Exact law and further properties of X(p)

Determine the exact distribution of X(p), or establish further properties including whether the laws of X(p) and X(p') are distinct for p≠p', the expectation of X(p), and the tail behavior of X(p).

Background

The scaling-limit theorem identifies X(p) only through its existence and qualitative properties. No explicit distributional formula is obtained.

The authors list several concrete unresolved questions: parameter identifiability from the law, computation of the expectation, and quantitative tail asymptotics.

References

In this work, we do not determine the exact law of the limiting random variable $X(p)$ appearing in \cref{thm:main_permutations}. It would be interesting to identify the distribution of $X(p)$ or at least establish further properties. For instance, three natural questions are whether the laws of $X(p)$ and $X(p')$ are distinct for all $p \neq p'$, what the expectation of $X(p)$ is, or what the tail behavior of $X(p)$ is.

— The longest increasing subsequence of Brownian separable permutons  (2506.19123 - Adhikari et al., 23 Jun 2025) in Section 1.3, paragraph “The limiting random variable X(p)”