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We give a proof of the Universality Conjecture for orthogonal (β=1) and symplectic (β=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5, 1.7). They concern the appropriately rescaled kernels Kn, β, correlation and cluster functions,...
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