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abstract: 'In this article, we study the fluctuation of linear eigenvalue statistics of symmetric circulant matrices $(SC_n)$ with independent entries which satisfy some moment conditions. We show that $\frac{1}{\sqrt{n}} \Tr \phi(SC_n)$ obey the central limit theorem (CLT) type result, where $\phi$ is a nice test function.'
address:
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    Department of Mathematics\
    Indian Institute of Technology Bombay\
    Powai, Mumbai, Maharashtra 400076, India
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    Department of Mathematics\
    Indian Institute of Technology Bombay\
    Powai, Mumbai, Maharashtra 400076, India
author:
- Shambhu Nath Maurya
- Koushik Saha
date:
- 
- 
title: Fluctuation of eigenvalues of symmetric circulant matrices with independent entries
---

[**Keywords :**]{} Symmetric circulant matrix, linear statistics of eigenvalues, weak convergence, central limit theorem, Trace formula, Wick’s formula.

 introduction and main results
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Let $A_n$ be an $n\times n$ matrix with real or complex entries. The linear statistics of eigenvalues $\lambda_1,\lambda_2,\ldots, \lambda_n$ of $A_n$ is a function of the form $$\label{eqn:1}
\frac{1}{n}\sum_{k=1}^{n}f(\lambda_k)$$ where $f$ is some fixed function. The function $f$ is known as the test function. One of the interesting object to study in random matrix theory is the fluctuation of linear statistics of eigenvalues of random matrices. The study of fluctuation of linear statistics of eigenvalues was 