Searcharxiv⌕ Search

arXiv subjects

Sukrit Chakraborty

Publications and source records attributed to Sukrit Chakraborty.

14 recordsLinked to original sources
arXiv2026

Ideal Zero-Product Probability in Finite Commutative Rings

We introduce the \emph{ideal zero-product probability}, a new probabilistic invariant associated with a finite commutative ring $R$, defined by \[ ζ_k(R) = \frac{ |\{(I_1,\ldots,I_k)\in\mathcal I(R)^k : I_1\cdots I_k=(0)\}| } {|\mathcal I(R)|^k}, \] where $\mathcal I(R)$ denotes the set of all ideals of $R$. This quantity gives the probability that the product of $k$ independently and uniformly chosen ideals is the zero ideal. We establish its basic properties, including invariance under ring isomorphisms, monotonicity in $k$, and multiplicativity with respect to finite direct products. Explicit formulas are obtained for finite fields, finite Boolean rings, finite chain rings, and finite principal ideal rings. For finite chain rings, we prove that the invariant depends only on the Loewy length. We also derive closed formulas using inclusion--exclusion and bounded compositions, together with generating functions. Finally, we study the asymptotic behaviour of $ζ_k(R)$ and prove that it converges to $1$ as $k\to\infty$. We also compare the ideal zero-product probability with the classical zero-product probability of ring elements on finite chain rings, obtaining explicit formulas and showing that the two invariants capture fundamentally different structural information.

math.AC↗
arXiv2025

Shortest Paths in a Weighted Simplicial Complex

Simplicial complexes are extensively studied in the field of algebraic topology. They have gained attention in recent time due to their applications in fields like theoretical distributed computing and simplicial neural networks. Graphs are mono-dimensional simplicial complex. Graph theory has application in topics like theoretical computer science, operations research, bioinformatics and social sciences. This makes it natural to try to adapt graph-theoretic results for simplicial complexes, which can model more intricate and detailed structures appearing in real-world systems. Though seemingly obvious, we did not find any previous work that looked into this prospect of simplicial complexes. In this article, we define the concept of weighted simplicial complex and $d$-path in a simplicial complex. Both these concepts have the potential to have numerous real-life applications. We start by adapting the Depth-First Search and Breadth-First Search algorithms for our setup. Next, we provide two novel algorithms to find the shortest paths in a weighted simplicial complex. The core principles of our algorithms align with those of Dijkstra$^\prime$s algorithm and Bellman-Ford algorithm for graphs. Hence, this work lays a building block for the sake of integrating graph-theoretic concepts with abstract simplicial complexes.

cs.DM↗
arXiv2025

Boolean and Free Symmetrization of Bernoulli Distributions

We investigate variance bounds under symmetry constraints in classical, free, and Boolean probability, focusing on Bernoulli distributions and their noncommutative analogues, projections with trace \(p\). We show that symmetrizers under classical, free, and Boolean convolution satisfy a sharp variance bound of \(pq\), with equality for the reflection law. Additionally, we highlight phenomena specific to Boolean convolution, demonstrating that non-symmetric measures can produce symmetric convolutions and that symmetrizers may be non-unique for certain measures. These results unify variance inequalities across probabilistic frameworks and offer insights for quantum information and noncommutative stochastic modeling.

math.PR↗
arXiv2025

Random Finite Sumsets and Product Sets in Subsets of the Natural Numbers

We investigate the occurrence of additive and multiplicative structures in random subsets of the natural numbers. Specifically, for a Bernoulli random subset of $\mathbb{N}$ where each integer is included independently with probability $p\in (0,1)$, we prove that almost surely such a set contains finite sumsets (FS-sets) and finite product sets (FP-sets) of every finite length. In addition, we establish a novel connection between Hindman's partition theorem and the central limit theorem, providing a probabilistic perspective on the asymptotic Gaussian behavior of monochromatic finite sums and products. These results can be interpreted as probabilistic analogues of finite-dimensional versions of Hindman's theorem. Applications, implications, and open questions related to infinite FS-sets and FP-sets are discussed.

math.CO↗
arXiv2024

Eigenvalues outside the bulk of inhomogeneous Erdős-Rënyi random graphs

The article considers an inhomogeneous Erdős-Rënyi 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 Erdős-Rényi 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 Erdős et al.(2013).

math.PR↗
arXiv2023

Shotgun Assembly of Random Geometric Graphs

In a recent work, Huang and Tikhomirov considered the shotgun assembly for Erd\H os-Rényi graphs $\mathcal G(n,p_n)$ with $p_n=n^{-α}$, and showed that the graph is reconstructable if $0<α< \frac{1}{2}$ and not reconstructable if $\frac{1}{2}<α<1$ from its $1$-neighbourhoods. In this article, we consider random geometric graphs $G(n,r)$, where $r^2=n^{-α}$ and $ 0<α<1$, on flat torus. Interestingly, unlike the results for the Erd\H os-Rényi random graphs, we show that the random geometric graph is always reconstructable from its 1-neighbourhoods.

math.PR↗
arXiv2022

Shotgun Assembly of Linial-Meshulam Model

In a recent paper [6], J. Gaudio and E. Mossel studied the shotgun assembly of the Erdős-Rényi graph $\mathcal G(n,p_n)$ with $p_n=n^{-α}$, and showed that the graph is reconstructable form its $1$-neighbourhoods if $0<α< 1/3$ and not reconstructable from its $1$-neighbourhoods if $1/2 <α<1$. In this article, we generalise the notion of reconstruction of graphs to the reconstruction of simplicial complexes. We show that the Linial-Meshulam model $Y_{d}(n,p_n)$ on $n$ vertices with $p_n=n^{-α}$ is reconstructable from its $1$-neighbourhoods when $0< α< 1/3$ and is not reconstructable form its $1$-neighbourhoods when $1/2 < α< 1$.

math.CO↗
arXiv2022

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↗
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

Boolean convolutions and regular variation

In this article we study the influence of regularly varying probability measures on additive and multiplicative Boolean convolutions. We introduce the notion of Boolean subexponentiality (for additive Boolean convolution), which extends the notion of classical and free subexponentiality. We show that the distributions with regularly varying tails belong to the class of Boolean subexponential distributions. As an application we also study the behaviour of the Belinschi-Nica map. Breiman's theorem study the classical product convolution between regularly varying measures. We derive an analogous result to Breiman's theorem in case of multiplicative Boolean convolution. In proving these results we exploit the relationship of regular variation with different transforms and their Taylor series expansion.

math.PR↗
arXiv2018

C-image partition regularity near zero

In \cite{dehind1}, the concept of image partition regularity near zero was first instigated. In contrast to the finite case , infinite image partition regular matrices near zero are very fascinating to analyze. In this regard the abstraction of Centrally image partition regular matrices near zero was introduced in \cite{biswaspaul}. In this paper we propose the notion of matrices that are C-image partition regular near zero for dense subsemigropus of $((0,\infty),+)$.

math.GN↗
arXiv2018

Infinite image partition regular matrices - Solution in C-sets

A finite or infinite matrix $A$ is image partition regular provided that whenever $\mathbb{N}$ is finitely colored, there must be some $\overset{\rightarrow}{x}$ with entries from $\mathbb{N}$ such that all entries of $A \overset{\rightarrow}{x}$ are in the same color class. Comparing to the finite case, infinite image partition regular matrices seem more harder to analyze. The concept of centrally image partition regular matrices were introduced to extend the results of finite image partition regular matrices to infinite one. In this paper, we shall introduce the notion of C-image partition regular matrices, an interesting subclass of centrally image partition regular matrices. Also we shall see that many of known centrally image partition regular matrices are C-image partition regular.

math.CO↗
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↗