Distributional convergence for the uniform separable models

Establish the two convergences in distribution for a uniform separable permutation and a uniform separable cograph stated in Conjecture 1.1, namely convergence of their normalized longest increasing subsequence and largest independent-set sizes to c·X(1/2).

Background

The conjecture gives a precise formulation of the expected extension from Brownian-permuton and Brownian-cographon sampling to the uniform discrete models. It asserts the same nontrivial exponent α(1/2) and the same limiting random variable X(1/2), modified by a deterministic scaling factor.

The numerical simulations suggest that the factor is approximately 0.901, but the convergence itself remains unresolved.

References

Then there is a deterministic constant $c>0$ such that we have the convergences in distribution \frac{\LIS(\overline{\sigma}_n)}{n{\alpha(1/2)} \xlongrightarrow[n\rightarrow \infty]{\mathrm{d} c \cdot X(1/2) \qquad\text{and}\qquad \frac{\LIN(\overline{G}_n)}{n{\alpha(1/2)} \xlongrightarrow[n\rightarrow \infty]{\mathrm{d} c\cdot X(1/2). \end{conj}

— The longest increasing subsequence of Brownian separable permutons  (2506.19123 - Adhikari et al., 23 Jun 2025) in Conjecture 1.1, Section 1.3