notion of quantum integrability. We report here on the scientific motivations and background for this activity and on its main outputs.
We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall bo... more We consider the six-vertex model in an L-shaped domain of the square lattice, with domain wall boundary conditions, in the case of freefermion 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 five-vertex model on an M × 2N lattice with fixed boundary conditions of special ... more 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 MacMahon formula for the number of boxed plane partitions, while for generic weights (outside the free-fermion point), it describes enumerations with the weight depending on the cumulative number of jumps along vertical (or horizontal) rows. Various representations for the partition function which describes such enumerations are obtained. Bibliography: 31 titles.
ABSTRACT Quantum nonrelativistic two-component Bose and Fermi gases with an infinitely strong δ-f... more ABSTRACT 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.
The one-dimensional XXZ Heizenberg magnet for ∆ = −∞ is considered, and the time-dependent temper... more 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 Hamilton... more 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 asymptotic of this correlation function, the correlation radius is calculated and its low-temperature behavior is studied. Bibliography: 11 titles.
ABSTRACT For the one-dimensional XXZ Heisenberg chain of spin 1/2 at = –, we calculate the two-ti... more ABSTRACT 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.
ABSTRACT The results of taking into account quark masses in effective low-energy theory featuring... more ABSTRACT 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.
The problem of the limit shape of large alternating sign matrices (ASMs) is addressed by studying... more 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 representation for EFP, the limit shape of large ASMs is found. The case of 3-enumerated ASMs is also considered.
The partition function of the six vertex model on the finite lattice with domain wall boundary co... more 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 method applied to certain fermionic model.
An effective boson Hamiltonian applicable to atomic beam splitters, coupled Bose-Einstein condens... more 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 tight-binding lattice particle. Through exact results for dynamic correlation functions, it is shown how the previous weak interaction dynamics of these systems are extended to strong interactions, now becoming relevant in the experiments. The effect of the number of boson components is also analyzed.
Temperature and time dependent correlation functions of the spin?-1/2 ladder model with infinitel... more Temperature and time dependent correlation functions of the spin?-1/2 ladder model with infinitely strong coupling are calculated. Bibliography: 34 titles.
Journal of Physics A: Mathematical and Theoretical, 2011
The arctic curve, i.e. the spatial curve separating ordered (or 'frozen') and disordered (or 'tem... more 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 rootof-unity vertex weights. In these cases the curve is described by algebraic equations which can be worked out explicitly from the parametric solution for this curve. Some interesting examples are discussed in detail. The upper bound on the maximal degree of the equation in a generic root-of-unity case is obtained.
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Papers by Andrei Pronko