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Arijit Chakrabarty

Publications and source records attributed to Arijit Chakrabarty.

At least 19 recordsLinked to original sources
arXiv2026

The Generalized Friendship Paradox for Eigenvectors

In this paper, we investigate the generalized friendship paradox for eigenvectors (alternatively called the eigen friendship paradox and abbreviated hereafter as EFP) in the setting of inhomogeneous Erd\H{o}s--R\'enyi random graphs whose edge probabilities are generated by a continuous graphon. We consider the adjacency matrix of the graph and take the entries of the eigenvector corresponding to its largest eigenvalue as the vertex attributes. It was shown in \cite{hazra2026generalized} that the generalized friendship paradox holds in this setting. We study the empirical distribution of the resulting bias values across the vertices and derive its limiting distribution explicitly in terms of the principal eigenvalue and the corresponding eigenfunction of the integral operator whose kernel is the underlying graphon.

math.PR↗
arXiv2026

Outlier eigenvalues and eigenvectors of generalized Wigner matrices with finite-rank perturbations

A generalized Wigner matrix perturbed by a finite-rank deterministic matrix is considered. The fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors, are studied. Under certain assumptions on the perturbation and the matrix structure, we derive the first-order behavior of these eigenvalues and show that they are well separated from the bulk. The fluctuations of these eigenvalues are shown to follow a multivariate Gaussian distribution, and the asymptotic behavior of the associated eigenvectors is also studied. We prove central limit theorems that describe the asymptotic alignment of these eigenvectors with the perturbation's eigenvectors, as well as their Gaussian fluctuations around the origin for non-aligned components. Furthermore, we discuss the convergence of the eigenvector process in a Sobolev space framework.

math.PR↗
arXiv2025

Shape Characterization of Ferrous Burden Material of Blast Furnace Feed using Image Analysis

In this study, we attempt to characterize the shape of three different types of grains commonly used in the iron and steel-making industry, namely pellet, sinter and iron ore lump. We choose particles over the entire size ranges used in industrial-scale blast furnace and consider two different size ranges of pellet particles and four different size ranges for sinter and iron ore lumps. We perform image analysis to calculate size and shape-related properties of the grains. We select some of the common length scales used to measure the size of the particles and categorize them in different classes. We show that the length scales of a particular category class are well correlated with each other. We identify the independent, uncorrelated length scales for the particles from different categories. Using these uncorrelated length scales, we define different shape descriptors and obtain the distribution of these shape descriptors for each component of the blast furnace feed. Our image analysis results show that the cumulative distribution curves for these shape descriptors turn out to be nearly independent of the size range for a given type of material. Our study identifies the three key shape descriptors that are required to characterize the shape of the blast furnace feed. Two of these shape descriptors, namely the aspect ratio and the circularity, have been considered important by the researchers earlier as well. The third shape descriptor, the average contact eccentricity to the projected particle diameter ratio, usually not considered to be an important shape descriptor in previous studies, is of high relevance for Discrete Element Method simulations of granular materials.

cond-mat.soft↗
arXiv2024

Largest eigenvalue of positive mean Gaussian matrices

This short note studies the fluctuations of the largest eigenvalue of symmetric random matrices with correlated Gaussian entries having positive mean. Under the assumption that the covariance kernel is absolutely summable, it is proved that the largest eigenvalue, after centering, converges in distribution to normal with an explicitly defined mean and variance. This result generalizes known findings for Wigner matrices with independent entries.

math.PR↗
arXiv2023

Clustering of large deviations in moving average processes: the long memory regime

We investigate how large deviations events cluster in the framework of an infinite moving average process with light-tailed noise and long memory. The long memory makes clusters larger, and the asymptotic behaviour of the size of the cluster turns out to be described by the first hitting time of a randomly shifted fractional Brownian motion with drift.

math.PR↗
arXiv2021

Length of stationary Gaussian excursions

Given that a stationary Gaussian process is above a high threshold, the length of time it spends before going below that threshold is studied. The asymptotic order is determined by the smoothness of the sample paths, which in turn is a function of the tails of the spectral measure. Two disjoint regimes are studied - one in which the second spectral moment is finite and the other in which the tails of the spectral measure are regularly varying and the second moment is infinite.

math.PR↗
arXiv2020

Large deviation principle for the maximal eigenvalue of inhomogeneous Erdős-Rényi random graphs

We consider an inhomogeneous Erdős-Rényi random graph $G_N$ with vertex set $[N] = \{1,\dots,N\}$ for which the pair of vertices $i,j \in [N]$, $i\neq j$, is connected by an edge with probability $r(\tfrac{i}{N},\tfrac{j}{N})$, independently of other pairs of vertices. Here, $r\colon\,[0,1]^2 \to (0,1)$ is a symmetric function that plays the role of a reference graphon. Let $λ_N$ be the maximal eigenvalue of the adjacency matrix of $G_N$. It is known that $λ_N/N$ satisfies a large deviation principle as $N \to \infty$. The associated rate function $ψ_r$ is given by a variational formula that involves the rate function $I_r$ of a large deviation principle on graphon space. We analyse this variational formula in order to identify the properties of $ψ_r$, specially when the reference graphon is of rank 1.

math.PR↗
arXiv2019

Eigenvalues outside the bulk of inhomogeneous Erd\H{o}s-R\"enyi random graphs

The article considers an inhomogeneous Erd\H{o}s-R\"enyi random graph on $\{1,\ldots, N\}$, where an edge is placed between vertices $i$ and $j$ with probability $\varepsilon_N f(i/N,j/N)$, for $i\le j$, the choice being made independent for each pair. The function $f$ is assumed to be non-negative definite, symmetric, bounded and of finite rank $k$. We study the edge of the spectrum of the adjacency matrix of such an inhomogeneous Erd\H{o}s-R\'enyi random graph under the assumption that $N\varepsilon_N\to \infty$ sufficiently fast. Although the bulk of the spectrum of the adjacency matrix, scaled by $\sqrt{N\varepsilon_N}$, is compactly supported, the $k$-th largest eigenvalue goes to infinity. It turns out that the largest eigenvalue after appropriate scaling and centering converge to a Gaussian law, if the largest eigenvalue of $f$ has multiplicity $1$. If $f$ has $k$ distinct non-zero eigenvalues, then the joint distribution of the $k$ largest eigenvalues converge jointly to a multivariate Gaussian law. The first order behaviour of the eigenvectors is derived as a by-product of the above results. The results complement the homogeneous case derived by Erd\H{o}s et al.(2013).

math.PR↗
arXiv2019

Spectra of Adjacency and Laplacian Matrices of Inhomogeneous Erdős-Rényi Random Graphs

Inhomogeneous Erdős-Rényi random graphs $\mathbb G_N$ on $N$ vertices in the non-dense regime are considered in this paper. The edge between the pair of vertices $\{i,j\}$ is retained with probability $\varepsilon_N\,f(\frac{i}{N},\frac{j}{N})$, $1 \leq i \neq j \leq N$, independently of other edges, where $f\colon\,[0,1] \times [0,1] \to [0,\infty)$ is a continuous function such that $f(x,y)=f(y,x)$ for all $x,y \in [0,1]$. We study the empirical distribution of both the adjacency matrix $A_N$ and the Laplacian matrix $Δ_N$ associated with $\mathbb G_N$ in the limit as $N \to \infty$ when $\lim_{N\to\infty} \varepsilon_N = 0$ and $\lim_{N\to\infty} N\varepsilon_N = \infty$. In particular, it is shown that the empirical spectral distributions of $A_N$ and $Δ_N$, after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where $f(x,y) = r(x)r(y)$ with $r\colon\,[0,1] \to [0,\infty)$ a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, applications of the results to constrained random graphs, Chung-Lu random graphs and social networks are shown.

math.PR↗
arXiv2019

High minima of non-smooth Gaussian processes

In this short note we study the asymptotic behaviour of the minima over compact intervals of Gaussian processes, whose paths are not necessarily smooth. We show that, beyond the logarithmic large deviation Gaussian estimates, this problem is closely related to the classical small-ball problem. Under certain conditions we estimate the term describing the correction to the large deviation behaviour. In addition, the asymptotic distribution of the location of the minimum, conditionally on the minimum exceeding a high threshold, is also studied.

math.PR↗
arXiv2018

Regular variation and free regular infinitely divisible laws

In this article the relation between the tail behaviours of a free regular infinitely divisible (positively supported) probability measure and its Lévy measure is studied. An important example of such a measure is the compound free Poisson distribution, which often occurs as a limiting spectral distribution of certain sequences of random matrices. We also describe a connection between an analogous classical result of Embrechts et al. [1979] and our result using the Bercovici-Pata bijection.

math.PR↗
arXiv2018

A note on the folklore of free independence

It is shown that a Wishart matrix of standard complex normal random variables is asymptotically freely independent of an independent random matrix, under minimal conditions, in two different sense of asymptotic free independence.

math.PR↗
arXiv2017

Asymptotic behaviour of high Gaussian minima

We investigate what happens when an entire sample path of a smooth Gaussian process on a compact interval lies above a high level. Specifically, we determine the precise asymptotic probability of such an event, the extent to which the high level is exceeded, the conditional shape of the process above the high level, and the location of the minimum of the process given that the sample path is above a high level.

math.PR↗
arXiv2017

Large deviations for truncated heavy-tailed random variables: a boundary case

This paper investigates the decay rate of the probability that the row sum of a triangular array of truncated heavy tailed random variables is larger than an integer (k) times the truncating threshold, as both - the number of summands and the threshold go to infinity. The method of attack for this problem is significantly different from the one where k is not an integer, and requires much sharper estimates.

math.PR↗
arXiv2016

From random matrices to long range dependence

Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process, a phenomenon resembling that in the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues in a completely different way. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.

math.PR↗
arXiv2015

The Hadamard product and the free convolutions

It is shown that if a probability measure $ν$ is supported on a closed subset of $(0,\infty)$, that is, its support is bounded away from zero, then the free multiplicative convolution of $ν$ and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stationary Gaussian process, is obtained.

math.PR↗
arXiv2015

Remarks on absolute continuity in the context of free probability and random matrices

In this note, we show that the limiting spectral distribution of symmetric random matrices with stationary entries is absolutely continuous under some sufficient conditions. This result is applied to obtain sufficient conditions on a probability measure for its free multiplicative convolution with the semicircle law to be absolutely continuous.

math.PR↗