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## Students

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### PhD

- Sunita Rani [January, 2021 - ] -
*ongoing*

### MSc

- Sudeshna Goswami [July, 2023 - May, 2024] -
*A theoretical bridge between random matrix and neural network* - Sagar Ratnakar [July, 2022 - May, 2023] -
*Market volatility via stochastic differential equations* - Amit Ghosh [August, 2021 - May, 2022] -
*Maximal length of increasing subsequences* - Harshit Jain [August, 2020 - May, 2021] -
*The circular law*

## Publications and Preprints

- I. Jana, S. Rani,
*"Spectrum of random centrosymmetric matrices; CLT and Circular law",*preprint, 2024## Abstract

We analyze the asymptotic fluctuations of linear eigenvalue statistics of random centrosymmetric matrices with i.i.d. entries. We prove that for a complex analytic test function, the centered and normalized linear eigenvalue statistics of random centrosymmetric matrices converge to a normal distribution. We find the exact expression of the variance of the limiting normal distribution via combinatorial arguments. Moreover, we also argue that the limiting spectral distribution of properly scaled centrosymmetric matrices follows the circular law. - I. Jana, F. Montorsi, P. Padmanabhan, D. Trancanelli,
*"Topological quantum computation on supersymmetric spin chains",*Journal of High Energy Physics, 2023, (251), 2023## Abstract

Quantum gates built out of braid group elements form the building blocks of topological quantum computation. They have been extensively studied in SU (2) k quantum group theories, a rich source of examples of non-Abelian anyons such as the Ising $(k= 2)$, Fibonacci $(k= 3)$ and Jones-Kauffman $(k= 4)$ anyons. We show that the fusion spaces of these anyonic systems can be precisely mapped to the product state zero modes of certain Nicolai-like supersymmetric spin chains. As a result, we can realize the braid group in terms of the product state zero modes of these supersymmetric systems. These operators kill all the other states in the Hilbert space, thus preventing the occurrence of errors while processing information, making them suitable for quantum computing. - V. K. Singh, A. Sinha, P. Padmanabhan, I. Jana,
*"Dyck Paths and Topological Quantum Computation",*preprint, 2023## Abstract

The fusion basis of Fibonacci anyons supports unitary braid representations that can be utilized for universal quantum computation. We show a mapping between the fusion basis of three Fibonacci anyons, $\{|1\rangle, |\tau\rangle\}$, and the two length $4$ Dyck paths via an isomorphism between the two dimensional braid group representations on the fusion basis and the braid group representation built on the standard $(2, 2)$ Young diagrams using the Jones construction. This correspondence helps us construct the fusion basis of the Fibonacci anyons using Dyck paths as the number of standard $(N, N)$ Young tableaux is the Catalan number, $C_{N}$. We then use the local Fredkin moves to construct a spin chain that contains precisely those Dyck paths that correspond to the Fibonacci fusion basis, as a degenerate set. We show that the system is gapped and examine its stability to random noise thereby establishing its usefulness as a platform for topological quantum computation. Finally, we show braidwords in this rotated space that efficiently enable the execution of any desired single-qubit operation, achieving the desired level of precision$(10^{-3})$. - I. Jana,
*"CLT for non-Hermitian random band matrices with variance profiles",*Journal of Statistical Physics, 187, (13), 2022## Abstract

We show that the fluctuations of the linear eigenvalue statistics of a non-Hermitian 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 $\nu\downarrow 0$. - P. Padmanabhan, I. Jana,
*"Groupoid Toric Codes ",*preprint, 2022## Abstract

The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an inverse. We show that a consistent system can be constructed for an arbitrary groupoid and analyze the simplest example that can be seen as the analog of the Abelian $\mathbb{Z}_{2}$ toric code. We find several exactly solvable models that have fracton-like features which include an extensive ground state degeneracy and excitations that are either immobile or have restricted mobility. Among the possibilities we study in detail the one where the ground state degeneracy scales as $2\times 2^{N_{v}}$, where $N_{v}$ is the number of vertices in the lattice. The origin of this degeneracy can be traced to loop operators supported on both contractible and non-contractible loops. In particular, different non-contractible loops, along the same direction on a torus, result in different ground states. This is an exponential increase in the number of logical qubits that can be encoded in this code. Moreover the face excitations in this system are deconfined, free to move without an energy cost along certain directions of the lattice, whereas in certain other directions their movement incurs an energy cost. This places a restriction on the types of loop operators that contribute to the ground state degeneracy. The vertex excitations are immobile. The results are also extended to the groupoid analogs of Abelian $\mathbb{Z}_{N}$ toric codes. - K. Adhikari, I. Jana, K. Saha,
*"Linear eigenvalue statistics of random matrices with a variance profile",*Random Matrices: Theory and Applications, 10, (3), 2250004, 2021## Abstract

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 re-establish some existing results on fluctuations of linear eigenvalue statistics of some well known random matrix ensembles by choosing appropriate variance profiles. - V. Jain, I. Jana, K. Luh, S. O'Rourke,
*"Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth",*Journal of Mathematical Physics, 62, (8), 083306, 2021## Abstract

We prove the circular law for a class of non-Hermitian 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. - I. Jana, A. Soshnikov,
*"Distribution of singular values of random band matrices; Marchenko-Pastur law and more",*Journal of Statistical Physics, 168, (5), 964-985, 2017## Abstract

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 non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For $R=0$, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law. - I. Jana, K. Saha, A. Soshnikov,
*"Fluctuations of Linear Eigenvalue Statistics of Random Band Matrices",*Theory of Probablity and Applications, 60, (3), 407--443, 2016## Abstract

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 elements are up to 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.