arXiv subjects

Debraj Das

Publications and source records attributed to Debraj Das.

At least 19 recordsLinked to original sources

arXiv2026

Restart and first detection in a lackadaisical quantum walk with flat-band localization

We study stochastic and sharp restart in a one-dimensional lackadaisical discrete-time quantum walk with self-loop weight $\ell$. In the absence of restart, the dynamics has a flat band responsible for intrinsic localization and two dispersive bands supporting ballistic propagation. We compare two initially localized benchmark states: a flat-band-active state with finite flat-band overlap and a flat-band-dark state with zero flat-band overlap. For geometric stochastic restart with per-step restart probability $q$, the stationary mean-squared displacement scales as $q^{-2}$ as $q\to0$. In the same limit, the restart-site occupation probability approaches the restart-free intrinsic localized value for the flat-band-active state, whereas for the flat-band-dark state it vanishes as $q\ln(1/q)$. For power-law restart, where $p_m\propto m^{-s}$ is the probability that the waiting time to the next restart is $m$ steps, a normalized stationary site-occupation distribution exists only for $s>2$, while the stationary absolute spatial moment of order $p$ is finite only for $s>p+2$. In the regime $1<s\leq2$, at every fixed lattice site, the flat-band-active occupation converges to the intrinsic flat-band profile, while the flat-band-dark occupation tends to zero. We also consider monitored first detection with sharp restart, in which the walk is reinitialized after a fixed number $r$ of consecutive unsuccessful measurements. For fixed $r$, the mean first-detected-passage time of the flat-band-active state exhibits a minimum at an intermediate self-loop weight, whereas the flat-band-dark state approaches a ballistic detection limit as $\ell\to\infty$.

quant-ph↗

arXiv2026

Fast high-dimensional mean testing via logistic regression

We propose computationally efficient tests for equality of mean vectors of two or more high-dimensional populations. Central to our approach is an equivalence between equality of means and a zero population logistic regression parameter. We establish this equivalence for independently distributed observations without imposing common distributional assumptions across populations. Our procedure uses logistic Lasso to screen informative variables and an unpenalized logistic refit for inference in the reduced dimension, yielding asymptotically correct size and consistency. For a specified two-sample Gaussian submodel and sparse discriminative class, the test also attains the minimax separation rate. The framework extends to multiple populations through multi-class logistic regression. Simulations demonstrate accurate size control, strong power, and favorable computational scaling compared with existing tests under unbalanced designs and variance heterogeneity. Applications to gene-expression data with more than twenty-two thousand variables illustrate the practical scalability of the proposed procedures.

stat.ME↗

arXiv2026

Retained hidden excess generates memory in price-limited markets

The daily return of a stock is often restricted to an exchange-imposed band to curb extreme fluctuations. Any attempted price movement beyond this band is clipped, leaving an unobserved excess. We introduce a minimal stochastic latent-state model in which a fraction of this hidden excess is retained for the next day. This retention generates memory, even though the daily stochastic driving shocks are independent. For symmetric driving shocks with regularly varying tails, the stationary latent return preserves the tail index of the noise, but has an enhanced tail amplitude. In the wide-band limit, a close of the daily return at either limit of the band admits a single-dominant-shock description. We show that after such an event, the mean return on the following day has the same sign and grows proportionally to the band width, while the probability of reaching the same limit again approaches a finite value. Reaching the opposite band limit on the following day requires a second extreme shock of opposite sign and is power-law suppressed. Simulations support these analytical predictions. Empirical data from stocks subject to daily price limits are qualitatively consistent with the predicted same-sign response and its increase across wider price bands.

cond-mat.stat-mech↗

arXiv2026

Dephasing-induced relaxation in tight-binding chains with linear and nonlinear defects

We investigate thermalization in a tight-binding chain with an on-site defect subject to local dephasing noise implemented as random phase kicks. For a single linear defect of strength $\epsilon$, we obtain an exact analytical description of the system spectrum and formulate the dephasing-induced dynamics in the eigenstate basis. We derive an approximate kinetic equation for mode populations that describes a continuous-time random walk in action space. The walk transition rates are defined by the overlap matrix encoding the spatial structure of eigenstates that can be computed exactly. Analyzing the spectral properties of the equation, we show that defect-induced localized modes act as bottlenecks that strongly slow down relaxation, with rates scaling as $\epsilon^{-2}$ for strong defects. Using large-deviation theory, we characterize rare dynamical trajectories and identify distinct relaxation pathways associated with low- and high-activity regimes in action space. We provide numerical evidence that the large-deviation function exhibits a dynamical phase transition in the limit $\epsilon \to \infty$. We then extend our analysis to the nonlinear case, considering a single nonlinear defect embedded in either a linear or a fully nonlinear discrete Schr\"odinger equation. Numerical simulations reveal a qualitatively faster approach to equilibrium than in the corresponding linear-defect model, driven by the amplitude-dependent weakening of the defect. Our results provide a unified framework for understanding thermalization, rare fluctuations, and relaxation pathways in stochastic tight-binding systems.

cond-mat.stat-mech↗

arXiv2026

Asymptotic Theory of Tail Dependence and Bootstrap for Checkerboard Copulas

A comprehensive asymptotic and bootstrap theory is established for checkerboard-based estimation of the copula and its lower and upper tail copula counterparts under unknown marginal distributions. The proposed estimator of the tail copula extends a local bilinear interpolation of the empirical copula to the tail region, providing a flexible nonparametric approach for modeling extremal dependence. Almost sure uniform consistency is established under mild conditions on the checkerboard grid. Weak convergence of the checkerboard copula process is derived, showing that smoothing preserves the first-order asymptotic limit of the empirical copula process, including the effect of marginal estimation. These results are further extended to lower and upper tail copula processes, yielding asymptotic normality for tail dependence measures. Since the limiting processes depend on unknown characteristics of the underlying true copula, a multiplier bootstrap procedure adapted to the checkerboard structure is proposed and shown to be asymptotically valid. Simulation studies and statistical applications validate our theoretical findings under a range of dependence structures. Although the limiting processes match with that observed for the empirical copula, the finite sample performance shows a noticeable improvement under checkerboard smoothing.

stat.ME↗

arXiv2026

High Dimensional Gaussian and Bootstrap Approximations in Generalized Linear Models

Generalized Linear Model (or GLM) extends the ordinary linear regression by linking the mean of the response variable to covariates through appropriate link functions. GLM is widely used in the analysis of datasets arising from diverse fields including medical sciences, clinical trials, population surveys and risk analysis. In this paper, we investigate the Gaussian and Bootstrap approximations of GLM under two separate high dimensional regimes: (I) when the dimension $d$ grows slower than $n$ and (II) when $d$ grows exponentially with $n$. Under regime (I), we essentially show that the Gaussian approximation holds over the collection of Borel convex sets when $d = o\big(n^{2/5}\big)$ and over the collection of Euclidean balls when $d = o\big(n^{1/2}\big)$. We further devise two high dimensional Bootstrap methods which are valid over the collections of Borel convex sets and Euclidean balls under the same dimension growth rates. Then we move to regime (II) where we invoke sparsity to GLM through Lasso. We show that the high dimensional Gaussian approximation fails under regime (II). However, the Bootstrap approximations over convex sets and Euclidean balls are valid for the relevant part of the GLM estimator provided $\log d = o\big(n^{2\tau/3}\big)$ and the number of non-zero regression parameters is $o\big(n^{1/3- 4\tau/3}\big)$, when the Lasso penalty $\lambda_n \sim n^{1/2 + \tau}$, for some $\tau \in (0, 1/4)$. Simulation studies confirm the strong finite-sample performance of our proposed Bootstrap methods under both regime (I) and (II). We also implement our methods on real datasets.

stat.ME↗

arXiv2025

Tethering effects on first-passage variables of lattice random walks in linear and quadratic focal point potentials

Diffusion in a confining potential offers a minimal setting to understand the interplay between random motion and deterministic forces driving a particle towards a focal point or potential minimum. In continuous space and time, two extensively studied examples are Brownian motion in a linear (V-shaped) or a quadratic (U-shaped) potential. The deterministic bias towards the minimum is represented, respectively, by a constant force for the former and by an elastic restoring force that increases proportionally with distance for the latter. Surprisingly, unlike Brownian walks, random walks under focal point potentials in discrete space and time have received little attention. Here, we bridge this gap by analysing the dynamics of lattice random walkers in the presence of a V-shaped potential, both in a finite and an infinite spatial domain, and a finite U-shaped potential. For the V-potential in unbounded space, we find the generating function of the occupation probability and analyse the time dependence of the mean number of distinct sites visited, demonstrating that its long-time growth is logarithmic. We also study the first-passage probability and show that its mean may display a minimum as a function of bias strength, depending on the location of the initial and target sites relative to the focal point. Qualitatively similar dependencies in the first-passage probability and its mean appear for the finite U-potential. As a comparative analysis to the U-potential, we construct the bounded V-potential and superimpose in both cases a resetting process, in which the walker returns at random times to a site distinct from the focal point with some probability. We quantify the different effects of resetting on the steady-state probability and the first-passage dynamics in the two cases, and show a motion-limited regime emerges even for relatively moderate resetting probabilities.

cond-mat.stat-mech↗

arXiv2025

Necessary and sufficient conditions for high dimensional Central Limit Theorem under moment conditions

High dimensional central limit theorems (the CLTs) have been extensively studied in recent years under a variety of sufficient moment conditions connecting the dimension growth rate with the tail decay rate. In this article, we investigate whether the existing moment conditions are also necessary under the independence of the components. We consider four exhaustive classes, viz. when underlying random variables (I) have all polynomial moments, (II) have some polynomial moment of order higher than two, (III) have only second moment but no polynomial moment higher than two exists, and (IV) have infinite second moment, but belong to the domain of attraction of normal distribution. We find the optimal growth rate of the dimension with respect to sample size in the high dimensional CLTs over hyper-rectangles. More precisely, we derive necessary and sufficient moment conditions for the validity of the the CLT over hyper-rectangles in each of the four regimes listed above, showing that the CLT may hold under much weaker conditions compared to those considered in the existing literature.

math.PR↗

arXiv2025

Asymptotic Theory of $K$-fold Cross-validation in Lasso and the validity of Bootstrap

Least absolute shrinkage and selection operator or Lasso is one of the widely used regularization methods in regression. Statisticians usually implement Lasso in practice by choosing the penalty parameter in a data-dependent way, the most popular being the $K-$fold cross-validation (or $K-$fold CV). However, inferential properties, such as the variable selection consistency and $n^{1/2}-$consistency, of the $K-$fold CV based Lasso estimator and validity of the Bootstrap approximation are still unknown. In this paper, we consider the heteroscedastic linear regression model and show only under some moment type conditions that the Lasso estimator with $K$-fold CV based penalty is $n^{1/2}-$consistent, but not variable selection consistent. Additionally, we establish the validity of Bootstrap in approximating the distribution of the $K-$fold CV based Lasso estimator. Therefore, our results theoretically justify the use of $K-$fold CV based Lasso estimator to perform statistical inference in linear regression. We validate our Bootstrap method for the $K-$fold CV based Lasso estimator in finite samples based on simulations. We also implement our Bootstrap based inference on a real data set.

stat.ME↗

arXiv2025

First-Passage-Time Asymmetry for Biased Run-and-Tumble Processes

We explore first-passage phenomenology for biased active processes with a renewal-type structure, focusing in particular on paradigmatic run-and-tumble models in both discrete and continuous state spaces. In general, we show there is no equality between distributions of conditional first-passage times to symmetric barriers positioned in and against the bias direction. However, we give conditions for such a duality to be restored asymptotically (in the limit of a large barrier distance) and highlight connections to the Gallavotti-Cohen fluctuation relation and the method of images. Our general trajectory arguments of first-passage-time distributions for asymmetric run-and-tumble processes to escape from an interval of arbitrary width are supported by exact analytical results, which we derive extending Montroll's defect technique. Furthermore, we quantify the degree of violation of first-passage duality using Kullback-Leibler divergence and signal-to-noise ratios associated with the first-passage times to the two barriers. We reveal an intriguing dependence of such measures of first-passage asymmetry on the underlying often hidden tumbling dynamics which may inspire inference techniques based on first-passage-time statistics in active systems.

cond-mat.stat-mech↗

arXiv2024

Run-and-tumble exact work statistics in a lazy quantum measurement engine: stochastic information processing

We introduce a single-qubit quantum measurement engine fuelled by backaction energy input. To reduce energetic costs associated with information processing, the measurement outcomes are only used with a prescribed laziness probability in the feedback step. As a result, we show that the work extracted over consecutive cycles is a second-order Markov process, analogous to a run-and-tumble process with transient anomalous diffusion. We derive exact analytical expressions for the work finite-time moments and first-passage-time statistics. Furthermore, we find the optimal laziness probability maximizing the mean power extracted per cycle.

quant-ph↗

arXiv2024

Bootstrapping Lasso in Generalized Linear Models

Generalized linear model or GLM constitutes a large class of models and essentially extends the ordinary linear regression by connecting the mean of the response variable with the covariate through appropriate link functions. On the other hand, Lasso is a popular and easy-to-implement penalization method in regression when not all covariates are relevant. However, the asymptotic distributional properties the Lasso estimator in GLM is still unknown. In this paper, we show that the Lasso estimator in GLM does not have a tractable form and subsequently, we develop two Bootstrap methods, namely the Perturbation Bootstrap and Pearson's Residual Bootstrap methods, for approximating the distribution of the Lasso estimator in GLM. As a result, our Bootstrap methods can be used to draw valid statistical inferences for any sub-model of GLM. We support our theoretical findings by showing good finite-sample properties of the proposed Bootstrap methods through a moderately large simulation study. We also implement one of our Bootstrap methods on a real data set.

stat.ME↗

arXiv2023

On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities

Using martingale theory, we compute, in very few lines, exact analytical expressions for various first-exit-time statistics associated with one-dimensional biased diffusion. Examples include the distribution for the first-exit time from an interval, moments for the first-exit site, and functionals of the position, which involve memory and time integration. As a key example, we compute analytically the mean area swept by a biased diffusion until it escapes an interval that may be asymmetric and have arbitrary length. The mean area allows us to derive the hitherto unexplored cross-correlation function between the first-exit time and the first-exit site, which vanishes only for exit problems from symmetric intervals. As a colophon, we explore connections of our results with gambling, showing that betting on the time-integrated value of a losing game it is possible to design a strategy that leads to a net average win.

cond-mat.stat-mech↗

arXiv2023

Human perceptual decision making of nonequilibrium fluctuations

To better characterize the statistical processes underlying human decision-making, we performed experiments where human participants visualized fluctuations of physical nonequilibrium stationary states, and we analyzed responses in the context of stochastic thermodynamics. A total of forty five participants viewed hundreds of movies of a particle endowed with drifted Brownian dynamics and were tasked with judging the motion as leftward or rightward in a quick and reliable manner. Overall, the results uncover fundamental performance limits, consistent with recently established thermodynamic trade-offs (uncertainty relations, TURs) involving speed, accuracy, and dissipation; specifically, lower rates of entropy production lead to longer decision times. Moreover, to achieve a given level of observed accuracy, participants require more time than predicted by Wald's optimal sequential probability ratio test, indicating suboptimal integration of available information. In view of such suboptimality, we develop an alternative account equipped with non-Markovian evidence integration with a memory time constant, and find tight fits. Our results suggest that humans adapt their memory relaxation time to the rate of dissipation of the observed phenomenon, favouring memory over momentary evidence for effective decisions in scenarios where stimuli are far from equilibrium. Furthermore, we identify the effects of the environmental stability on decision-making performance and memory by comparing the results of the two sets of experiments: blocked (stationary) versus intermixed (non-stationary) conditions. Our study illustrates that perceptual psychophysics using stimuli rooted in nonequilibrium physical processes provides a robust platform for understanding how the human brain makes decisions on stochastic information inputs.

cond-mat.stat-mech↗

arXiv2023

Multi-target search in bounded and heterogeneous environments: a lattice random walk perspective

For more than a century lattice random walks have been employed ubiquitously, both as a theoretical laboratory to develop intuition about more complex stochastic processes and as a tool to interpret a vast array of empirical observations. Recent advances in lattice random walk theory in bounded and heterogeneous environments have opened up opportunities to cope with the finely resolved spatio-temporal nature of modern movement data. We review such advances and their formalisms to represent analytically the walker spatio-temporal dynamics in arbitrary dimensions and geometries. As new findings, we derive the exact spatio-temporal representation of biased walks in a periodic hexagon, we use the discrete Feynman-Kac equation to describe a walker's interaction with a radiation boundary, and we unearth a disorder indifference phenomenon. To demonstrate the power of the formalism we uncover the appearance of multiple first-passage peaks with biased walkers in a periodic hexagon, we display the dependence of the first-transmission probability on the proximity transfer efficiency between two resetting walkers in a one-dimensional periodic lattice, we present an example of spatial disorder in a two-dimensional square lattice that strongly affects the splitting probabilities to either of two targets, and we study the first-reaction dynamics to a single lattice site in an unbounded one-dimensional lattice.

cond-mat.stat-mech↗

arXiv2023

Misconceptions about quantifying animal encounter and interaction processes

Quantifying animal interactions is crucial for understanding various ecological processes, including social community structures, predator-prey dynamics, spreading of pathogens and information. Despite the ubiquity of interaction processes among animals and the advancements in tracking technologies enabling simultaneous monitoring of multiple individuals, a common theoretical framework to analyse movement data is still lacking. The diverse mechanisms governing how organisms perceive the proximity of others have led to species-specific theoretical approaches, hindering a common currency with which to evaluate and compare findings across taxa. We propose a general framework, borrowing tools from statistical physics, specifically from the theory of reaction diffusion processes. While some of these tools have been employed to predict pathogen transmission events, they have not yet pervaded the movement ecology literature. Using both continuous and discrete variables, we demonstrate the suitability of our framework to study interaction processes. Defining interactions as the transfer of information between individuals, we show that the probability of information transfer for the first time is equivalent to the first-passage probability of reacting in a multi-target environment. As interaction events reduce to encounter events for perfectly efficient information transfer, we compare our formalism to a recent approach that takes the joint occupation probability of two animals over a region of interaction as a measure of the encounter probability, rather than the first-encounter probability. We show the discrepancy between the two approaches by comparing analytically their predictions with continuous variables, while with discrete variables we quantify their difference over time. We conclude by pointing to some of the open problems that reaction diffusion formalism might be able to tackle.

cond-mat.stat-mech↗

arXiv2022

Dynamics of lattice random walk within regions composed of different media and interfaces

We study the lattice random walk dynamics in a heterogeneous space of two media separated by an interface and having different diffusivity and bias. Depending on the position of the interface, there exist two exclusive ways to model the dynamics: (1) Type A dynamics whereby the interface is placed between two lattice points, and (2) Type B dynamics whereby the interface is placed on a lattice point. For both types, we obtain exact results for the one-dimensional generating function of the Green's function or propagator for the composite system in unbounded domain as well as domains confined with reflecting, absorbing, and mixed boundaries. For the case with reflecting confinement in the absence of bias, the steady-state probability shows a step-like behavior for the Type A dynamics, while it is uniform for the Type B dynamics. We also derive explicit expressions for the first-passage probability and the mean first-passage time, and compare the hitting time dependence to a single target. Finally, considering the continuous-space continuous-time limit of the propagator, we obtain the boundary conditions at the interface. At the interface, while the flux is the same, the probability density is discontinuous for Type A and is continuous for Type B. For the latter we derive a generalized version of the so-called leather boundary condition in the appropriate limit.

cond-mat.stat-mech↗

arXiv2022

Stochastic resets in the context of a tight-binding chain driven by an oscillating field

In this work, we study in the framework of the so-called driven tight-binding chain (TBC) the issue of quantum unitary dynamics interspersed at random times with stochastic resets mimicking non-unitary evolution due to interactions with the external environment, The driven TBC involves a quantum particle hopping between the nearest-neighbour sites of a one-dimensional lattice and subject to an external forcing field that is periodic in time. We consider the resets to be taking place at exponentially-distributed random times. Using the method of stochastic Liouville equation, we derive exact results for the probability at a given time for the particle to be found on different sites and averaged with respect to different realizations of the dynamics. We establish the remarkable effect of localization of the TBC particle on the sites of the underlying lattice at long times. The system in the absence of stochastic resets exhibits delocalization of the particle, whereby the particle does not have a time-independent probability distribution of being found on different sites even at long times, and, consequently, the mean-squared displacement of the particle about its initial location has an unbounded growth in time. One may induce localization in the bare model only through tuning the ratio of the strength to the frequency of the field to have a special value, namely, equal to one of the zeros of the zeroth order Bessel function of the first kind. We show here that localization may be induced by a far simpler procedure of subjecting the system to stochastic resets.

quant-ph↗