## Brownian excursions, critical random graphs and the multiplicative coalescent.(English)Zbl 0877.60010

Consider the critical random graph on $$n$$ vertices with probability $$n^{-1}+ tn^{-4/3}$$ for each edge, and denote by $$C_n^t(j)$$ the size of its $$j$$-th largest component. Consider also the “surplus” of this same $$j$$-th component: $$\sigma_n^t(j): =1+$$ number of edges – number of vertices. Consider on the other hand the reflecting inhomogeneous Brownian motion $$B^t(s)$$ with drift $$(t-s)$$ at time $$s$$, marked with a point process of intensity $$B^t(s)$$ at time $$s$$. Denote by $$|\gamma_j |$$ the length of the $$j$$-th largest excursion of $$B^t(s)$$ (away from 0), and by $$y(\gamma_j)$$ the number of marks that $$\gamma_j$$ contains. Then the sharp main result asserts that the vector $$(n^{-2/3} C^t_n(j))$$ converges in $$\ell^2$$ to some limit which has the law of $$(|\gamma_j |,y (\gamma_j))$$. The link between random graphs and Brownian motion arises from some deterministic graph-exploration procedure, the so-called “breadth-first walk”. The dynamics of merging of components as $$t$$ increases are abstracted to define a fine Markov process on $$\ell^2$$, shown to be Feller, the so called “multiplicative coalescent”, for which clusters of sizes $$x_i$$ and $$x_j$$ merge at rate $$x_ix_j$$.
Reviewer: J.Franchi (Paris)

### MSC:

 60C05 Combinatorial probability 60J65 Brownian motion 60J50 Boundary theory for Markov processes
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### References:

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