arXiv subjects

Kirill Polovnikov

Publications and source records attributed to Kirill Polovnikov.

15 recordsLinked to original sources

arXiv2025

Universal scaling of segment fluctuations in polymer and chromatin dynamics

We demonstrate how center-of-mass (COM) motion influences polymer segment fluctuations. Cancellation of internal forces, together with spatially uncorrelated external noise, generally yields COM diffusivity scaling as $1/s$ with segment length $s$, regardless of fractal dimension, viscoelasticity, or activity. This introduces distinct dynamic scaling corrections to two-point fluctuations and quenched-induced tangential correlations, validated by theory, simulations, and chromatin imaging data. In the latter, the extracted dynamic exponent reveals topological constraints, thereby resolving the discrepancy between chromatin's crumpled structure and its Rouse-like dynamics.

cond-mat.soft↗

arXiv2024

Generative inpainting of incomplete Euclidean distance matrices of trajectories generated by a fractional Brownian motion

Fractional Brownian motion (fBm) features both randomness and strong scale-free correlations, challenging generative models to reproduce the intrinsic memory characterizing the underlying stochastic process. Here we examine a zoo of diffusion-based inpainting methods on a specific dataset of corrupted images, which represent incomplete Euclidean distance matrices (EDMs) of fBm at various memory exponents $H$. Our dataset implies uniqueness of the data imputation in the regime of low missing ratio, where the remaining partial graph is rigid, providing the ground truth for the inpainting. We find that the conditional diffusion generation readily reproduces the built-in correlations of fBm paths in different memory regimes (i.e., for sub-, Brownian and super-diffusion trajectories), providing a robust tool for the statistical imputation at high missing ratio. Furthermore, while diffusion models have been recently shown to memorize samples from the training database, we demonstrate that diffusion behaves qualitatively different from the database search and thus generalize rather than memorize the training dataset. As a biological application, we apply our fBm-trained diffusion model for the imputation of microscopy-derived distance matrices of chromosomal segments (FISH data) - incomplete due to experimental imperfections - and demonstrate its superiority over the standard approaches used in bioinformatics.

cs.CV↗

arXiv2023

Largest eigenvalue statistics of sparse random adjacency matrices

We investigate the statistics of the largest eigenvalue, $\lambda_{\rm max}$, in an ensemble of $N\times N$ large ($N\gg 1$) sparse adjacency matrices, $A_N$. The most attention is paid to the distribution and typical fluctuations of $\lambda_{\rm max}$ in the vicinity of the percolation threshold, $p_c=\frac{1}{N}$. The overwhelming majority of subgraphs representing $A_N$ near $p_c$ are exponentially distributed linear subchains, for which the statistics of the normalized largest eigenvalue can be analytically connected with the Gumbel distribution. For the ensemble of {\rm all} subgraphs near $p_c$ we suggest that under an appropriate modification of the normalization constant the Gumbel distribution provides a reasonably good approximation. Using numerical simulations we demonstrate that the proposed transformation of $\lambda_{\rm max}$ is indeed Gumbel-distributed and the leading finite-size corrections in the vicinity of $p_c$ scale with $N$ as $\sim \ln^{-2}N$. All together, our results reveal a previously unknown universality in eigenvalue statistics of sparse matrices close to the percolation threshold.

cond-mat.stat-mech↗

arXiv2023

Topological and non-topological mechanisms of loops formation in chromosomes: effects on the contact probability

Chromosomes are crumpled polymer chains further folded into a sequence of stochastic loops via loop extrusion. While extrusion has been verified experimentally, the particular means by which the extruding complexes bind DNA polymer remains controversial. Here we analyze the behaviour of the contact probability function for a crumpled polymer with loops for the two possible modes of cohesin binding, topological and non-topological mechanisms. As we show, in the non-topological model the chain with loops resembles a comb-like polymer that can be solved analytically using the quenched disorder approach. In contrast, in the topological binding case the loop constraints are statistically coupled due to long-range correlations present in a non-ideal chain, which can be described by the perturbation theory in the limit of small loop densities. As we show, the quantitative effect of loops on a crumpled chain in the case of topological binding should be stronger, which is translated into a larger amplitude of the log-derivative of the contact probability. Our results highlight a physically different organization of a crumpled chain with loops by the two mechanisms of loops formation.

cond-mat.soft↗

arXiv2022

Communities in C.elegans connectome through the prism of non-backtracking walks

The fundamental relationship between the mesoscopic structure of neuronal circuits and organismic functions they subserve is one of the major challenges in contemporary neuroscience. Formation of structurally connected modules of neurons enacts the conversion from single-cell firing to large-scale behaviour of an organism, highlighting the importance of their accurate profiling in the data. While connectomes are typically characterized by significant sparsity of neuronal connections, recent advances in network theory and machine learning have revealed fundamental limitations of traditionally used community detection approaches in cases where the network is sparse. Here we studied the optimal community structure in the structural connectome of C.elegans, for which we exploited a non-conventional approach that is based on non-backtracking random walks, virtually eliminating the sparsity issue. In full agreement with the previous asymptotic results, we demonstrated that non-backtracking walks resolve the ground truth annotation into clusters on stochastic block models (SBM) with the size and density of the connectome better than the spectral methods related to simple random walks. Based on the cluster detectability threshold, we determined that the optimal number of modules in a recently mapped connectome of C.elegans is 10, which precisely corresponds to the number of isolated eigenvalues in the spectrum of the non-backtracking flow matrix. Broadly, our work provides a robust network-based framework to reveal mesoscopic structures in sparse connectomic datasets, paving way to further investigation of connectome mechanisms for different functions.

q-bio.NC↗

arXiv2021

{\alpha}-Indirect Control in Onion-like Networks

Tens of thousands of parent companies control millions of subsidiaries through long chains of intermediary entities in global corporate networks. Conversely, tens of millions of entities are directly held by sole owners. We propose an algorithm for identification of ultimate controlling entities in such networks that unifies direct and indirect control and allows for continuous interpolation between the two concepts via a factor damping long paths. By exploiting onion-like properties of ownership networks the algorithm scales linearly with the network size and handles circular ownership by design. We apply it to the universe of 4.2 mln UK companies and 4 mln of their holders to understand the distribution of control in the country. Furthermore, we provide the first independent evaluation of the control identification results. We reveal that the proposed $\alpha$-ICON algorithm identifies more than 96% of beneficiary entities from the evaluation set and supersedes the existing approaches reported in the literature. We refer the superiority of $\alpha$-ICON algorithm to its ability to correctly identify the parents long away from their subsidiaries in the network.

physics.soc-ph↗

arXiv2019

Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime

The following question is the subject of our work: could a two-dimensional random path pushed by some constraints to an improbable "large deviation regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang (KPZ) fluctuations? The answer is positive, though non-universal, since the fluctuations depend on the underlying geometry. We consider in details two examples of 2D systems for which imposed external constraints force the underlying stationary stochastic process to stay in an atypical regime with anomalous statistics. The first example deals with the fluctuations of a stretched 2D random walk above a semicircle or a triangle. In the second example we consider a 2D biased random walk along a channel with forbidden voids of circular and triangular shapes. In both cases we are interested in the dependence of a typical span $\left< d(t) \right> \sim t^γ$ of the trajectory of $t$ steps above the top of the semicircle or the triangle. We show that $γ= \frac{1}{3}$, i.e. $\left< d(t) \right>$ shares the KPZ statistics for the semicircle, while $γ=0$ for the triangle. We propose heuristic derivations of scaling exponents $γ$ for different geometries, justify them by explicit analytic computations and compare with numeric simulations. For practical purposes, our results demonstrate that the geometry of voids in a channel might have a crucial impact on the width of the boundary layer and, thus, on the heat transfer in the channel.

cond-mat.stat-mech↗

arXiv2017

Dynamics of polymers: classic results and recent developments

In this chapter we review concepts and theories of polymer dynamics. We think of it as an introduction to the topic for scientists specializing in other subfields of statistical mechanics and condensed matter theory, so, for the readers reference, we start with a short review of the equilibrium static properties of polymer systems. Most attention is paid to the dynamics of unentangled polymer systems, where apart from classical Rouse and Zimm models we review some recent scaling and analytical generalizations. The dynamics of systems with entanglements is also briefly reviewed. Special attention is paid to the discussion of comparatively weakly understood topological states of polymer systems and possible approaches to the description of their dynamics.

cond-mat.soft↗

arXiv2016

From geometric optics to plants: eikonal equation for buckling

Optimal embedding in the three-dimensional space of exponentially growing squeezed surfaces, like plants leaves, or 2D colonies of exponentially reproducing cells, is considered in the framework of conformal approach. It is shown that the boundary profile of a growing tissue is described by the 2D eikonal equation, which provides the geometric optic approximation for the wave front propagating in the media with inhomogeneous refraction coefficient. The variety of optimal surfaces embedded in 3D is controlled by spatial dependence of the refraction coefficient which, in turn, is dictated by the local growth protocol.

cond-mat.soft↗

arXiv2015

Buckling and wrinkling from geometric and energetic viewpoints

We discuss shape profiles emerging in inhomogeneous growth of squeezed tissues. Two approaches are used simultaneously: i) conformal embedding of two-dimensional domain with hyperbolic metrics into the plane, and ii) a pure energetic consideration based on the minimization of the total energy functional. In the latter case the non-uniformly pre-stressed plate, which models the inhomogeneous two-dimensional growth, is analyzed in linear regime under small stochastic perturbations. It is explicitly demonstrated that both approaches give consistent results for buckling profiles and reveal self-similar behavior. We speculate that fractal-like organization of growing squeezed structure has a far-reaching impact on understanding cell proliferation in various biological tissues.

cond-mat.soft↗

arXiv2009

A new class of solutions to the WDVV equation

The known prepotential solutions F to the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation are parametrized by a set {alpha} of covectors. This set may be taken to be indecomposable, since F_{alpha oplus beta}=F_{alpha}+F_{beta}. We couple mutually orthogonal covector sets by adding so-called radial terms to the standard form of F. The resulting reducible covector set yields a new type of irreducible solution to the WDVV equation.

hep-th↗

arXiv2009

N=4 mechanics, WDVV equations and roots

N=4 superconformal multi-particle quantum mechanics on the real line is governed by two prepotentials, U and F, which obey a system of partial differential equations linear in U and generalizing the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation for F. Putting U=0 yields a class of models (with zero central charge) which are encoded by the finite Coxeter root systems. We extend these WDVV solutions F in two ways: the A_n system is deformed n-parametrically to the edge set of a general orthocentric n-simplex, and the BCF-type systems form one-parameter families. A classification strategy is proposed. A nonzero central charge requires turning on U in a given F background, which we show is outside of reach of the standard root-system ansatz for indecomposable systems of more than three particles. In the three-body case, however, this ansatz can be generalized to establish a series of nontrivial models based on the dihedral groups I_2(p), which are permutation symmetric if 3 divides p. We explicitly present their full prepotentials.

hep-th↗

arXiv2008

N=4 superconformal n-particle mechanics via superspace

We revisit the (untwisted) superfield approach to one-dimensional multi-particle systems with N=4 superconformal invariance. The requirement of a standard (flat) bosonic kinetic energy implies the existence of inertial (super-)coordinates, which is nontrivial beyond three particles. We formulate the corresponding integrability conditions, whose solution directly yields the superpotential, the two prepotentials and the bosonic potential. The structure equations for the two prepotentials, including the WDVV equation, follow automatically. The general solution for translation-invariant three-particle models is presented and illustrated with examples. For the four-particle case, we take advantage of known WDVV solutions to construct a D_3 and a B_3 model, thus overcoming a previously-found barrier regarding the bosonic potential. The general solution and classification remain a challenge.

hep-th↗

arXiv2007

N=4 superconformal Calogero models

We continue the research initiated in hep-th/0607215 and apply our method of conformal automorphisms to generate various N=4 superconformal quantum many-body systems on the real line from a set of decoupled particles extended by fermionic degrees of freedom. The su(1,1|2) invariant models are governed by two scalar potentials obeying a system of nonlinear partial differential equations which generalizes the Witten-Dijkgraaf-Verlinde-Verlinde equations. As an application, the N=4 superconformal extension of the three-particle (A-type) Calogero model generates a unique G_2-type Hamiltonian featuring three-body interactions. We fully analyze the N=4 superconformal three- and four-particle models based on the root systems of A_1 + G_2 and F_4, respectively. Beyond Wyllard's solutions we find a list of new models, whose translational non-invariance of the center-of-mass motion fails to decouple and extends even to the relative particle motion.

hep-th↗

arXiv2006

Calogero models and nonlocal conformal transformations

We propose a universal method of relating the Calogero model to a set of decoupled particles on the real line, which can be uniformly applied to both the conformal and nonconformal versions as well as to supersymmetric extensions. For conformal models the simplification is achieved at the price of a nonlocal realization of the full conformal symmetry in the Hilbert space of the resulting free theory. As an application, we construct two different N=2 superconformal extensions.

hep-th↗