Limiting Curves for IID records simulation

Consider $$\{Z_i=(X_i,Y_i), i=1,\ldots, n\}$$ a (random) set of $$n$$ i.i.d. points on $$[0,1]^2$$, with common distribution $$P_{X,Y}$$. A subset $$\{ Z_{i} \}_{i\in I}$$ is said to be increasing if there exists a permutation $$\pi:I\to I$$ such that $$X_{\pi(i)} < X_{\pi(i+1)}$$ and $$Y_{\pi(i)} <Y_{\pi(i+1)}$$ for all $$i\in I$$. In other words, the set $$\{ Z_{i} \}_{i\in I}$$ is increasing if its points can be connected by an up/right path. We are interested in the number of points of the longest increasing subset of $$\{Z_i\}_{i=1,\ldots, n}$$, that we denote $$\ell_n$$. One motivation is that in the case where $$P_{X,Y}$$ is uniform on $$[0,1]^2$$, this is equivalent to Ulam's problem of the longest increasing subsequence of a random permutation $$\sigma$$ taken uniformly in $$\mathfrak{S}_n$$, which has turned out to have many connection with different areas of probability theory and combinatorics.

We offer here an illustration of a result by Deuschel and Zeitouni (Annals of Probability, 1995): under some mild conditions on $$P_{X,Y}$$, we have

1. $$\ell_n /\sqrt{n}$$ converges in probability to a constant (which is explicit);
2. maximal increasing subsets of $$\{Z_i\}_{i=1,\ldots, n}$$ converge to a limiting curve (also explicit).

For the sake of the simulations, we focus on the simpler case when $$X$$ and $$Y$$ are independent: in that specific case, $$\ell_n /\sqrt{n} \to 2$$ in probability (as found in Ulam's problem), and the limiting curve is parametrized as $$(F_X(t), F_Y(t))_{t\in[0,1]}$$, with $$F_X$$ (resp. $$F_Y$$) the c.d.f. of $$X$$ (resp. of $$Y$$). Let us consider here the case $$X\sim \text{Beta}(a,b)$$, $$Y\sim \text{Beta}(c,d)$$ independent (we recover the case where $$P_{X,Y}$$ is uniform on $$[0,1]^2$$ by taking $$a=b=c=d=1$$).