Continuity of Limit Surfaces of Locally Uniform Random Permutations

A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\rho$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th point from below. As $n$ tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences gives rise to a surface. In a recent paper by the author it was shown that, for any $r\ge0$, under the correct scaling as $n$ tends to infinity, the surface of the largest union of $\lfloor r\sqrt{n}\rfloor$ decreasing subsequences approaches a limit in the sense that it will come close to a maximizer of a specific variational integral (and, under reasonable assumptions, that the maximizer is essentially unique). In the present paper we show that there exists a continuous maximizer, provided that $\rho$ has bounded density and support. The key ingredient in the proof is a new theorem about real functions of two variables that are increasing in both variables: We show that, for any constant $C$, any such function can be made continuous without increasing the diameter of its image or decreasing anywhere the product of its partial derivatives clipped by $C$, that is the minimum of the product and $C$.