Spectral rigidity for addition of random matrices at the regular edge
| Autor: | László Erdős, Kevin Schnelli, Zhigang Bao |
|---|---|
| Rok vydání: | 2017 |
| Předmět: |
010102 general mathematics
Mathematical analysis Probability (math.PR) Haar FOS: Physical sciences Unitary matrix Mathematical Physics (math-ph) 01 natural sciences Unitary state Hermitian matrix Rate of convergence 0103 physical sciences FOS: Mathematics 010307 mathematical physics Orthogonal matrix 0101 mathematics Random matrix Analysis Eigenvalues and eigenvectors Mathematics - Probability Mathematical Physics Mathematics |
| DOI: | 10.48550/arxiv.1708.01597 |
| Popis: | We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4] , [5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix. |
| Databáze: | OpenAIRE |
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