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 ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the jth 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 $r\ge 0$, 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 ρ 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.