Given $n>0$, let $S\subset [0,1]^2$ be a set of $n$ points, chosen uniformly at random. Let $R\cup B$ be a random partition, or coloring, of $S$ in which each point of $S$ is included in $R$ uniformly at random with probability $1/2$. Corujo et al.~(JOCO 2023) studied the random variable $M(n)$ equal to the number of points of $S$ that are covered by the rectangles of a maximum matching of $S$ using pairwise-disjoint rectangles. Each rectangle is axis-aligned and covers exactly two points of $S$ of the same color. They designed a deterministic algorithm to match points of $S$, and the algorithm was modeled as a discrete stochastic process over a finite set of states. After claiming that the stochastic process is a Markov chain, they proved that almost surely $M(n)\ge 0.83\,n$ for $n$ large enough. The issue is that such a process is not actually a Markov one, as we discuss in this note. We argue this issue, and correct it by obtaining the same result but considering that the stochastic process is not Markov, but satisfies some kind of first-order homogeneity property that allows us to compute its marginal distributions.

翻译：暂无翻译