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Kartick Adhikari

Publications and source records attributed to Kartick Adhikari.

At least 19 recordsLinked to original sources
arXiv2026

Semicircle law for multi-parameter random simplicial complexes

In this paper, we consider the multi-parameter random simplicial complex model, which generalizes the Linial-Meshulam model and random clique complexes by allowing simplices of different dimensions to be included with distinct probabilities. For $n,d \in \mathbb{N}$ and $\mathbf{p}=(p_1,p_2,\ldots, p_d)\in (0,1]^d$, the multi-parameter random simplicial complex $Y_d(n,\mathbf{p})$ is constructed inductively. Starting with $n$ vertices, edges (1-cells) are included independently with probability $p_1$, yielding the Erdős-Rényi graph $G(n,p_1)$, which forms the $1$-skeleton. Conditional on the $(k-1)$-skeleton, each possible $k$-cell is included independently with probability $p_k$, for $2 \leq k \leq d$. We study the signed and unsigned adjacency matrices of $d$-dimensional multi-parameter random simplicial complexes $Y_d(n,\mathbf{p})$, under the assumptions $\min_{i=1,\ldots d-1}\liminf p_i >0$ and $np_d \rightarrow \infty$ with $p_d=o(1)$. In general, these matrices have random dimensions and exhibit dependency among their entries. We prove that the empirical spectral measures of both matrices converge weakly to the semicircle law in probability. We also prove that the limiting spectral measure of the signed and unsigned adjacency matrices of another associated model, the multi-parameter upper model, is the semicircle law, under the assumption $np_d(1-p_d) \rightarrow \infty$. Further, we establish convergence results for the number of $(d-1)$-cells and the number of maximal $(d-1)$-cells of multi-parameter simplicial complexes.

math.PR↗
arXiv2026

Large deviations of crowding in finite $β$-ensembles

We consider finite $β$-ensembles $\mathcal X_{n,β}^{\mathbb F}$ with $n$ points on $\mathbb F$, where $\mathbb F$ denotes either the real line or the complex plane. Let $U$ be a bounded subset of $ \mathbb F$ such that $\partial U$ (the boundary of $U$) is polar for $\mathbb F=\mathbb R$ and $\partial U$ is a closed $1$--rectifiable set with finite $1$-dimensional Hausdorff measure. Suppose $\mathcal X_{n,β}^{\mathbb F}(U)$ denotes the number of points in the region $U$. We show that the sequence of laws of $\{n^{-1}\mathcal X_{n,β}^{\mathbb F}(U); n\ge 1\}$ satisfies the large deviation type bound with speed $n^2$ and with a good rate function. For $\mathbb{F} = \mathbb{R}$, this result can be derived using the contraction principle. However, when $\mathbb{F} = \mathbb{C}$, the contraction principle does not yield the desired outcome. Therefore, we adopt a direct approach to establish our results.

math.PR↗
arXiv2026

Star-collision in random hypergraphs

We study star-based symmetries in uniform hypergraphs and their consequences for matrices whose entries depend only on vertex stars. Such matrices admit a deterministic decomposition into a global component and a local component supported on equivalence classes of vertices with identical stars, known as units. While nontrivial units may exist at finite size in hypergraphs of uniformity greater than two, their persistence in random settings has remained unclear. We analyze star collisions in random $k$-uniform hypergraphs and show that, in some particular regimes, nontrivial units disappear with high probability as the number of vertices grows. As a consequence, star-dependent matrices exhibit asymptotically trivial local structure, and their spectral behavior, invariant subspaces, and associated linear dynamics are governed by a reduced quotient object obtained by contracting vertex stars. These results identify star collisions as a finite-size phenomenon in random hypergraphs and clarify the asymptotic irrelevance of star-based symmetries for operator behavior in large random systems in particular regimes.

math.CO↗
arXiv2025

On the diameter of random uniform hypergraphs in dense regime

For a fixed natural number $t\geq 2$, we consider $t$-uniform random hypergraphs $\mathscr{H} (n,t,p)$ on $n$ vertices $[n]=\{1,\ldots, n\}$, where each $t$-subset of $[n]$ is included as a hyperedge with probability $p$ and independently. We show that the diameter of $\mathscr{H} (n,t,p)$ is concentrated only at two points in the dense regime. More precisely, suppose $diam(\mathcal H)$ denotes the diameter of a hypergraph $\mathcal H$ on $n$ vertices. We show that, for fixed $t,c,d$ constants, if $n$ and $p$ (depends on $t,c,d,n$) satisfy $$ \frac{ (t-1)^ {d} N^{d} p^{d}} {n}= \log \left( \frac{n^2}{c} \right), \mbox{ where } N={n-1\choose t-1}, $$ $c$ is a positive constant and $d\geq2$ is a natural number, then $$ \lim_{n \to \infty} \mathbb{P} \left( diam( \mathcal{H}) = d \right) = e^{- \frac{c}{2}} \text{ and } \lim_{n \to \infty} \mathbb{P} \left( diam(\mathcal{H}) = d+1 \right) = 1- e^{- \frac{c}{2}}. $$ In particular, the case where $t = 2$ corresponds to the diameter of the Erdős-Rényi graph, as established by Bollobás in \cite[Theorem~6]{bollobas1981diameter}. Bollob\' as's result was proven using the moments method, which is challenging to apply in our context due to the complexity of the model. In this paper, we utilize the Stein-Chen method along with coupling techniques to prove our result. This approach can potentially be used to solve various problems, in particular diameter problems, in more complex networks.

math.PR↗
arXiv2025

Spectrum and local weak convergence of sparse random uniform hypergraphs

The notion of local weak convergence, or Benjamini--Schramm convergence, was introduced by Benjamini and Schramm. The local weak limit of sparse Erd\H os--Rényi graphs is the Galton--Watson measure with Poisson offspring almost surely. Recently, Adhikari, Kumar, and Saha showed that the line graph of sparse Linial--Meshulam complexes converges to the $d$-block Galton--Watson measure. We study a unified model: weighted line graphs of sparse $k$-uniform random hypergraphs on $n$ vertices. Let $H(n,k,p)$ be the $k$-uniform random hypergraph where each $k$-subset of $[n]$ is included as a hyperedge independently with probability $p$. For a $k$-uniform hypergraph $H=(V,E)$ and $1\le r\le k-1$, define the $r$-set weighted line graph $G_r(H)=(\mathcal V_r,\mathcal E_r,w_H)$ by $$ \mathcal V_r=\bigl[\tbinom{n}{r}\bigr],\quad \mathcal E_r=\bigl\{\{τ_1,τ_2\}:τ_1,τ_2\in\mathcal V_r,\ \exists e\in E\text{ s.t. }τ_1,τ_2\subset e\bigr\}, $$ with weight $w_H(\{τ_1,τ_2\})=\bigl|\{e\in E:τ_1,τ_2\subset e\}\bigr|$. In particular, $G_1(H_n)$ generalizes Erd\H os--Rényi graphs and $G_{k-1}(H_n)$ is the line graph of the Linial--Meshulam complex. We show that if $\tbinom{n-r}{k-r}\to λ$ as $n\to\infty$, then $G_r(H_n)$ converges locally to the $(\tbinom{k}{r}-1)$-block Galton--Watson measure with $\operatorname{Poisson}(λ)$ offspring almost surely. As a consequence, we obtain the limiting spectral distribution of the adjacency matrices of $G_r(H_n)$.

math.CO↗
arXiv2025

Convergence and asymptotic freeness of missing data matrices

We consider a random matrix of the form $D_n \odot X_n$ (known as a variance profile matrix), where $\odot$ denotes the Hadamard product of the two matrices, $D_n$ is a deterministic matrix, and $X_n$ is a random matrix. We call $D_n\odot X_n$ as a missing data matrix of $X_n$ when the entries of $D_n$ are either $0$ or $1$. This framework is commonly used in various applied fields, such as biology, neuroscience, and network data analysis. We study the convergence and asymptotic freeness of missing data matrices of iid, elliptic, and covariance random matrices. Specifically, it is known that independent iid, elliptic, and covariance matrices converge to freely independent circular, elliptic, and Marčenko-Pastur variables, respectively. In this article, we provide the necessary and sufficient conditions on deterministic matrices $D_n$ for which these results hold true for independent missing data matrices of these three types of random matrices.

math.PR↗
arXiv2024

Convergence of high dimensional Toeplitz and related matrices with correlated inputs

We investigate the joint convergence of independent random Toeplitz matrices with complex input entries that have a pair-correlation structure, along with deterministic Toeplitz matrices and the backward identity permutation matrix. Further, we study the joint convergence of independent generalized Toeplitz matrices along with other related matrices. The limits depend only on the correlation structure but are universal otherwise, in that they do not depend on the underlying distributions of the entries. In particular, these results provide the joint convergence of asymmetric Hankel matrices. Earlier results in the literature on the joint convergence of random symmetric Toeplitz and symmetric Hankel matrices with real entries follow as special cases.

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↗
arXiv2023

On the spectrum of Random Simplicial Complexes in Thermodynamic Regime

Linial-Meshulam complex is a random simplicial complex on $n$ vertices with a complete $(d-1)$-dimensional skeleton and $d$-simplices occurring independently with probability p. Linial-Meshulam complex is one of the most studied generalizations of the Erdos-Renyi random graph in higher dimensions. In this paper, we discuss the spectrum of adjacency matrices of the Linial-Meshulam complex when $np \rightarrow λ$. We prove the existence of a non-random limiting spectral distribution(LSD) and show that the LSD of signed and unsigned adjacency matrices of Linial-Meshulam complex are reflections of each other. We also show that the LSD is unsymmetric around zero, unbounded and under the normalization $1/\sqrt{λd}$, converges to standard semicircle law as $λ\rightarrow \infty$. In the later part of the paper, we derive the local weak limit of the line graph of the Linial-Meshulam complex and study its consequence on the continuous part of the LSD.

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

Fluctuation and Entropy in Spectrally Constrained random fields

We investigate the statistical properties of translation invariant random fields (including point processes) on Euclidean spaces (or lattices) under constraints on their spectrum or structure function. An important class of models that motivate our study are hyperuniform and stealthy hyperuniform systems, which are characterised by the vanishing of the structure function at the origin (resp., vanishing in a neighbourhood of the origin). We show that many key features of two classical statistical mechanical measures of randomness - namely, fluctuations and entropy, are governed only by some particular local aspects of their structure function. We obtain exponents for the fluctuations of the local mass in domains of growing size, and show that spatial geometric considerations play an important role - both the shape of the domain and the mode of spectral decay. In doing so, we unveil intriguing oscillatory behaviour of spatial correlations of local masses in adjacent box domains. We describe very general conditions under which we show that the field of local masses exhibit Gaussian asymptotics, with an explicitly described limit. We further demonstrate that stealthy hyperuniform systems with joint densities exhibit degeneracy in their asymptotic entropy per site. In fact, our analysis shows that entropic degeneracy sets in under much milder conditions than stealthiness, as soon as the structure function fails to be logarithmically integrable.

math.PR↗
arXiv2020

On the Spectrum of Dense Random Geometric Graphs

In this paper we study the spectrum of the random geometric graph $G(n,r)$, in a regime where the graph is dense and highly connected. In the \erdren $G(n,p)$ random graph it is well known that upon connectivity the spectrum of the normalized graph Laplacian is concentrated around $1$. We show that such concentration does not occur in the $G(n,r)$ case, even when the graph is dense and almost a complete graph. In particular, we show that the limiting spectral gap is strictly smaller than $1$. In the special case where the vertices are distributed uniformly in the unit cube and $r=1$, we show that for every $0\le k \le d$ there are at least $\binom{d}{k}$ eigenvalues near $1-2^{-k}$, and the limiting spectral gap is exactly $1/2$. We also show that the corresponding eigenfunctions in this case are tightly related to the geometric configuration of the points.

math.PR↗
arXiv2019

Linear eigenvalue statistics of random matrices with a variance profile

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centred, of a random matrix with a variance profile and the standard Gaussian random variable. The second order Poincaré inequality type result is used to establish the bound. Using this bound we prove Central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well known random matrix ensembles by choosing appropriate variance profiles.

math.PR↗
arXiv2017

Brown measure and asymptotic freeness of elliptic and related matrices

We show that independent elliptic matrices converge to freely independent elliptic elements. Moreover, the elliptic matrices are asymptotically free with deterministic matrices under appropriate conditions. We compute the Brown measure of the product of elliptic elements. It turns out that this Brown measure is same as the limiting spectral distribution.

math.PR↗
arXiv2017

Fluctuations of eigenvalues of patterned random matrices

In this article we study the fluctuation of linear statistics of eigenvalues of circulant, symmetric circulant, reverse circulant and Hankel matrices. We show that the linear spectral statistics of these matrices converges to the Gaussian distribution in total variation norm when the matrices are constructed using i.i.d. normal random variables. We also calculate the limiting variance of the linear spectral statistics for circulant, symmetric circulant and reverse circulant matrices.

math.PR↗
arXiv2016

Hole probabilities for finite and infinite Ginibre ensembles

We study the hole probabilities of the infinite Ginibre ensemble ${\mathcal X}_{\infty}$, a determinantal point process on the complex plane with the kernel $\mathbb K(z,w)= \frac{1}πe^{z\bar w-\frac{1}{2}|z|^2-\frac{1}{2}|w|^2}$ with respect to the Lebesgue measure on the complex plane. Let $U$ be an open subset of open unit disk $\mathbb D$ and ${\mathcal X}_{\infty}(rU)$ denote the number of points of ${\mathcal X}_{\infty}$ that fall in $rU$. Then, under some conditions on $U$, we show that $$ \lim_{r\to \infty}\frac{1}{r^4}\log\mathbb P[\mathcal X_{\infty}(rU)=0]=R_{\emptyset}-R_{U}, $$ where $\emptyset$ is the empty set and $$ R_U:=\inf_{μ\in \mathcal P(U^c)}\left\{\iint \log{\frac{1}{|z-w|}}dμ(z)dμ(w)+\int |z|^2dμ(z) \right\}, $$ $\mathcal P(U^c)$ is the space of all compactly supported probability measures with support in $U^c$. Using potential theory, we give an explicit formula for $R_U$, the minimum possible energy of a probability measure compactly supported on $U^c$ under logarithmic potential with a quadratic external field. Moreover, we calculate $R_U$ explicitly for some special sets like annulus, cardioid, ellipse, equilateral triangle and half disk.

math.PR↗