arXiv subjects

Sergei Nechaev

Publications and source records attributed to Sergei Nechaev.

At least 19 recordsLinked to original sources

arXiv2026

Motif enrichment as a driver of scale-free behavior in rewired random regular graphs

We study the statistics of rewired random regular graphs (RRGs) in a mixed ensemble, where the average number of triangles is controlled by the fugacity $\lambda$, while the number of vertices and the vertex degree are fixed. This model exhibits a phase transition at critical fugacity $\lambda_{cr}$ from a triangle-poor phase (TPP), in which the number of triangles is independent of the system size, to a triangle-rich phase (TRP), in which the number of triangles scales linearly with the system size. We estimate $\lambda_{cr}$ by comparing the entropy of TPP with the energy of TRP. Above $\lambda_{cr}$, the RRG becomes a two-phase system in which dense clusters are connected by a sparse scale-free sub-network characterized by a degree distribution, $P(d) \sim d^{-\gamma}$, with $\gamma \approx 2$, independent of the size of the whole graph and its degree. We attribute this behavior to an "emergent preferential attachment" induced by triangle motifs, describe the mechanism underlying its formation, and derive the exponent $\gamma$ within a mean-field approach. We show that most inter-cluster triangles are isosceles, with the base lying inside one cluster and the apex belonging to the inter-cluster network. Finally, we speculate on a possible connection between these triangles and Efimov states in a conformally invariant potential.

cond-mat.dis-nn↗

arXiv2025

Three faces of random walks in hyperbolic domain: BKT, Lifshitz tails, and KPZ

We show that continuous random walks (diffusion) in the Poincar\'{e} hyperbolic upper halfplane $\mathbb{H}^2 = \{(x,y)|y>0\}$ provide a unifying description of three seemingly unrelated phenomena: (i) the non-analytic divergence of the correlation length at the Berezinskii--Kosterlitz--Thouless (BKT) transition; (ii) the appearance of the Kardar--Parisi--Zhang (KPZ) exponent in the fluctuational behavior of stretched random walks constrained above an impermeable disc; and (iii) the emergence of Lifshitz tails (LT) in 1D statistics of rare events. We adapt the renormalization-group equations originally developed for the Efimov effect in a 2D conformally invariant potential to the case of diffusion in $\mathbb{H}^2$, thereby reproducing the BKT--type divergence of the correlation length. In frameworks of the same model we derive the KPZ--type behavior for the survival probability of stretched random walks near the boundary of $\mathbb{H}^2$ using scaling arguments, WKB--type approach, and numerical analysis. We demonstrate that LT emerge naturally in a deterministic large-deviation random walks' statistics in $\mathbb{H}^2$ via instanton approach, which rhymes with the rare-event behavior of 1D diffusion in the array of traps with the Poisson distribution. We conjecture that the dominant contribution to the statistics of paths responsible for BKT--like physics emerges from trajectories pushed to large-deviation stretched regime.

cond-mat.stat-mech↗

arXiv2025

Collective communication in a transparent world: Phase transitions in a many-body Potts model and social-quantum duality

Digitally connected societies approach a \enquote{transparent} regime where all agents can interact without geographic or social barriers -- a limit realized by complete graph topologies. We solve exactly a $q$-state Potts model with many-body interactions on this geometry, modeling agents from $q$ distinct communities. Analyzing the illustrative case of competing pairwise and three-body couplings, we identify three key phases in the thermodynamic limit: democratic (all communities equal), marginalized ($q-1$ communities surviving), and consensus (one dominant group). For two-community systems, we identify a special coupling regime where interaction energies cancel, yielding purely entropy-driven dynamics -- a statistical physics representation of atomized societies without structured influence. Monte Carlo simulations confirm these phases and reveal metastable switching dynamics in finite systems. Furthermore, we establish an exact correspondence between this social model and mean-field $SU(N)$ quantum spin systems with quadratic and cubic Casimir interactions, revealing a \enquote{social-quantum} duality. This duality enables quantitative classification of social structures via Young diagrams and reinterprets quantum symmetry breaking as opinion stratification, bridging statistical sociology and quantum many-body physics.

physics.soc-ph↗

arXiv2025

Probing phase transitions of finite directed polymers near a corrugated wall via two-replica analysis

We study the pinning transition in a (1+1)-dimensional lattice model of a fluctuating interface interacting with a corrugated impenetrable wall. The interface is modeled as an $N$-step directed one-dimensional random walk on the half-line $x \ge 0$. Its interaction with the wall is described by a quenched, site-dependent, short-ranged random potential $u_j$ ($j = 1,\ldots,N$), distributed according to $Q(u_j)$ and localized at $x = 0$. By computing the first two disorder--averaged moments of the partition function, $\langle G_N \rangle$ and $\langle G_N^2 \rangle$, and by analyzing the analytic structure of the resulting expressions, we derive an explicit criterion for the coincidence or distinction of the pinning transitions in annealed and quenched systems. We show that, although the transition points of the annealed and quenched systems are always different in the thermodynamic limit, for finite systems there exists a "gray zone" in which this difference is hardly detectable. Our results may help reconcile conflicting views on whether quenched disorder is marginally relevant.

cond-mat.stat-mech↗

arXiv2025

Non-algebraic first return probability of a stretched random walk near a convex boundary and its effect on adsorption

The $N$-step random walk, elongated in the vicinity of a disc (in 2D) or a sphere (in 3D) of radius $R$, demonstrates a non-algebraic stretched exponential decay $P_N\sim \exp\left(-{\rm const}\, N^{1/3}\right)$ for the first return probability $P_N$ in the double-scaling limit $N=\frac{L}{a}\gg 1, \frac{R}{a}\gg 1$ conditioned that $\frac{L}{R}=c={\rm const}$. Stretching means that the length of the walk, $L=Na$ (where $a$ is the unit step length) satisfies the condition $L = cR$, where $c > \pi$ and under "first return" we understand the radial first arrival to a boundary. Both analytic and numerical evidences of the non-algebraic behavior of $P_N$ are provided. Considering the model of a polymer loop stretched ("inflated") by external force, we show that non-algebraic behavior of $P_N$ affects the adsorption of a polymer at the boundary of a sticky disc in 2D, manifesting in a first order localization transition.

cond-mat.stat-mech↗

arXiv2024

KPZ-like scaling on a high-dimensional hypersphere

We consider the orientational diffusion controlled by the hyperspherical Laplacian, $\nabla^2_D$, on the surface of the $D$--dimensional hypersphere in the limit $D \to \infty$. We find that for stretched paths with lengths relatively short compared to the hypersphere's radius, the finite-size corrections in orientational correlations are controlled by the Kardar-Parisi-Zhang (KPZ) scaling exponent, $\gamma = 1/3$. In addition, we speculate about the topology of the orientational target space representing the surface of the hypersphere.

cond-mat.stat-mech↗

arXiv2024

KPZ scaling from the Krylov space

Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling in late-time correlators and autocorrelators of certain interacting many-body systems has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis. We focus on the Heisenberg time scale, which approximately corresponds to the ramp--plateau transition for the Krylov complexity in systems with a large but finite number degrees of freedom. Two frameworks are under consideration: i) the system with growing Lanczos coefficients and an artificial cut-off, and ii) the system with the finite Hilbert space. In both cases via numerical analysis, we observe the transition from Gaussian to KPZ-like scaling at the critical Euclidean time $t_{E}^*=c_{cr}K$, for the Krylov chain of finite length $K$, and $c_{cr}=O(1)$. In particular, we find a scaling $\sim K^{1/3}$ for fluctuations in the one-point correlation function and a dynamical scaling $\sim K^{-2/3}$ associated with the return probability (Loschmidt echo) corresponding to autocorrelators in physical space. In the first case, the transition is of the 3rd order and can be considered as an example of dynamical quantum phase transition (DQPT), while in the second, it is a crossover. For case ii), utilizing the relationship between the spectrum of tridiagonal matrices at the spectral edge and the spectrum of the stochastic Airy operator, we demonstrate analytically the origin of the KPZ scaling for the particular Krylov chain using the results of the probability theory. We argue that there is some outcome of our study for the double scaling limit of matrix models. For the case of topological gravity, the white noise $O(\frac{1}{N})$ term is identified, which should be taken into account in the controversial issue of ensemble averaging in 2D/1D holography.

hep-th↗

arXiv2024

Math behind everyday life: "black days", their manifestation as traffic jams, and beyond

In our daily lives, we encounter numerous independent events, each occurring with varying probabilities over time. This letter delves into the scientific background behind the inhomogeneous distribution of these events over time, often resulting in what we refer to as ``black days'', where multiple events seem to converge at once. In the first part of the work we performed an analysis involving $D$ independent periodic and random sequences of events. Using the Uniform Manifold Approximation and Projection (UMAP) technique, we observed a clustering of event sequences on a two-dimensional manifold ${\cal M}$ at a certain large $D$. We interpret this clustering as a signature of ``black days'', which bears a clear resemblance to traffic jams in vehicle flow. In the second part of the work we examined in detail clustering patterns of independently distributed $N$ points within the corners of a $D$-dimensional cube when $1\ll N<D$. Our findings revealed that a transition to a single-component cluster occurs at a critical dimensionality, $D_{cr}$, via a nearly third-order phase transition. Analyzing the spectral density, $\rho(\lambda)$, of the corresponding adjacency graph in the vicinity of the clustering transition we recover the singular ''Lifshitz tail'' behavior at the spectral boundary of $\rho(\lambda)$.

physics.soc-ph↗

arXiv2024

Metric structural human connectomes: localization and multifractality of eigenmodes

In this study, we explore the fundamental principles behind the architecture of the human brain's structural connectome, from the perspective of spectral analysis of Laplacian and adjacency matrices. Building on the idea that the brain strikes a balance between efficient information processing and minimizing wiring costs, we aim to understand the impact of the metric properties of the connectome and how they relate to the existence of an inherent scale. We demonstrate that a simple generative model, combining nonlinear preferential attachment with an exponential penalty for spatial distance between nodes, can effectively reproduce several key characteristics of the human connectome, including spectral density, edge length distribution, eigenmode localization and local clustering properties. We also delve into the finer spectral properties of the human structural connectomes by evaluating the inverse participation ratios ($\text{IPR}_q$) across various parts of the spectrum. Our analysis reveals that the level statistics in the soft cluster region of the Laplacian spectrum deviate from a purely Poisson distribution due to interactions between clusters. Additionally, we identified scar-like localized modes with large IPR values in the continuum spectrum. We identify multiple fractal eigenmodes distributed across different parts of the spectrum, evaluate their fractal dimensions and find a power-law relationship in the return probability, which is a hallmark of critical behavior. We discuss the conjectures that a brain operates in the Griffiths or multifractal phases.

q-bio.NC↗

arXiv2023

Unconventional critical behavior of polymers at sticky boundaries

We discuss the generalization of a classical problem involving an $N$-step ideal polymer adsorption at a sticky boundary (potential well of depth $U$). It is known that as $N$ approaches infinity, the path undergoes a 2nd-order localization transition at a certain value of $U_{\text{tr}}$. By considering the random walk on a half-line with a sticky boundary (Model I), we demonstrate that the order of the phase transition can be altered by adjusting the scaling of the first return probability to the boundary. Additionally, we present a model of a random path on a discrete 1D lattice with non-uniform local hopping amplitudes and a potential well at the boundary (Model II). We illustrate that one can tailor such amplitudes so that the polymer undergoes a 3rd-order phase transition.

cond-mat.stat-mech↗

arXiv2023

Generalized Devil's staircase and RG flows

We discuss a two-parameter renormalization group (RG) flow when parameters are organized in a single complex variable, $τ$, with modular properties. Throughout the work we consider a special limit when the imaginary part of $τ$ characterizing the disorder strength tends to zero. We argue that generalized Riemann-Thomae (gRT) function and the corresponding generalized Devil's staircase emerge naturally in a variety of physical models providing a universal behavior. In 1D we study the Anderson-like probe hopping in a weakly disordered lattice, recognize the origin of the gRT function in the spectral density of the probe and formulate specific RG procedure which gets mapped onto the discrete flow in the fundamental domain of the modular group $SL(2,Z)$. In 2D we consider the generalization of the phyllotaxis crystal model proposed by L. Levitov and suggest the explicit form of the effective potential for the probe particle propagating in the symmetric and asymmetric 2D lattice of defects. Analyzing the structure of RG flow equations in the vicinity of saddle points we claim emergence of BKT-like transitions at ${\rm Im}\,τ\to 0$. We show that the RG-like dynamics in the fundamental domain of $SL(2,Z)$ for asymmetric lattices asymptotically approaches the "Silver ratio". For a Hubbard model of particles on a ring interacting via long-ranged potentials we investigate the dependence of the ground state energy on the potential and demonstrate by combining numerical and analytical tools the emergence of the generalized Devil's staircase. Also we conjecture a bridge between a Hubbard model and a phyllotaxis.

cond-mat.stat-mech↗

arXiv2023

BKT in Phyllotaxis

We discuss a two-parameter renormalization group (RG) consideration of a phyllotaxis model in the framework of the ``energetic approach'' proposed by L. Levitov in 1991. Following L. Levitov, we consider an equilibrium distribution of strongly repulsive particles on the surface of a finite cylinder and study the redistribution of these particles when the cylinder is squeezed along its axis. We construct explicitly the $β$-function of a given system in terms of the modular Dedekind $η$-function. On basis of this $β$-function we derive the equations describing the RG flow in the vicinity of the bifurcation points between different lattices. Analyzing the structure of RG equations, we claim emergence of Berezinskii-Kosterlitz-Thouless (BKT) transitions at strong compression of the cylinder.

cond-mat.stat-mech↗

arXiv2023

Largest eigenvalue statistics of sparse random adjacency matrices

We investigate the statistics of the largest eigenvalue, $λ_{\rm max}$, 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 $λ_{\rm max}$ 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 $λ_{\rm max}$ 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↗

arXiv2022

Eigenvalue detachment, BBP transition and constrained Brownian motion

We discuss the eigenvalue detachment transition in terms of scaling of fluctuations in ensembles of paths located near convex boundaries of various physical nature. We consider numerically the BBP-like (Baik-Ben Arous-Péché) transition from the Gaussian to the Tracy-Widom scaling of fluctuations in several statistical systems for both canonical and microcanonical ensembles and identify the corresponding control parameter in each case. In particular, for fixed path length (microcanonical) ensemble of paths located in the vicinity of a partially permeable semicircle, the transition occurs at the critical value of a permeability. The Tracy-Widom regime and the BBP-like transition for fluctuations are discussed in terms of the Jakiw-Teitelbom (JT) gravity with a radial cutoff which, in turn, has an interpretation as a ensemble of fixed length world-line geometrically constrained trajectories of a charged particle in an effective transversal magnetic field.

cond-mat.stat-mech↗

arXiv2022

From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity

We demonstrate here a series of exact mappings between particular cases of four statistical physics models: equilibrium 1-dimensional lattice gas with nearest-neighbor repulsion, $(1+1)$-dimensional combinatorial heap of pieces, random walks on half-plane and totally asymmetric simple exclusion process (TASEP) in one dimension (1D). In particular, we show that generating function of a steady state of one-dimensional TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of (i) a representation of the steady-state TASEP configurations in terms of $(1+1)$-dimensional heaps of pieces and (ii) a theorem connecting the partition function of $(1+1)$-dimensional heaps of pieces with that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.

cond-mat.stat-mech↗

arXiv2021

Fixman problem revisited: When fluctuations of inflated ideal polymer loop are non-Gaussian?

We consider statistics of a planar ideal polymer loop of length $L$ in a large deviation regime, when a gyration radius, $R_g$, is slightly less than the radius of a fully inflated ring, $\frac{L}{2π}$. Specifically, we study analytically and via off-lattice Monte-Carlo simulations relative fluctuations of chain monomers in ensemble of Brownian loops. We have shown that these fluctuations in the regime with fixed large gyration radius are Gaussian with the critical exponent $γ= \frac{1}{2}$. However, if we insert inside the inflated loop the impenetrable disc of radius $R_d=R_g$, the fluctuations become non-Gaussian with the critical exponent $γ=\frac{1}{3}$ typical for the Kardar-Parisi-Zhang universality class.

cond-mat.stat-mech↗

arXiv2020

Lifshitz tails at spectral edge and holography with a finite cutoff

We propose the holographic description of the Lifshitz tail typical for one-particle spectral density of bounded disordered system in $D=1$ space. To this aim the "polymer representation" of the Jackiw-Teitelboim (JT) 2D dilaton gravity at a finite cutoff is used and the corresponding partition function is considered as the weighted sum over paths of fixed length in an external magnetic field. We identify the regime of small loops, responsible for emergence of a Lifshitz tail in the Gaussian disorder, and relate the strength of disorder to the boundary value of the dilaton. The geometry corresponding to the Poisson disorder in the boundary theory involves random paths fluctuating in the vicinity of the hard impenetrable cut-off disc in a 2D plane. It is shown that the ensemble of "stretched" paths evading the disc possesses the Kardar-Parisi-Zhang (KPZ) scaling for fluctuations, which is the key property that ensures the dual description of the Lifshitz tail in the spectral density for the Poisson disorder.

cond-mat.dis-nn↗

arXiv2020

Combinatorics of 3D directed animals on a simple cubic lattice

We provide combinatorial arguments based on a two-dimensional extension of a locally-free semigroup allowing us to compute the growth rate, $Λ$, of the partition function $Z_N=N^θΛ^N$ of the $N$-particle directed animals ($N\gg 1$) on a simple cubic lattice in a three-dimensional space. Establishing the bijection between the particular configuration of the lattice animal and a class of equivalences of words in the 2D projective locally-free semigroup, we find we find $\ln Λ= \lim_{N\to\infty} \ln Z_N / N$ with $Λ= 2(\sqrt{2}+1) \approx 4.8284$.

cond-mat.stat-mech↗