Absolute continuity and parameter distinguishability of the limiting variable

Establish that the law of the limiting random variable X(p) is absolutely continuous with respect to Lebesgue measure and that the laws of X(p) and X(p') are distinct whenever p and p' are different.

Background

The paper proves that the normalized longest increasing subsequence converges almost surely to a non-deterministic, almost surely positive and finite random variable X(p), which is a measurable function of the Brownian separable permuton. However, the distributional properties of X(p) are not determined.

The authors specifically identify absolute continuity and distinguishability across parameters as properties supported by numerical simulations but left unproved.

References

We believe that the law of $X(p)$ is absolutely continuous with respect to the Lebesgue measure and that for all $p \neq p'$, the laws of $X(p)$ and $X(p')$ are distinct -- although we do not prove this here.

— The longest increasing subsequence of Brownian separable permutons  (2506.19123 - Adhikari et al., 23 Jun 2025) in Section 1.1, immediately after the discussion of the exponent equation