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The limiting spectral distribution of large random permutation matrices
- Part of
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- Journal:
- Journal of Applied Probability / Volume 61 / Issue 4 / December 2024
- Published online by Cambridge University Press:
- 12 April 2024, pp. 1301-1318
- Print publication:
- December 2024
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- Article
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We explore the limiting spectral distribution of large-dimensional random permutation matrices, assuming the underlying population distribution possesses a general dependence structure. Let
$\textbf X = (\textbf x_1,\ldots,\textbf x_n)$ 
$\in \mathbb{C} ^{m \times n}$ be an 
$m \times n$ data matrix after self-normalization (n samples and m features), where 
$\textbf x_j = (x_{1j}^{*},\ldots, x_{mj}^{*} )^{*}$. Specifically, we generate a permutation matrix 
$\textbf X_\pi$ by permuting the entries of 
$\textbf x_j$ 
$(j=1,\ldots,n)$ and demonstrate that the empirical spectral distribution of 
$\textbf {B}_n = ({m}/{n})\textbf{U} _{n} \textbf{X} _\pi \textbf{X} _\pi^{*} \textbf{U} _{n}^{*}$ weakly converges to the generalized Marčenko–Pastur distribution with probability 1, where 
$\textbf{U} _n$ is a sequence of 
$p \times m$ non-random complex matrices. The conditions we require are 
$p/n \to c >0$ and 
$m/n \to \gamma > 0$.
