For a random symmetric matrix with Bernoulli or subgaussian entries, the paper proves a product-form joint lower-tail estimate for the least singular value at two separated bulk locations, and shows the bulk singular values are distinct with probability 1 minus exponentially small error.
Extreme gaps between eigenvalues of Wig ner matrices
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Repeated singular values of a random symmetric matrix and decoupled singular value estimates
For a random symmetric matrix with Bernoulli or subgaussian entries, the paper proves a product-form joint lower-tail estimate for the least singular value at two separated bulk locations, and shows the bulk singular values are distinct with probability 1 minus exponentially small error.