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Largest Component in Random Combinatorial Structures
Résumé disponible dans le fichier PD
Approximation by the Dickman distribution and quasi-logarithmic combinatorial structures
Quasi-logarithmic combinatorial structures are a class of decomposable
combinatorial structures which extend the logarithmic class considered by
Arratia, Barbour and Tavar\'{e} (2003). In order to obtain asymptotic
approximations to their component spectrum, it is necessary first to establish
an approximation to the sum of an associated sequence of independent random
variables in terms of the Dickman distribution. This in turn requires an
argument that refines the Mineka coupling by incorporating a blocking
construction, leading to exponentially sharper coupling rates for the sums in
question. Applications include distributional limit theorems for the size of
the largest component and for the vector of counts of the small components in a
quasi-logarithmic combinatorial structure.Comment: 22 pages; replaces earlier paper [arXiv:math/0609129] with same title
by Bruno Nietlispac
The continuum random tree is the scaling limit of unlabelled unrooted trees
We prove that the uniform unlabelled unrooted tree with n vertices and vertex
degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable
rescaling to the Brownian continuum random tree. This proves a conjecture by
Aldous. Moreover, we establish Benjamini-Schramm convergence of this model of
random trees
Random enriched trees with applications to random graphs
We establish limit theorems that describe the asymptotic local and global
geometric behaviour of random enriched trees considered up to symmetry. We
apply these general results to random unlabelled weighted rooted graphs and
uniform random unlabelled -trees that are rooted at a -clique of
distinguishable vertices. For both models we establish a Gromov--Hausdorff
scaling limit, a Benjamini--Schramm limit, and a local weak limit that
describes the asymptotic shape near the fixed root
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