arXiv subjects

Gleb Oshanin

Publications and source records attributed to Gleb Oshanin.

At least 19 recordsLinked to original sources

arXiv2026

Ornstein-Uhlenbeck Process Driven by Multiple Dichotomous Noises

We study a generalized Ornstein-Uhlenbeck process driven by a superposition of $K$ independent dichotomous noises with arbitrary fixed amplitudes and switching rates. Unlike the classical Ornstein-Uhlenbeck process driven by equilibrium Gaussian white noise, the present system is governed by bounded nonequilibrium fluctuations with finite correlation times. We obtain exact expressions for the stationary position distribution and all cumulants, and show that the stationary state possesses an unexpectedly rich structure, including compact support, algebraic branch-point singularities, edge divergences, and multiple extrema. We establish a mapping onto a heterogeneous random-flight process with bounded jumps, yielding a transparent probabilistic interpretation of the stationary measure. We further analyze several limiting regimes, including the crossover to Gaussian statistics for large numbers of noise sources. For ensembles with exponentially-distributed quenched amplitudes, we derive exact disorder-averaged stationary distributions and show that disorder fundamentally alters the stationary state, producing exponential tails decorated by algebraic prefactors with non-trivial exponents.

cond-mat.stat-mech↗

arXiv2026

Power spectral density of trajectories of active Ornstein-Uhlenbeck particles

The power spectral density (PSD) is a central frequency-domain descriptor of stochastic processes. While PSDs have been studied for Brownian motion and a few anomalous diffusion processes, the spectral densities of active nonequilibrium processes remain almost unexplored. Here, we present an exact theory for the PSDs of active diffusion using the model of active Ornstein-Uhlenbeck particles (AOUPs). We investigate the spectral densities of AOUPs in free space and under harmonic confinement. In free space, active motion does not alter the Brownian $f^{-2}$ spectrum, but only modifies its amplitude and introduces a crossover at the persistence frequency. Under confinement, the spectrum exhibits a rich variety of features depending on the persistence, trap relaxation, and activity strength, including two characteristic signatures that are absent in both thermal systems and free AOUPs. These are a two-plateau structure from a double-trapping mechanism due to two noise sources, and the new $f^{-4}$ spectral scaling associated with transient ballistic motion. We also investigate the finite time effects through the finite-time PSD, and find that the low-frequency plateau and high frequency oscillation exhibit distinct dependences on the observation time $T$ in free and confined systems. Finally, we discuss our results in connection with previously reported experimental studies of active systems. Our results provide an analytically tractable framework for interpreting such systems.

cond-mat.stat-mech↗

arXiv2026

Lack of self-averaging of the critical internal energy in a weakly-disordered Baxter model

We investigate the first two moments of the critical internal energy $E$ in a weakly disordered two-dimensional Baxter eight-vertex model as a function of the system size $L$, evaluated at the pseudo-critical point. Disorder is introduced via an equivalent representation of the pure eight-vertex model in terms of two ferromagnetic Ising models coupled by a four-spin interaction of strength $g_0$, where the Ising couplings consist of a uniform ferromagnetic part $J>0$ supplemented by weak Gaussian spatial disorder. In the critical regime, the model is formulated in terms of interacting Grassmann-Majorana spinor fields with quartic interactions and analyzed, for small positive $g_0$, using a combination of replica and renormalization-group methods. We also run extensive numerical simulations measuring the critical internal energy. Our results show that its relative variance increases with $L$ and approaches a finite constant as $L \to \infty$ for both $\pm g_0$. Hence, fluctuations remain relevant independently of the sign of $g_0$ (and thus of the specific-heat exponent), implying a lack of self-averaging of both the critical internal energy and the free energy. Consequently, reliable estimates of these quantities require averaging over many disorder realizations. In addition, we numerically confirm earlier predictions concerning the absence of self-averaging of the critical internal energy in the disordered Ising model.

cond-mat.stat-mech↗

arXiv2026

Diffusing diffusivity model with dichotomous noise

We study Langevin dynamics with stochastic diffusivity arising from fluctuations of the surrounding medium. The diffusivity is modeled as Ornstein-Uhlenbeck process driven by symmetric dichotomous noise, which confines it to a finite interval. We derive analytical expressions for the short-time probability density function (PDF) of the particle displacement and analyse its asymptotic behaviour. While the PDF retains the characteristic logarithmic divergence at the origin, its tails differ from the Gaussian white-noise case: exponential tails are replaced by Gaussian ones modulated by a power-law with a switching-rate-dependent exponent. At long times, the dynamics converges to ordinary Gaussian diffusion. We determine the variance and covariance of the time-averaged stochastic diffusivity and show that it is self-averaging. The model provides a minimal analytically tractable framework for stochastic transport in environments with bounded or switching fluctuations.

cond-mat.stat-mech↗

arXiv2026

Fastest first-passage time for multiple searchers with finite speed

We study analytically and numerically the mean fastest first-passage time (fFPT) to an immobile target for an ensemble of $N$ independent finite-speed random searchers driven by dichotomous noise and described by the telegrapher's equation. In stark contrast to the well-studied case of Brownian particles -- for which the mean fFPT vanishes logarithmically with $N$ -- we uncover that the mean fFPT is bounded from below by the minimal ballistic travel time, with an exponentially fast convergence to this bound as $N \to \infty$. This behavior reveals a dramatic efficiency advantage of physically realistic, finite-speed searchers over Brownian ones and illustrates how diffusive macroscopic models may be conceptually misleading in predicting the short-time behavior of a physical system. We extend our analysis to anomalous diffusion generated by Riemann-Liouville-type dichotomous noises and find that target detection is more efficient in the superdiffusive regime, followed by normal and then subdiffusive regimes, in agreement with physical intuition and contrary to earlier predictions.

cond-mat.stat-mech↗

arXiv2026

Inferring Concepts from Noisy Examples in Hopfield-like Neural Networks

We study a variant of the pseudo-inverse learning rule for Hopfield-like Neural Networks, which allows the network to infer archetypal concepts on the basis of a limited number of examples. The mean-field replica theory for this model reveals how this generalization ability is mediated by a multitude of states, with diverse thermodynamic properties, coexisting with the standard Hopfield ones. They appear and vanish through smooth transitions or discontinuous jumps and, interestingly, show much stronger Replica Symmetry Breaking (RSB) effects than the standard Hopfield model, as captured by our 1RSB analysis. Our results, in excellent agreement with numerical simulations, provide deeper insight into the interplay between memory storage and generalization in attractor neural networks.

cond-mat.dis-nn↗

arXiv2025

Path-integrals and optimal paths for the fractional Ornstein-Uhlenbeck process

We derive the path-integral representation of the fractional Ornstein-Uhlenbeck process driven by Riemann-Liouville fractional Gaussian noise, for both the subdiffusive and superdiffusive regimes. We express the corresponding action, which is a quadratic functional of individual trajectories of the process, in two alternative but equivalent forms: either as a fractional integral or as a double integral with a nonlocal kernel. Moreover, we determine in closed form the optimal (action-minimizing) paths conditioned to reach a prescribed point at a fixed time moment and discuss their behavior, which appears to be non-intuitive for subdiffusive processes in the presence of a strong confining potential.

cond-mat.stat-mech↗

arXiv2025

Complex Eigenvalues in a pseudo-Hermitian \b{eta}-Laguerre ensemble

Non-Hermitian PT-symmetric models have been extensively studied in recent years. Following the seminal work that reduced classical random matrix ensembles to a tridiagonal form, several efforts have aimed to generalize this framework to non-Hermitian extensions of the so-called \b{eta}-ensembles. In particular, while the transition of eigenvalues from the real axis to the complex plane has been well characterized for the \b{eta}-Hermite ensemble under symmetry breaking, the behavior of the \b{eta}-Laguerre ensemble in a similar non-Hermitian setting remains less understood. In this work, we investigate an ensemble of unstable matrices isospectral to the \b{eta}-Laguerre ensemble. Introducing a small non-Hermitian perturbation breaks the symmetry and drives the eigenvalues into the complex plane. We derive analytical expressions for the loci of complex-conjugate eigenvalue pairs, which organize into a balloon-like structure in the complex plane, followed by a discrete finite line of real eigenvalues. The asymptotic behavior of these eigenvalues is analyzed in the large matrix-size limit, and our theoretical predictions are supported by numerical simulations.

cond-mat.stat-mech↗

arXiv2025

Stochastic gyration driven by dichotomous noises

We consider stochastic dynamics of a particle on a plane in presence of two noises and a confining parabolic potential - an analog of the experimentally-relevant Brownian Gyrator (BG) model. In contrast to the standard BG model, we suppose here that the time-evolution of the position components is driven not by Gaussian white-noises, but by two statistically-independent dichotomous noises. We calculate analytically the position variances and cross-correlations, as well as the mean angular momentum, which permits us to establish the conditions in which a spontaneous rotational motion of the particle around the origin takes place. We also present a numerical analysis of the mean angular velocity. Lastly, we calculate analytically some marginal position probability density functions revealing a remarkably rich behavior that emerges in such a system of two coupled linear stochastic differential equations. We show that depending on the values of parameters characterizing noises these distributions approach the steady-state forms defined on a finite support, having very unusual shapes, possessing multiple maxima and minima, plateaus and exhibiting a discontinuous behavior.

cond-mat.stat-mech↗

arXiv2025

One-dimensional lattice random walks in a Gaussian random potential

We study random walks evolving in continuous time on a one-dimensional lattice where each site $x$ hosts a quenched random potential $U_x$. The potentials on different sites are independent, identically distributed Gaussian random variables. We analyze three distinct models that specify how the transition rates depend on $U_x$: the random-force-like model, random walks with randomized stepping times, and the Gaussian trap model. Our analysis focuses on five key disorder-dependent quantities defined for a finite chain with $N$ sites: the probability current, its reciprocal (the resistance), the splitting probability $E_-$, the mean first-passage time $T_N$, and the diffusion coefficient $D_N$ in a periodic chain. By determining the moments of these random variables, we demonstrate that the probability current and resistance are not self-averaging, which leads to pronounced differences between their average and typical behaviors. In contrast, $E_-$, $T_N$ and $D_N$ become self-averaging when $N \to \infty$, though they exhibit strong sample-to-sample fluctuations for finite $N$.

cond-mat.stat-mech↗

arXiv2025

Discrete-space and -time analogue of a super-diffusive fractional Brownian motion

We discuss how to construct reliably well "a lattice and an integer time" version of a super-diffusive continuous-space and -time fractional Brownian motion (fBm) -- an experimentally-relevant non-Markovian Gaussian stochastic process with an everlasting power-law memory on the time-evolution of thermal noises extending over the entire past. We propose two algorithms, which are both validated by extensive numerical simulations showing that the ensuing lattice random walks have not only the same power-law covariance function as the standard fBm, but also individual trajectories follow those of the super-diffusive fBm. Finding a lattice and an integer time analogue of a sub-diffusion fBm, which is an anti-persistent process, remains a challenging open problem. Our results also clarify the relevant difference between sub-diffusive and super-diffusive fBm, that are frequently seen as two very analogous realizations of processes with memory. They are indeed substantially different.

cond-mat.stat-mech↗

arXiv2025

Fastest first-passage time statistics for time-dependent particle injection

A common scenario in a variety of biological systems is that multiple particles are searching in parallel for an immobile target located in a bounded domain, and the fastest among them that arrives to the target first triggers a given desirable or detrimental process. The statistics of such extreme events -- the \textit{fastest\/} first-passage to the target -- is well-understood by now through a series of theoretical analyses, but exclusively under the assumption that all $N$ particles start \textit{simultaneously\/}, i.e., all are introduced into the domain instantly, by $\delta$-function-like pulses. However, in many practically important situations this is not the case: in order to start their search, the particles often have to enter first into a bounded domain, e.g., a cell or its nucleus, penetrating through gated channels or nuclear pores. This entrance process has a random duration so that the particles appear in the domain sequentially and with a time delay. Here we focus on the effect of such an extended-in-time injection of multiple particles on the fastest first-passage time (fFPT) and its statistics. We derive the full probability density function $H_N(t)$ of the fFPT with an arbitrary time-dependent injection intensity of $N$ particles. Under rather general assumptions on the survival probability of a single particle and on the injection intensity, we derive the large-$N$ asymptotic formula for the mean fFPT, which is quite different from that obtained for the instantaneous $\delta$-pulse injection. The extended injection is also shown to considerably slow down the convergence of $H_N(t)$ to the large-$N$ limit -- the Gumbel distribution -- so that the latter may be inapplicable in the most relevant settings with few tens to few thousands of particles.

cond-mat.stat-mech↗

arXiv2025

A simple model of a sequence-reading diffusion: non-self-averaging and self-averaging properties

Motivated by a question about the sensitivity of knots' diffusive motion to the actual sequence of nucleotides placed on a given DNA, here we study a simple model of a sequence-reading diffusion on a stretched chain with a frozen sequence of "letters" $A$ and $B$, having different interaction energies. The chain contains a single distortion - a hernia - which brings the two letters at its bottom together such that they interact. Due to interactions with the solvent, the hernia performs a random hopping motion along the chain with the transition rates dependent on its actual position. Our two focal questions are a) the dependence of various transport properties on the letters' interaction energy and b) whether these properties are self-averaging with respect to different realizations of sequences. We show that the current through a finite interval, the resistance of this interval and the splitting probabilities on this interval lack self-averaging. On the contrary, the mean first-passage time through a finite interval with $N$ sites and the diffusion coefficient in a periodic chain are self-averaging in the limit $N \to \infty$. Concurrently, two latter properties exhibit sample-to-sample fluctuations for finite $N$, as evidenced by numerical simulations.

cond-mat.stat-mech↗

arXiv2023

Irregular gyration of a two-dimensional random-acceleration process in a confining potential

We study the stochastic dynamics of a two-dimensional particle assuming that the components of its position are two coupled random-acceleration processes evolving in a confining parabolic potential and are the subjects of independent Gaussian white noises with different amplitudes (temperatures). We determine the standard characteristic properties, i.e., the moments of position's components and their velocities, mixed moments and two-time correlations, as well as the position-velocity probability density function (pdf). We show that if the amplitudes of the noises are not equal, then the particle experiences a non-zero (on average) torque, such that the angular momentum L and the angular velocity W have non-zero mean values. Both are (irregularly) oscillating with time t, such that the characteristics of a rotational motion are changing their signs. We also evaluate the pdf-s of L and W and show that the former has exponential tails for any fixed t, and hence, all moments. In addition, in the large-time limit this pdf converges to a uniform distribution with a diverging variance. The pdf of W possesses heavy power-law tails such that the mean W is the only existing moment. This pdf converges to a limiting form which, surprisingly, is completely independent of the amplitudes of noises.

cond-mat.stat-mech↗

arXiv2023

Destructive effect of fluctuations on the performance of a Brownian gyrator

The Brownian gyrator (BG) is often called a minimal model of a nano-engine performing a rotational motion, judging solely upon the fact that in non-equilibrium conditions its torque, angular momentum ${\cal L}$ and angular velocity $\cal W$ have non-zero mean values. For a time-discretized model, which is most adapted for the analysis of an essentially discrete-time data garnered in experiments or numerical simulations, we calculate the previously unknown probability density functions (PDFs) of ${\cal L}$ and $\cal W$. For finite time-step $\delta t$, the PDF of ${\cal L}$ has exponential tails and all moments are therefore well-defined, but the noise-to-signal ratio can attain big values for small $\delta t$. Conversely, the PDF of ${\cal W}$ exhibits heavy power-law tails and its mean ${\cal W}$ is the only existing moment. The BG is therefore not an engine in the common sense: it does not exhibit regular rotations on each run and its fluctuations are not only a minor nuisance -- on contrary, their effect is completely destructive for the performance. Our theoretical predictions are confirmed by numerical simulations and experimental data. We discuss some plausible improvements

cond-mat.stat-mech↗

arXiv2023

Out-of-equilibrium dynamics of two interacting optically-trapped particles

We present a theoretical analysis of a non-equilibrium dynamics in a model system consisting of two particles which move randomly on a plane. The two particles interact via a harmonic potential, experience their own (independent from each other) noises characterized by two different temperatures $T_1$ and $T_2$, and each particle is being held by its own optical tweezer. Such a system with two particle coupled by hydrodynamic interactions was previously realised experimentally in Bérut et al. [EPL {\bf 107}, 60004 (2014)], and the difference between two temperatures has been achieved by exerting an additional noise on either of the tweezers. Framing the dynamics in terms of two coupled over-damped Langevin equations, we show that the system reaches a non-equilibrium steady-state with non-zero (for $T_1 \neq T_2$) probability currents that possess non-zero curls. As a consequence, in this system the particles are continuously spinning around their centers of mass in a completely synchronised way - the curls of currents at the instantaneous positions of two particles have the same magnitude and sign. Moreover, we demonstrate that the components of currents of two particles are strongly correlated and undergo a rotational motion along closed elliptic orbits.

cond-mat.stat-mech↗

arXiv2022

Path integrals for fractional Brownian motion and fractional Gaussian noise

The Wiener's path integral plays a central role in the studies of Brownian motion. Here we derive exact path-integral representations for the more general \emph{fractional} Brownian motion (fBm) and for its time derivative process -- the fractional Gaussian noise (fGn). These paradigmatic non-Markovian stochastic processes, introduced by Kolmogorov, Mandelbrot and van Ness, found numerous applications across the disciplines, ranging from anomalous diffusion in cellular environments to mathematical finance. Still, their exact path-integral representations were previously unknown. Our formalism exploits the Gaussianity of the fBm and fGn, relies on theory of singular integral equations and overcomes some technical difficulties by representing the action functional for the fBm in terms of the fGn for the sub-diffusive fBm, and in terms of the derivative of the fGn for the super-diffusive fBm. We also extend the formalism to include external forcing. The exact and explicit path-integral representations open new inroads into the studies of the fBm and fGn.

cond-mat.stat-mech↗

arXiv2022

Towards a robust criterion of anomalous diffusion

Anomalous-diffusion, the departure of the spreading dynamics of diffusing particles from the traditional law of Brownian-motion, is a signature feature of a large number of complex soft-matter and biological systems. Anomalous-diffusion emerges due to a variety of physical mechanisms, e.g., trapping interactions or the viscoelasticity of the environment. However, sometimes systems dynamics are erroneously claimed to be anomalous, despite the fact that the true motion is Brownian -- or vice versa. This ambiguity in establishing whether the dynamics as normal or anomalous can have far-reaching consequences, e.g., in predictions for reaction- or relaxation-laws. Demonstrating that a system exhibits normal- or anomalous-diffusion is highly desirable for a vast host of applications. Here, we present a criterion for anomalous-diffusion based on the method of power-spectral analysis of single trajectories. The robustness of this criterion is studied for trajectories of fractional-Brownian-motion, a ubiquitous stochastic process for the description of anomalous-diffusion, in the presence of two types of measurement errors. In particular, we find that our criterion is very robust for subdiffusion. Various tests on surrogate data in absence or presence of additional positional noise demonstrate the efficacy of this method in practical contexts. Finally, we provide a proof-of-concept based on diverse experiments exhibiting both normal and anomalous-diffusion.

cond-mat.stat-mech↗