arXiv subjects

Debapratim Banerjee

Publications and source records attributed to Debapratim Banerjee.

14 recordsLinked to original sources

arXiv2026

Semicircular law with a few independent entries in a random matrix

It is well known in random matrix literature that the limiting spectral distribution of a Wigner matrix is the semi circular law while the limiting spectral distributions of other patterned matrices like Toeplitz, Hankel, symmetric circulant and reverse circulant matrices have unbounded supports. One fundamental difference between the Wigner matrices and the other matrices mentioned above is that the Wigner matrices have $O(n^2)$ independent random variables while the others have $O(n)$ independent random variables. In this paper, we show that this is not true in general. In particular, we form matrices with $O(n)$ independent random variables whose empirical spectral distributions are arbitrary close to the semi circular law.

math.PR↗

arXiv2025

Overlap distribution of spherical spin glass models with general eigenvalue distribution of the interaction matrix

In this paper, we show that the replica symmetry of the Gibbs measure of spherical spin systems is a property of the eigenvalue spacing at the edge of the interaction matrix. In particular, our interaction matrix has \textbf{two} large outlier eigenvalues with mutual distance $\frac{c}{n}$. The empirical measure of the rest of the eigenvalues is close to the semicircular law with some rigidity conditions. We prove that in this scenario the overlap distribution of two independent samples from the Gibbs measure has a continuous density at a low enough temperature. Hence, the model is a full replica symmetry-breaking model. One might compare this result with only one outlier eigenvalue. This model comes for the Sherrington-Kirkpatrick model with Curie-Weiss interaction in the ferromagnetic case. Here, it is well known that the model is replica symmetric, although the free energy limit of this model is the same as the free energy limit of our model. In our limited understanding, we believe that this kind of phenomenon cannot be explained by the Parisi approach.

math.PR↗

arXiv2025

A short proof of free energy limit of two spin spherical Sherrington-Kirkpatrick model at any temperature

In this paper, we present a short proof of the limit of free energy of spherical 2 spin Sherrington-Kirkpatrick (SSK) model without external field. This proof works for all temperatures and is based on the Laplace method of integration and considering a discrete approximation of the whole space. This proof is general enough and can be adapted to any interaction matrix where the eigenvalue distribution has some nice properties. The proof in the high-temperature case is the same as the proof given in [BK19]. However, the low-temperature case is new.

math.PR↗

arXiv2025

Operator limit of Wigner matrices I

We consider the Wigner matrix $W_{n}$ of dimension $n \times n$ as $n \to \infty$. The objective of this paper is two folds: first we construct an operator $\mathcal{W}$ on a suitable Hilbert space $\mathcal{H}$ and then define a suitable notion of convergence such that the matrices $W_{n}$ converge in that notion of convergence to $\mathcal{W}$. We further investigate some properties of $\mathcal{W}$ and $\mathcal{H}$. We show that $\mathcal{H}$ is a nontrivial extension of $L^{2}[0,1]$ with respect to the Lebesgue measure and the spectral measure of $\mathcal{W}$ at any function $f \in L^{2}[0,1]$ is almost surely the semicircular law.

math.PR↗

arXiv2024

Edge spectra of Gaussian random symmetric matrices with correlated entries

We study the largest eigenvalue of a Gaussian random symmetric matrix $X_n$, with zero-mean, unit variance entries satisfying the condition $\sup_{(i, j) \ne (i', j')}|\mathbb{E}[X_{ij} X_{i'j'}]| = O(n^{-(1 + \varepsilon)})$, where $\varepsilon > 0$. It follows from Catalano et al. (2024) that the empirical spectral distribution of $n^{-1/2} X_n$ converges weakly almost surely to the standard semi-circle law. Using a F\"{u}redi-Koml\'{o}s-type high moment analysis, we show that the largest eigenvalue $\lambda_1(n^{-1/2} X_n)$ of $n^{-1/2} X_n$ converges almost surely to $2$. This result is essentially optimal in the sense that one cannot take $\varepsilon = 0$ and still obtain an almost sure limit of $2$. We also derive Gaussian fluctuation results for the largest eigenvalue in the case where the entries have a common non-zero mean. Let $Y_n = X_n + \frac{\lambda}{\sqrt{n}}\mathbf{1} \mathbf{1}^\top$. When $\varepsilon \ge 1$ and $\lambda \gg n^{1/4}$, we show that \[ n^{1/2}\bigg(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\bigg) \xrightarrow{d} \sqrt{2} Z, \] where $Z$ is a standard Gaussian. On the other hand, when $0 < \varepsilon < 1$, we have $\mathrm{Var}(\frac{1}{n}\sum_{i, j}X_{ij}) = O(n^{1 - \varepsilon})$. Assuming that $\mathrm{Var}(\frac{1}{n}\sum_{i, j} X_{ij}) = \sigma^2 n^{1 - \varepsilon} (1 + o(1))$, if $\lambda \gg n^{\varepsilon/4}$, then we have \[ n^{\varepsilon/2}\bigg(\lambda_1(n^{-1/2} Y_n) - \lambda - \frac{1}{\lambda}\bigg) \xrightarrow{d} \sigma Z. \] While the ranges of $\lambda$ in these fluctuation results are certainly not optimal, a striking aspect is that different scalings are required in the two regimes $0 < \varepsilon < 1$ and $\varepsilon \ge 1$.

math.PR↗

arXiv2022

Independence of linear spectral statistics and the point process at the edge of Wigner matrices

In the current paper we consider a Wigner matrix and consider an analytic function of polynomial growth on a set containing the support of the semicircular law in its interior. We prove that the linear spectral statistics corresponding to the function and the point process at the edge of the Wigner matrix are asymptotically independent when the entries of the Wigner matrix are sub-Gaussian. The main ingredient of the proof is based on a recent paper by Banerjee [6]. The result of this paper can be viewed as a first step to find the joint distribution of eigenvalues in the bulk and the edge.

math.PR↗

arXiv2022

A new combinatorial approach for edge universality of Wigner matrices

In this paper we introduce a new combinatorial approach to analyze the trace of large powers of Wigner matrices. Our approach is motivated from the paper by \citet{sosh}. However the counting approach is different. We start with classical word sentence approach similar to \citet{AZ05} and take the motivation from \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} to encode the words to objects similar to Dyck paths. To be precise the map takes a word to a Dyck path with some edges removed from it. Using this new counting we prove edge universality for large Wigner matrices with sub-Gaussian entries. One novelty of this approach is unlike \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} we do not need to assume the entries of the matrices are symmetrically distributed around $0$. The main technical contribution of this paper is two folded. Firstly we produce an encoding of the ``contributing words" (for definition one might look at Section \ref{sec:word}) of the Wigner matrix which retrieves the edge universality. Hence this is the best one can do. We hope this method will be applicable to many other scenarios in random matrices. Secondly in course of the paper we give a combinatorial description of the GOE Tracy Widom law. The explanation for GUE is very similar. This explanation might be important for the models where exact calculations are not available but some combinatorial structures are present.

math.PR↗

arXiv2021

Fluctuations of the free energy of the mixed $p$-spin mean field spin glass model

We prove the convergence in distribution of the fluctuations of the free energy of the mixed $p$-spin Sherrington-Kirkpatrick model with non-vanishing $2$-spin component at high enough temperature. The limit is Gaussian, and the fluctuations are seen to arise from weighted cycle counts in the complete graph on the spin indices weighted by the $2$-spin interaction matrix.

math.PR↗

arXiv2019

Fluctuation of the free energy of Sherrington-Kirkpatrick model with Curie-Weiss interaction: the paramagnetic regime

We consider a spin system with pure two spin Sherrington-Kirkpatrick Hamiltonian with Curie-Weiss interaction. The model where the spins are spherically symmetric was considered by \citet{Baiklee16} and \citet{Baikleewu18} which shows a two dimensional phase transition with respect to temperature and the coupling constant. In this paper we prove a result analogous to \citet{Baiklee16} in the "paramagnetic regime" when the spins are i.i.d. Rademacher. We prove the free energy in this case is asymptotically Gaussian and can be approximated by a suitable linear spectral statistics. Unlike the spherical symmetric case the free energy here can not be written as a function of the eigenvalues of the corresponding interaction matrix. The method in this paper relies on a dense sub-graph conditioning technique introduced by \citet{Ban16}. The proof of the approximation by the linear spectral statistics part is taken from \citet{Banerjee2017}.

math.PR↗

arXiv2019

A Bernstein type inequality for sums of selections from three dimensional arrays

We consider the three dimensional array $\mathcal{A} = \{a_{i,j,k}\}_{1\le i,j,k \le n}$, with $a_{i,j,k} \in [0,1]$, and the two random statistics $T_{1}:= \sum_{i=1}^n \sum_{j=1}^n a_{i,j,σ(i)}$ and $T_{2}:= \sum_{i=1}^{n} a_{i,σ(i),π(i)}$, where $σ$ and $π$ are chosen independently from the set of permutations of $\{1,2,\ldots,n \}.$ These can be viewed as natural three dimensional generalizations of the statistic $T_{3}=\sum_{i=1}^{n} a_{i,σ(i)}$, considered by Hoeffding \cite{Hoe51}. Here we give Bernstein type concentration inequalities for $T_{1}$ and $T_{2}$ by extending the argument for concentration of $T_{3}$ by Chatterjee \cite{Cha05}.

math.PR↗

arXiv2019

High-dimensional CLT: Improvements, Non-uniform Extensions and Large Deviations

Central limit theorems (CLTs) for high-dimensional random vectors with dimension possibly growing with the sample size have received a lot of attention in the recent times. Chernozhukov et al. (2017) proved a Berry--Esseen type result for high-dimensional averages for the class of hyperrectangles and they proved that the rate of convergence can be upper bounded by $n^{-1/6}$ upto a polynomial factor of $\log p$ (where $n$ represents the sample size and $p$ denotes the dimension). Convergence to zero of the bound requires $\log^7p = o(n)$. We improve upon their result which only requires $\log^4p = o(n)$ (in the best case). This improvement is made possible by a sharper dimension-free anti-concentration inequality for Gaussian process on a compact metric space. In addition, we prove two non-uniform variants of the high-dimensional CLT based on the large deviation and non-uniform CLT results for random variables in a Banach space by Bentkus, Ra{\v c}kauskas, and Paulauskas. We apply our results in the context of post-selection inference in linear regression and of empirical processes.

math.ST↗

arXiv2018

Asymptotic normality and analysis of variance of log-likelihood ratios in spiked random matrix models

The present manuscript studies signal detection by likelihood ratio tests in a number of spiked random matrix models, including but not limited to Gaussian mixtures and spiked Wishart covariance matrices. We work directly with multi-spiked cases in these models and with flexible priors on the signal component that allow dependence across spikes. We derive asymptotic normality for the log-likelihood ratios when the signal-to- noise ratios are below certain thresholds. In addition, we show that the variances of the log-likelihood ratios can be asymptotically decomposed as the sums of those of a collection of statistics which we call bipartite signed cycles.

math.ST↗

arXiv2017

Optimal hypothesis testing for stochastic block models with growing degrees

The present paper considers testing an Erdos--Renyi random graph model against a stochastic block model in the asymptotic regime where the average degree of the graph grows with the graph size n. Our primary interest lies in those cases in which the signal-to-noise ratio is at a constant level. Focusing on symmetric two block alternatives, we first derive joint central limit theorems for linear spectral statistics of power functions for properly rescaled graph adjacency matrices under both the null and local alternative hypotheses. The powers in the linear spectral statistics are allowed to grow to infinity together with the graph size. In addition, we show that linear spectral statistics of Chebyshev polynomials are closely connected to signed cycles of growing lengths that determine the asymptotic likelihood ratio test for the hypothesis testing problem of interest. This enables us to construct a sequence of test statistics that achieves the exact optimal asymptotic power within $O(n^3 \log n)$ time complexity in the contiguous regime when $n^2 p_{n,av}^3 \to\infty$ where $p_{n,av}$ is the average connection probability. We further propose a class of adaptive tests that are computationally tractable and completely data-driven. They achieve nontrivial powers in the contiguous regime and consistency in the singular regime whenever $n p_{n,av} \to\infty$. These tests remain powerful when the alternative becomes a more general stochastic block model with more than two blocks.

math.ST↗

arXiv2016

Contiguity and non-reconstruction results for planted partition models: the dense case

We consider the two block stochastic block model on $n$ nodes with asymptotically equal cluster sizes. The connection probabilities within and between cluster are denoted by $p_n:=\frac{a_n}{n}$ and $q_n:=\frac{b_n}{n}$ respectively. Mossel et al.(2012) considered the case when $a_n=a$ and $b_n=b$ are fixed. They proved the probability models of the stochastic block model and that of Erd{ö}s-R{é}nyi graph with same average degree are mutually contiguous whenever $(a-b)^2<2(a+b)$ and are asymptotically singular whenever $(a-b)^2>2(a+b)$. Mossel et al.(2012) also proved that when $(a-b)^2<2(a+b)$ no algorithm is able to find an estimate of the labeling of the nodes which is positively correlated with the true labeling. It is natural to ask what happens when $a_n$ and $b_n$ both grow to infinity. We prove that their results extend to the case when $a_n=o(n)$ and $b_n=o(n)$. We also consider the case when $\frac{a_n}{n} \to p \in (0,1)$ and $(a_n-b_n)^2= Θ(a_n+b_n)$. Observe that in this case $\frac{b_n}{n} \to p$ also. We show that here the models are mutually contiguous if $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$ and they are asymptotically singular if $(a_n-b_n)^2 > 2(1-p)(a_n+b_n)$. Further we also prove it is impossible find an estimate of the labeling of the nodes which is positively correlated with the true labeling whenever $(a_n-b_n)^2< 2(1-p)(a_n+b_n)$. The results of this paper justify the negative part of a conjecture made in Decelle et al.(2011) for dense graphs.

math.PR↗