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My questions come from the pdf https://www.math.univ-toulouse.fr/~bordenave/coursSRG.pdf that I am reading. More specifically, on page 6, the author computes the spectral measure for finite graphs, and in particular for $C_n$ a cycle of length $n$.


If $G = (V, E)$ is a finite graph, $|V| = n$, the author defines the spectral measure of $G$ as

$$\mu_G = \frac{1}{n} \sum_{i = 1}^{n} \delta_{\lambda_i}$$

where $(\lambda_1, \ldots, \lambda_n)$ are the eigenvalues (counting multiplicity) of $A$, the adjacency operator.


Let $C_n$ be a cycle of length $n$. The adjacency operator can be written as $A = B + B^\ast$ where $B$ is the permutation matrix of a cycle of length $n$. Since $BB^\ast = B^\ast B = I$, the eigenvalues of $B$ are the roots of unity and the eigenvalues of $A$ are $\lambda_k = 2 \cos(2\pi k /n)$, $1 \le k \le n$. We get

$$\mu_{C_n} = \frac{1}{n} \sum_{i = 1}^{n} \delta_{2 \cos(2 \pi k / n)}$$

As $n$ goes to infinity, $\mu_{C_n}$ converges weakly to a arcsine distribution $\nu$ with density on $[-2,2]$ given by

$$d \nu(x) = \frac{1}{\pi \sqrt{4 - x^2}} {\large\chi}_{|x|\leq 2} dx,$$

($\nu$ is the law of $2 \cos(\pi U)$ with $U$ uniform on [0,1]).


Question

I am not able to prove the weak convergence asserted by the author. Is someone able to prove it/elaborate on why this is true?

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Let $X_n$ be the discrete random variable that attains the value $k/n$ with probability $1/n$, for $k=1,\dotsc,n$. The sequence $(X_n)_n$ converges in distribution to $U\sim\mathrm{Uniform}(0,1)$.

Consider the continuous function $g(x)=2\cos(2\pi x)$. By the properties of convergence in distribution, namely the continuous mapping theorem, the variables $g(X_n)$ converge in distribution to $g(U)$.

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