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arXiv · 2306.07911

On the edge-density of the Brownian co-graphon and common ancestors of pairs in the CRT

Guillaume Chapuy
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Abstract

Bassino et al. (arXiv:1907.08517) have shown that uniform random co-graphs (graphs without induced $P_4$) of size $n$ converge to a certain non-deterministic graphon. The edge-density of this graphon is a random variable $\Lambda \in [0,1]$ whose first moments have been computed by these authors. The first purpose of this note is to observe that, in fact, these moments can be computed by a simple recurrence relation. The problem leads us to the following question of independent interest: given $k$ i.i.d. uniform pairs of points $(X_1,Y_1),$ $\dots$, $(X_k,Y_k)$ in the Brownian CRT, what is the size $S_k$ of the set $\{X_i\wedge Y_i, i=1\dots k\}$ formed by their pairwise last common ancestors? We show that ${S_k}\sim c{\sqrt{k}\log k}$ in probability, with $c=(\sqrt{2\pi})^{-1}$. The method to establish the recurrence relation is reminiscent of Janson's computation of moments of the Wiener index of (large) random trees. The logarithm factor in the convergence result comes from the estimation of Riemann sums in which summands are weighted by the integer divisor function -- such sums naturally occur in the problem. We have not been able to analyse the asymptotics of moments of $\Lambda$ directly from the recurrence relation, and in fact our study of $S_k$ is independent from it. Several things remain to be done, in particular we only scratch the question of large deviations of $S_k$, and the precise asymptotics of moments of $\Lambda$ is left open.

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Guillaume Chapuy. 2023-06-13. On the edge-density of the Brownian co-graphon and common ancestors of pairs in the CRT. https://arxiv.org/abs/2306.07911

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