# On generalization of Wigner semi circle

I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of permutation matrices, the entries are 0 or 1. Noise is introduced by replacing correct permutation matrix by a random permutation matrix. The eigenvalues of perturbed matrix clearly demonstrate semi circle behavior but I am stuck with analysis.

I tried to explain the eigenvalue behavior using Wigner semi circle law for a different matrix, lets say H. H is NxN and encode distance between two permutation matrices (Hamming distance). In unperturbed case all entries of H are zero. In perturbed case, H_ij can take integer value between [0,n]. Expected value and variance of H_ij is expected value of Hamming distance and variance over symmetric group Sn. Simulations with perturbed M agree greatly with theoretical support of eigenvalues using expectation and variance of H. But I do not know how to prove that analysis on H is directly related to noise analysis of M.

I would appreciate if anyone can suggest some pointers towards proving a relation between noise model of H and M.

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