## Long increasing subsequences in Brownian-type permutations

What is the behavior of the longest increasing subsequence of a uniformly random permutation? Its length is of order plus Tracy–Widom fluctuations of order . Its scaling limit is the directed geodesic of the directed landscape. This talk discusses how this behavior changes dramatically when one looks at universal Brownian-type permutations, i.e., permutations sampled from the Brownian separable permutons. We show that there are explicit constants such that …

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