# Circular law for the sum of random permutation matrices

## TL;DR

This paper proves that the eigenvalue distribution of the sum of many independent random permutation matrices, scaled appropriately, converges to the circular law as the matrix size grows large.

## Contribution

It establishes the circular law for the sum of independent random permutation matrices under specific growth conditions for the number of matrices.

## Key findings

- Empirical spectral distribution converges to the circular law.
- Convergence holds with high probability as matrix size increases.
- Requires the number of matrices to grow at least as fast as a polylogarithmic function of n.

## Abstract

Let $P_n^1,\dots, P_n^d$ be $n\times n$ permutation matrices drawn independently and uniformly at random, and set $S_n^d:=\sum_{\ell=1}^d P_n^\ell$. We show that if $\log^{12}n/(\log \log n)^{4} \le d=O(n)$, then the empirical spectral distribution of $S_n^d/\sqrt{d}$ converges weakly to the circular law in probability as $n \to \infty$.