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Oct 6, 2023 at 18:18 comment added Will Sawin Thanks, this is very interesting, but I don't think it's quite what I want. The goal is not just to avoid the Selberg integral but to use specifically the determinantal characterization (to give clues for how to prove similar theorems about other determinantal point process). This seems to use very heavily the description via the uniform measure on $N \times N$ matrices, considering the individual matrix entries.
Oct 6, 2023 at 17:45 history answered Carlo Beenakker CC BY-SA 4.0