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We investigate the maximal size of an increasing subset among points randomlysampled from certain probability densities. Kerov and Vershik's celebratedresult states that the largest increasing subset among $N$ uniformly randompoints on $[0,1]^2$ has size asymptotically $2\sqrt{N}$. More generally, theorder $\Theta(\sqrt{N})$ still holds if the sampling density is continuous. Inthis paper we exhibit two sufficient conditions on the density to obtain agrowth rate equivalent to any given power of $N$ greater than $\sqrt{N}$, up tologarithmic factors. Our proofs use methods of slicing the unit square intoappropriate grids, and investigating sampled points appearing in each box.

Locally uniform random permutations with large increasing subsequences