arXiv subjects

S. Nechaev

Publications and source records attributed to S. Nechaev.

At least 19 recordsLinked to original sources

arXiv2020

Growth rate of 3D heaps of pieces

We consider configurational statistics of three-dimensional heaps of $N$ pieces ($N\gg 1$) on a simple cubic lattice in a large 3D bounding box of base $n \times n$, and calculate the growth rate, $\Lambda(n)$, of the corresponding partition function, $Z_N\sim N^{\theta}[\Lambda(n)]^N$, at $n\gg 1$. Our computations rely on a theorem of G.X. Viennot \cite{viennot-rev}, which connects the generating function of a $(D+1)$-dimensional heap of pieces to the generating function of projection of these pieces onto a $D$-dimensional subspace. The growth rate of a heap of cubic blocks, which cannot touch each other by vertical faces, is thus related to the position of zeros of the partition function describing 2D lattice gas of hard squares. We study the corresponding partition function exactly at low densities on finite $n\times n$ lattice of arbitrary $n$, and extrapolate its behavior to the jamming transition density. This allows us to estimate the limiting growth rate, $\Lambda =\lim_{n\to\infty}\Lambda(n)\approx 9.5$. The same method works for any underlying 2D lattice and for various shapes of pieces: flat vertical squares, mapped to an ensemble of repulsive dimers, dominoes mapped to an ensemble of rectangles with hard-core repulsion, etc.

cond-mat.stat-mech↗

arXiv2020

Field-driven tracer diffusion through curved bottlenecks: Fine structure of first passage events

Using scaling arguments and extensive numerical simulations, we study dynamics of a tracer particle in a corrugated channel represented by a periodic sequence of broad chambers and narrow funnel-like bottlenecks enclosed by a hard-wall boundary. A tracer particle is affected by an external force pointing along the channel, and performs an unbiased diffusion in the perpendicular direction. We present a detailed analysis a) of the distribution function of the height above the funnel's boundary at which the first crossing of a given bottleneck takes place, and b) of the distribution function of the first passage time to such an event. Our analysis reveals several new features of the dynamical behaviour which are overlooked in the studies based on the Fick-Jacobs approach. In particular, trajectories passing through a funnel concentrate predominantly on its boundary, which makes first-crossing events very sensitive to the presence of binding sites and a microscopic roughness.

cond-mat.stat-mech↗

arXiv2020

Self-isolation or borders closing: what prevents epidemic spreading better?

Pandemic distribution of COVID-19 in the world has motivated us to discuss combined effects of network clustering and adaptivity on epidemic spreading. We address the question concerning the choice of optimal mechanism for most effective prohibiting disease propagation in a connected network: adaptive clustering, which mimics self-isolation (SI) in local communities, or sharp instant clustering, which looks like frontiers closing (FC) between cities and countries. SI-networks are "adaptively grown" under condition of maximization of small cliques in the entire network, while FC-networks are "instantly created". Running the standard SIR model on clustered SI- and FC-networks, we demonstrate that the adaptive network clustering prohibits the epidemic spreading better than the instant clustering in the network with similar parameters. We found that SI model has scale-free property for degree distribution $P(k)\sim k^{\eta}$ with small critical exponent $-2<\eta<-1$ and argue that scale-free behavior emerges due to the randomness in the initial degree distributions and is absent for random regular graphs.

physics.soc-ph↗

arXiv2020

Non-backtracking walks reveal compartments in sparse chromatin interaction networks

Chromatin communities stabilized by protein machinery play essential role in gene regulation and refine global polymeric folding of the chromatin fiber. However, treatment of these communities in the framework of the classical network theory (stochastic block model, SBM) does not take into account intrinsic linear connectivity of the chromatin loci. Here we propose the "polymer" block model, paving the way for community detection in polymer networks. On the basis of this new model we modify the non-backtracking flow operator and suggest the first protocol for annotation of compartmental domains in sparse single cell Hi-C matrices. In particular, we prove that our approach corresponds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking operator resolves the true compartmental structure up to the theoretical detectability threshold, while all commonly used operators fail above it. We test various operators on real data and conclude that the sizes of the non-backtracking single cell domains are most close to the sizes of compartments from the population data. Moreover, the found domains clearly segregate in the gene density and correlate with the population compartmental mask, corroborating biological significance of our annotation of the chromatin compartmental domains in single cells Hi-C matrices.

q-bio.MN↗

arXiv2019

Many-body contacts in fractal polymer chains and fBm trajectories

We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension $d_f$ visits the same spot $n \ge 3$ times, at given moments $t_1, ..., t_n$, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with $d_f = 2$, the resulting many-body contact probabilities cannot be factorized into a product of single loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently, the probabilities we calculate here can be interpreted as probabilities of multi-body contacts in a fractal polymer conformation with the same fractal dimension $d_f$. This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically-stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.

physics.chem-ph↗

arXiv2018

Spectral peculiarity and criticality of the human connectome

We have performed the comparative spectral analysis of structural connectomes for various organisms using open-access data. Our analysis indicates several new peculiar features of the human connectome. We found that the spectral density of human connectome has the maximal deviation from the spectral density of the randomized network compared to all other organisms. For many animals except human structural peculiarities of connectomes are well reproduced in the network evolution induced by the preference of 3-cycles formation. To get the reliable fit , we discovered the crucial role of the conservation of local clusterization in human connectome evolution. We investigated for the first time the level spacing distribution in the spectrum of human connectome graph Laplacian. It turns out that the spectral statistics of human connectome corresponds exactly to the critical regime familiar in the condensed matter physics which is hybrid of Wigner-Dyson and Poisson distributions. This observation provides the strong support for the much debated statement of the brain criticality.

q-bio.NC↗

arXiv2018

Phase transitions in social networks inspired by the Schelling model

We propose two models of social segregation inspired by the Schelling model. Agents in our models are nodes of evolving social networks. The total number of social connections of each node remains constant in time, though may vary from one node to the other. The first model describes a "polychromatic" society, in which colors designate different social categories of agents. The parameter $\mu$ favors/disfavors connected "monochromatic triads", i.e. connected groups of three individuals \emph{within the same social category}, while the parameter $\nu$ controls the preference of interactions between two individuals \emph{from different social categories}. The polychromatic model has several distinct regimes in $(\mu,\nu)$-parameter space. In $\nu$-dominated region, the phase diagram is characterized by the plateau in the number of the inter-color connections, where the network is bipartite, while in $\mu$-dominated region, the network looks as two weakly connected unicolor clusters. At $\mu>\mu_{crit}$ and $\nu >\nu_{crit}$ two phases are separated by a critical line, while at small values of $\mu$ and $\nu$, a gradual crossover between the two phases occurs. The second "colorless" model describes a society in which the advantage/disadvantage of forming small fully connected communities (short cycles or cliques in a graph) is controlled by a parameter $\gamma$. We analyze the topological structure of a social network in this model and demonstrate that above a critical threshold, $\gamma^+>0$, the entire network splits into a set of weakly connected clusters, while below another threshold, $\gamma^-<0$, the network acquires a bipartite graph structure. Our results propose mechanisms of formation of self-organized communities in international communication between countries, as well as in crime clans and prehistoric societies.

physics.soc-ph↗

arXiv2017

Effective Hamiltonian of topologically stabilized polymer states

Topologically stabilized polymer conformations observed in melts of nonconcatenated polymer rings and crumpled globules, are considered to be a good candidate for the description of the spatial structure of mitotic chromosomes. Despite significant efforts, the microscopic Hamiltonian capable of describing such systems, remains yet inaccessible. In this paper we consider a Gaussian network - a system with a simple Hamiltonian quadratic in all coordinates - and show that by tuning interactions, one can obtain fractal equilibrium conformations with any fractal dimension between 2 (ideal polymer chain) and 3 (crumpled globule). Monomer-to-monomer distances in topologically stabilized states, according to our analysis of available numerical data, fit very well the Gaussian distribution, giving an additional argument in support of the quadratic Hamiltonian model. Mathematically, the resulting polymer conformations can be mapped onto the trajectories of a subdiffusive fractal Brownian particle. As a by-product of our study, two novel continual integral representations of the fractal Brownian motion are proposed.

cond-mat.stat-mech↗

arXiv2017

Finite plateau in spectral gap of polychromatic constrained random networks

We consider the canonical ensemble of multilayered constrained Erdos-Renyi networks (CERN) and regular random graphs (RRG), where each layer represents graph vertices painted in a specific color. We study the critical behavior in such networks under changing the fugacity, $\mu$, which controls the number of monochromatic triads of nodes. The behavior of considered systems is investigated via the spectral properties of the adjacency and Laplacian matrices of corresponding networks. For some wide region of $\mu$ we find the formation of a finite plateau in the number of the intercolor links, which exactly matches the finite plateau for the algebraic connectivity of the network (the value of the first non-vanishing eigenvalue of the Laplacian matrix, $\lambda_2$). We claim that at the plateau the restoring of the spontaneously broken $Z_2$ symmetry by the mechanism of modes collectivization in clusters of different colors occurs. The phenomena of a finite plateau formation holds for the polychromatic (multilayer) networks with $M>2$ colors.

cond-mat.stat-mech↗

arXiv2017

Number-theoretic aspects of 1D localization: "popcorn function" with Lifshitz tails and its continuous approximation by the Dedekind $\eta$

We discuss the number-theoretic properties of distributions appearing in physical systems when an observable is a quotient of two independent exponentially weighted integers. The spectral density of ensemble of linear polymer chains distributed with the law $\sim f^L$ ($0<f<1$), where $L$ is the chain length, serves as a particular example. At $f\to 1$, the spectral density can be expressed through the discontinuous at all rational points, Thomae ("popcorn") function. We suggest a continuous approximation of the popcorn function, based on the Dedekind $\eta$-function near the real axis. Moreover, we provide simple arguments, based on the "Euclid orchard" construction, that demonstrate the presence of Lifshitz tails, typical for the 1D Anderson localization, at the spectral edges. We emphasize that the ultrametric structure of the spectral density is ultimately connected with number-theoretic relations on asymptotic modular functions. We also pay attention to connection of the Dedekind $\eta$-function near the real axis to invariant measures of some continued fractions studied by Borwein and Borwein in 1993.

math.NT↗

arXiv2016

Peculiar spectral statistics of ensembles of trees and star-like graphs

In this paper we investigate the eigenvalue statistics of exponentially weighted ensembles of full binary trees and $p$-branching star graphs. We show that spectral densities of corresponding adjacency matrices demonstrate peculiar ultrametric structure inherent to sparse systems. In particular, the tails of the distribution for binary trees share the "Lifshitz singularity" emerging in the one-dimensional localization, while the spectral statistics of $p$-branching star-like graphs is less universal, being strongly dependent on $p$. The hierarchical structure of spectra of adjacency matrices is interpreted as sets of resonance frequencies, that emerge in ensembles of fully branched tree-like systems, known as dendrimers. However, the relaxational spectrum is not determined by the cluster topology, but has rather the number-theoretic origin, reflecting the peculiarities of the rare-event statistics typical for one-dimensional systems with a quenched structural disorder. The similarity of spectral densities of an individual dendrimer and of ensemble of linear chains with exponential distribution in lengths, demonstrates that dendrimers could be served as simple disorder-less toy models of one-dimensional systems with quenched disorder.

cond-mat.stat-mech↗

arXiv2016

Many-body localization and new critical phenomena in regular random graphs and constrained Erd\H{o}s-Renyi networks

We consider from the localization perspective the new critical phenomena discovered recently for perturbed random regular graphs (RRG) and constrained Erd\H{o}s-R\'enyi networks (CERN) \cite{crit2}. At some critical value of the chemical potential of 3-cycles, $\mu$, the network decays into the maximally possible number of almost full subgraphs, and the spectrum of the Laplacian matrix acquires the two-zonal structure with a large gap. We find that the Laplacian eigenvalue statistics corresponds to delocalized states in one zone, and to the localized states in the second one. We interpret this behavior in terms of the many-body localization problem where the structure of the Fock space of some interacting many-body system is approximated by the RGG and/or by the CERN. We associate 3-cycles in RRGs and CERNs as resonant triples in the Fock space. We show that the scenario of the "localization without disorder", discussed previously in physical space, can be realized in the Fock space as well. We argue that it is natural to identify clusters in a RRG with particles in a many-body system above the phase transition. We discuss the controversial issue of an additional phase transition between ergodic and non-ergodic regimes in the delocalized phase in the Fock space and find a strong "memory dependence" of the states in the delocalized phase, thus advocating existence of non-ergodic delocalized states.

cond-mat.dis-nn↗

arXiv2016

Eigenvalue tunnelling and decay of quenched random networks

We consider the canonical ensemble of $N$-vertex Erd\H{o}s-R\'enyi (ER) random topological graphs with quenched vertex degree, and with fugacity $\mu$ for each closed triple of bonds. We claim complete defragmentation of large-$N$ graphs into the collection of $[p^{-1}]$ almost full subgraphs (cliques) above critical fugacity, $\mu_c$, where $p$ is the ER bond formation probability. Evolution of the spectral density, $\rho(\lambda)$, of the adjacency matrix with increasing $\mu$ leads to the formation of two-zonal support for $\mu>\mu_c$. Eigenvalue tunneling from one (central) zone to the other means formation of a new clique in the defragmentation process. The adjacency matrix of the ground state of a network has the block-diagonal form where number of vertices in blocks fluctuate around the mean value $Np$. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.

cond-mat.stat-mech↗

arXiv2015

Native ultrametricity of sparse random ensembles

We investigate the eigenvalue density in ensembles of large sparse Bernoulli random matrices. We demonstrate that the fraction of linear subgraphs just below the percolation threshold is about 95\% of all finite subgraphs, and the distribution of linear chains is purely exponential. We analyze in detail the spectral density of ensembles of linear subgraphs, discuss its ultrametric nature and show that near the spectrum boundary, the tail of the spectral density exhibits a Lifshitz singularity typical for Anderson localization. We also discuss an intriguing connection of the spectral density to the Dedekind $\eta$-function. We conjecture that ultrametricity is inherit to complex systems with extremal sparse statistics and argue that a number-theoretic ultrametricity emerges in any rare-event statistics.

cond-mat.dis-nn↗

arXiv2015

Spontaneous Symmetry Breaking and Phase Coexistence in Two-Color Networks

We have considered an equilibrium ensemble of large Erd\H{o}s-Renyi topological random networks with fixed vertex degree and two types of vertices, black and white, prepared randomly with the bond connection probability, $p$. The network energy is a sum of all unicolor triples (either black or white), weighted with chemical potential of triples, $\mu$. Minimizing the system energy, we see for some positive $\mu$ formation of two predominantly unicolor clusters, linked by a "string" of $N_{bw}$ black-white bonds. We have demonstrated that the system exhibits critical behavior manifested in emergence of a wide plateau on the $N_{bw}(\mu)$-curve, which is relevant to a spinodal decomposition in 1st order phase transitions. In terms of a string theory, the plateau formation can be interpreted as an entanglement between baby-universes in 2D gravity. We have conjectured that observed classical phenomenon can be considered as a toy model for the chiral condensate formation in quantum chromodynamics.

cond-mat.stat-mech↗

arXiv2014

Critical behavior in topological ensembles

We consider the relation between three physical problems: 2D directed lattice random walks, ensembles of $T_{n,n+1}$ torus knots, and instanton ensembles in 5D SQED with one compact dimension in $\Omega$ background and with 5D Chern-Simons term at the level one. All these ensembles exhibit the critical behavior typical for the "area+length+corners" statistics of grand ensembles of 2D directed paths. Using the combinatorial description, we obtain an explicit expression of the generating function for $q$-Narayana numbers which amounts to the new critical behavior in the ensemble of $T_{n,n+1}$ torus knots and in the ensemble of instantons in 5D SQED. Depending on the number of the nontrivial fugacities, we get either the critical point, or cascade of critical lines and critical surfaces. In the 5D gauge theory the phase transition is of the 3rd order, while in the ensemble of paths and ensemble of knots it is typically of the 1st order. We also discuss the relation with the integrable models.

hep-th↗

arXiv2012

From elongated spanning trees to vicious random walks

Given a spanning forest on a large square lattice, we consider by combinatorial methods a correlation function of $k$ paths ($k$ is odd) along branches of trees or, equivalently, $k$ loop--erased random walks. Starting and ending points of the paths are grouped in a fashion a $k$--leg watermelon. For large distance $r$ between groups of starting and ending points, the ratio of the number of watermelon configurations to the total number of spanning trees behaves as $r^{-\nu} \log r$ with $\nu = (k^2-1)/2$. Considering the spanning forest stretched along the meridian of this watermelon, we see that the two--dimensional $k$--leg loop--erased watermelon exponent $\nu$ is converting into the scaling exponent for the reunion probability (at a given point) of $k$ (1+1)--dimensional vicious walkers, $\tilde{\nu} = k^2/2$. Also, we express the conjectures about the possible relation to integrable systems.

cond-mat.stat-mech↗

arXiv2012

Random ballistic growth and diffusion in symmetric spaces

Sequential ballistic deposition (BD) with next-nearest-neighbor (NNN) interactions in a N-column box is viewed a time-ordered product of N\times N-matrices consisting of a single sl_2-block which has a random position along the diagonal. We relate the uniform BD growth with the diffusion in the symmetric space H_N=SL(N,R)/SO(N). In particular, the distribution of the maximal height of a growing heap is connected with the distribution of the maximal distance for the diffusion process in H_N. The coordinates of H_N are interpreted as the coordinates of particles of the one--dimensional Toda chain. The group-theoretic structure of the system and links to some random matrix models are also discussed.

math-ph↗