arXiv subjects

A. M. Povolotsky

Publications and source records attributed to A. M. Povolotsky.

At least 19 recordsLinked to original sources

arXiv2024

Exact loop densities in $O(1)$ dense loop model on the cylinder of odd circumference and clusters in half-turn self-dual critical percolation

We consider $O(1)$ dense loop model in a square lattice wrapped on a cylinder of odd circumference $L$ and calculate the exact densities of loops. These densities of loops are equal to the densities of critical bond percolation clusters on a forty-five-degree rotated square lattice rolled into a cylinder with special boundary conditions which we refer to as half-turn self-dual percolation. The solution is based on a correspondence between the $O(1)$ dense loop model and the six-vertex model at the Razumov-Stroganov point.

math-ph↗

arXiv2023

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice

This work continues the study started in \cite{Povolotsky2021}, where the exact densities of loops in the O(1) dense loop model on an infinite strip of the square lattice with periodic boundary conditions were obtained. These densities are also equal to the densities of critical percolation clusters on the forty five degree rotated square lattice rolled into a cylinder. Here, we extend those results to the square lattice with a tilt. This in particular allow us to obtain the densities of critical percolation clusters on the cylinder of the square lattice of standard orientation extensively studied before. We obtain exact densities of contractible and non-contractible loops or equivalently the densities of critical percolation clusters, which do not and do wrap around the cylinder respectively. The solution uses the mapping of O(1) dense loop model to the six-vertex model in the Razumov-Stroganov point, while the effective tilt is introduced via the the inhomogeneous transfer matrix proposed by Fujimoto. The further solution is based on the Bethe ansatz and Fridkin-Stroganov-Zagier's solution of the Baxter's T-Q equation. The results are represented in terms of the solution of two explicit systems of linear algebraic equations, which can be performed either analytically for small circumferences of the cylinder or numerically for larger ones. We present exact rational values of the densities on the cylinders of small circumferences and several lattice orientations and use the results of high precision numerical calculations to study the finite-size corrections to the densities, in particular their dependence on the tilt of the lattice.

math-ph↗

arXiv2022

Generalized TASEP on open chains with a modified injection condition

We report here our preliminary results in the study of a new version of the generalized Totally Asymmetric Simple Exclusion process (gTASEP) on open tracks. In the gTASEP an additional interaction between the particles is considered, besides the hard-core exclusion, which exists in the standard TASEP. It is modeled by the introduction of a second hopping probability $p_m$ for the particles, belonging to the same cluster, in addition to the standard hopping probability p, which now applies only for single particles and the head (rightmost) particle of a cluster. We briefly describe how one can obtain analytically solvable version of gTASEP by modifying the left (injection) boundary condition. A short comparison is made with the previously studied version of gTASEP on open tracks.

cond-mat.stat-mech↗

arXiv2022

Nonstationary generalized TASEP in KPZ and jamming regimes

We study the model of the totally asymmetric exclusion process with generalized update, which compared to the usual totally asymmetric exclusion process, has an additional parameter enhancing clustering of particles. We derive the exact multiparticle distributions of distances travelled by particles on the infinite lattice for two types of initial conditions: step and alternating once. Two different scaling limits of the exact formulas are studied. Under the first scaling associated to Kardar-Parisi-Zhang (KPZ) universality class we prove convergence of joint distributions of the scaled particle positions to finite-dimensional distributions of the universal Airy$_2$ and Airy$_1$ processes. Under the second scaling we prove convergence of the same position distributions to finite-dimensional distributions of two new random processes, which describe the transition between the KPZ regime and the deterministic aggregation regime, in which the particles stick together into a single giant cluster moving as one particle. It is shown that the transitional distributions have the Airy processes and fully correlated Gaussian fluctuations as limiting cases. We also give the heuristic arguments explaining how the non-universal scaling constants appearing from the asymptotic analysis in the KPZ regime are related to the properties of translationally invariant stationary states in the infinite system and how the parameters of the model should scale in the transitional regime.

math-ph↗

arXiv2021

Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder

We obtain exact densities of contractible and non-contractible loops in the O(1) model on a strip of the square lattice rolled into an infinite cylinder of finite even circumference $L$. They are also equal to the densities of critical percolation clusters on forty five degree rotated square lattice rolled into a cylinder, which do not or do wrap around the cylinder respectively. The results are presented as explicit rational functions of $L$ taking rational values for any even $L$. Their asymptotic expansions in the large $L$ limit have irrational coefficients reproducing the earlier results in the leading orders. The solution is based on a mapping to the six-vertex model and the use of technique of Baxter's T-Q equation.

math-ph↗

arXiv2020

Current statistics in the q-boson zero range process

We obtain exact formulas of the first two cumulants of particle current in the q-boson zero range process via exact perturbative solution of the TQ-relation. The result is represented as an infinite sum of double contour integrals. We perform the asymptotic analysis of the large system size limit $N\to\infty$ of the expressions obtained. For $|q|\neq1$ the leading terms of the second cumulant reproduce the $N^{3/2}$ scaling expected for models in the Kardar-Parisi-Zhang universality class. The scaling $q\asymp\exp(-α/\sqrt{N})\to1$ corresponds to the the crossover between the Kardar-Parisi-Zhang and Edwards-Wilkinson universality classes. Under this scaling the sum converges to an integral, resulting in the crossover scaling function derived previously for the asymmetric simple exclusion process and conjectured to be universal.

math-ph↗

arXiv2019

Laws of large numbers in the Raise and Peel model

We establish the exact laws of large numbers for two time additive quantities in the Raise and Peel model, the number of tiles removed by avalanches and the number of global avalanches happened by given time. The validity of conjectures for the related stationary state correlation functions then follow. The proof is based on the technique of Baxter's T-Q equation applied to the associated XXZ chain and on its solution at $Δ=-1/2$ obtained by Fridkin, Stroganov and Zagier.

math-ph↗

arXiv2015

Emergence of jams in the generalized totally asymmetric simple exclusion process

The generalized totally asymmetric exclusion process (TASEP) [J. Stat. Mech. P05014 (2012)] is an integrable generalization of the TASEP equipped with an interaction, which enhances the clustering of particles. The process interpolates between two extremal cases: the TASEP with parallel update and the process with all particles irreversibly merging into a single cluster moving as an isolated particle. We are interested in the large time behavior of this process on a ring in the whole range of the parameter $λ$ controlling the interaction. We study the stationary state correlations, the cluster size distribution and the large-time fluctuations of integrated particle current. When $λ$ is finite, we find the usual TASEP-like behavior: The correlation length is finite; there are only clusters of finite size in the stationary state and current fluctuations belong to the Kardar-Parisi-Zhang universality class. When $λ$ grows with the system size so does the correlation length. We find a nontrivial transition regime with clusters of all sizes on the lattice. We identify a crossover parameter and derive the large deviation function for particle current, which interpolates between the case considered by Derrida-Lebowitz and a single particle diffusion.

math-ph↗

arXiv2013

On integrability of zero-range chipping models with factorized steady state

Conditions of integrability of general zero range chipping models with factorized steady state, which were proposed in [Evans, Majumdar, Zia 2004 J. Phys. A 37 L275], are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis we conjecture an integral formula for the Green function of evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.

math-ph↗

arXiv2012

Universal exit probabilities in the TASEP

We study the joint exit probabilities of particles in the totally asymmetric simple exclusion process (TASEP) from space-time sets of given form. We extend previous results on the space-time correlation functions of the TASEP, which correspond to exits from the sets bounded by straight vertical or horizontal lines. In particular, our approach allows us to remove ordering of time moments used in previous studies so that only a natural space-like ordering of particle coordinates remains. We consider sequences of general staircase-like boundaries going from the northeast to southwest in the space-time plane. The exit probabilities from the given sets are derived in the form of Fredholm determinant defined on the boundaries of the sets. In the scaling limit, the staircase-like boundaries are treated as approximations of continuous differentiable curves. The exit probabilities with respect to points of these curves belonging to arbitrary space-like path are shown to converge to the universal Airy$_2$ process.

cond-mat.stat-mech↗

arXiv2012

The totally asymmetric exclusion process with generalized update

We consider the totally asymmetric exclusion process in discrete time with generalized updating rules. We introduce a control parameter into the interaction between particles. Two particular values of the parameter correspond to known parallel and sequential updates. In the whole range of its values the interaction varies from repulsive to attractive. In the latter case the particle flow demonstrates an apparent jamming tendency not typical for the known updates. We solve the master equation for $N$ particles on the infinite lattice by the Bethe ansatz. The non-stationary solution for arbitrary initial conditions is obtained in a closed determinant form.

cond-mat.stat-mech↗

arXiv2011

Generalized Green Functions and current correlations in the TASEP

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinantal formula for the generalized Green function which describes transitions between positions of particles at different individual time moments. In particular, the generalized Green function defines a probability measure at staircase lines on the space-time plane. The marginals of this measure are the TASEP correlation functions in the space-time region not covered by the standard Green function approach. As an example, we calculate the current correlation function that is the joint probability distribution of times taken by selected particles to travel given distance. An asymptotic analysis shows that current fluctuations converge to the ${Airy}_2$ process.

math-ph↗

arXiv2008

From Vicious Walkers to TASEP

We propose a model of semi-vicious walkers, which interpolates between the totally asymmetric simple exclusion process and the vicious walkers model, having the two as limiting cases. For this model we calculate the asymptotics of the survival probability for $m$ particles and obtain a scaling function, which describes the transition from one limiting case to another. Then, we use a fluctuation-dissipation relation allowing us to reinterpret the result as the particle current generating function in the totally asymmetric simple exclusion process. Thus we obtain the particle current distribution asymptotically in the large time limit as the number of particles is fixed. The results apply to the large deviation scale as well as to the diffusive scale. In the latter we obtain a new universal distribution, which has a skew non-Gaussian form. For $m$ particles its asymptotic behavior is shown to be $e^{-\frac{y^{2}}{2m^{2}}}$ as $y\to -\infty $ and $e^{-\frac{y^{2}}{2m}}y^{-\frac{m(m-1)}{2}}$ as $y\to \infty $.

cond-mat.stat-mech↗

arXiv2007

Determinant solution for the Totally Asymmetric Exclusion Process with parallel update II. Ring geometry

Using the Bethe ansatz we obtain the determinant expression for the time dependent transition probabilities in the totally asymmetric exclusion process with parallel update on a ring. Developing a method of summation over the roots of Bethe equations based on the multidimensional analogue of the Cauchy residue theorem, we construct the resolution of the identity operator, which allows us to calculate the matrix elements of the evolution operator and its powers. Representation of results in the form of an infinite series elucidates connection to other results obtained for the ring geometry. As a byproduct we also obtain the generating function of the joint probability distribution of particle configurations and the total distance traveled by the particles.

cond-mat.stat-mech↗

arXiv2006

Determinant solution for the Totally Asymmetric Exclusion Process with parallel update

We consider the totally asymmetric exclusion process in discrete time with the parallel update. Constructing an appropriate transformation of the evolution operator, we reduce the problem to that solvable by the Bethe ansatz. The non-stationary solution of the master equation for the infinite 1D lattice is obtained in a determinant form. Using a modified combinatorial treatment of the Bethe ansatz, we give an alternative derivation of the resulting determinant expression.

cond-mat.stat-mech↗

arXiv2006

Bethe ansatz solution of discrete time stochastic processes with fully parallel update

We present the Bethe ansatz solution for the discrete time zero range and asymmetric exclusion processes with fully parallel dynamics. The model depends on two parameters: $p$, the probability of single particle hopping, and $q$, the deformation parameter, which in the general case, $|q|<1$, is responsible for long range interaction between particles. The particular case $q=0$ corresponds to the Nagel-Schreckenberg traffic model with $v_{\mathrm{max}}=1$. As a result, we obtain the largest eigenvalue of the equation for the generating function of the distance travelled by particles. For the case $q=0$ the result is obtained for arbitrary size of the lattice and number of particles. In the general case we study the model in the scaling limit and obtain the universal form specific for the Kardar-Parisi-Zhang universality class. We describe the phase transition occurring in the limit $p\to 1$ when $q<0$.

cond-mat.stat-mech↗

arXiv2005

Organization of complex networks without multiple connections

We find a new structural feature of equilibrium complex random networks without multiple and self-connections. We show that if the number of connections is sufficiently high, these networks contain a core of highly interconnected vertices. The number of vertices in this core varies in the range between $const N^{1/2}$ and $const N^{2/3}$, where $N$ is the number of vertices in a network. At the birth point of the core, we obtain the size-dependent cut-off of the distribution of the number of connections and find that its position differs from earlier estimates.

cond-mat.stat-mech↗

arXiv2004

Bethe ansatz solution of zero-range process with nonuniform stationary state

The eigenfunctions and eigenvalues of the master-equation for zero range process with totally asymmetric dynamics on a ring are found exactly using the Bethe ansatz weighted with the stationary weights of particle configurations. The Bethe ansatz applicability requires the rates of hopping of particles out of a site to be the $q$-numbers $[n]_q$. This is a generalization of the rates of hopping of noninteracting particles equal to the occupation number $n$ of a site of departure. The noninteracting case can be restored in the limit $q\to 1$. The limiting cases of the model for $q=0,\infty$ correspond to the totally asymmetric exclusion process, and the drop-push model respectively. We analyze the partition function of the model and apply the Bethe ansatz to evaluate the generating function of the total distance travelled by particles at large time in the scaling limit. In case of non-zero interaction, $q \ne 1$, the generating function has the universal scaling form specific for the Kardar-Parizi-Zhang universality class.

cond-mat.stat-mech↗