I did my PhD on
Spectrum of Random Band Matrices under the supervision of professor
Alexander Soshnikov. My MS thesis was on
matchings between Point processes .
Publications and Preprints

[Abstract] Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth
(with Vishesh Jain, Kyle Luh, Sean O'Rourke),
Journal of Mathematical Physics, Volume 62, Issue 8
We prove the circular law for a class of nonHermitian random block band
matrices with genuinely sublinear bandwidth. Namely, we show there exists $\tau
\in (0,1)$ so that if the bandwidth of the matrix $X$ is at least $n^{1\tau}$
and the nonzero entries are iid random variables with mean zero and slightly
more than four finite moments, then the limiting empirical eigenvalue
distribution of $X$, when properly normalized, converges in probability to the
uniform distribution on the unit disk in the complex plane. The key technical
result is a least singular value bound for shifted random band block matrices
with genuinely sublinear bandwidth, which improves on a result of Cook in the
band matrix setting \cite{cook2018lower}.
[Abstract] CLT for nonHermitian random band matrices with variance profiles
arXiv preprint: arXiv: 1904.11098, 2019
We show that the fluctuations of the linear eigenvalue statistics of a nonHermitian random band
matrix of bandwidth $b_{n}$ with a continuous variance profile $w_{\nu}(x)$ converges to a
$N(0,\sigma_{f}^{2}(\nu))$, where $\nu=\lim_{n\to\infty}(2b_{n}/n)$. We obtain an explicit formula
for $\sigma_{f}^{2}(\nu)$, which depends on the test function, and $w_{\nu}$. When $\nu=1$, the
formula is consistent with Rider, and Silverstein (2006). We also compute
an explicit formula for $\sigma_{f}^{2}(0)$. We show that $\sigma_{f}^{2}(\nu)\to \sigma_{f}^{2}(0)$
as $\nu\downarrow 0$.
[Abstract] Linear eigenvalue statistics of random matrices with a variance profile, (with Kartick Adhikari, Koushik Saha)
Random Matrices: Theory and Applications
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 PoincarĂ© 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 reestablish some existing results on fluctuations of
linear eigenvalue statistics of some well known random matrix ensembles by choosing appropriate
variance profiles.
[Abstract] Distribution of singular values of random band matrices; MarchenkoPastur law and more, (with Alexander Soshnikov)
Journal of Statistical Physics, 168(5), 964985, doi:10.1007/s1095501718445
We consider the limiting spectral distribution of matrices of the form
$\frac{1}{2b_{n}+1} (R + X)(R + X)^{*}$, where $X$ is an $n\times n$ band matrix of bandwidth
$b_{n}$ and $R$ is a non random band matrix of bandwidth $b_{n}$. We show that the Stieltjes
transform of ESD of such matrices converges to the Stieltjes transform of a nonrandom measure.
And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral
equation yields the Stieltjes transform of the MarchenkoPastur law.
[Abstract] Fluctuations of Linear Eigenvalue Statistics of Random band Matrices, (with Koushik Saha, Alexander Soshnikov)
Theory of Probablity and Applications, Vol 60, No. 3, 407443, 2016.
In this paper, we study the fluctuation of linear eigenvalue statistics of Random Band Matrices
defined by $M_{n}=\frac{1}{\sqrt{b_{n}}}W_{n}$, where $W_{n}$ is a $n\times n$ band Hermitian
random matrix of bandwidth $b_{n}$, i.e., the diagonal elements and only first $b_{n}$ off
diagonal elements are nonzero. Also variances of the matrix elmements are upto a order of
constant. We study the linear eigenvalue statistics
$\mathcal{N}(\phi)=\sum_{i=1}^{n}\phi(\lambda_{i})$ of such matrices, where $\lambda_{i}$ are
the eigenvalues of $M_{n}$ and $\phi$ is a sufficiently smooth function. We prove that
$\sqrt{\frac{b_{n}}{n}}[\mathcal{N}(\phi)\mathbb{E} \mathcal{N}(\phi)]\stackrel{d}{\to} N(0,V(\phi))$
for $b_{n}>>\sqrt{n}$, where $V(\phi)$ is given in the Theorem 1.