10 research outputs found
Large Unicellular maps in high genus
We study the geometry of a random unicellular map which is uniformly
distributed on the set of all unicellular maps whose genus size is proportional
to the number of edges of the map. We prove that the distance between two
uniformly selected vertices of such a map is of order and the diameter
is also of order with high probability. We further prove that the map
is locally planar with high probability. The main ingredient of the proofs is
an exploration procedure which uses a bijection due to Chapuy, Feray and Fusy.Comment: 43 pages, 6 figures, revised file taking into account referee's
comment
A decorated tree approach to random permutations in substitution-closed classes
We establish a novel bijective encoding that represents permutations as
forests of decorated (or enriched) trees. This allows us to prove local
convergence of uniform random permutations from substitution-closed classes
satisfying a criticality constraint. It also enables us to reprove and
strengthen permuton limits for these classes in a new way, that uses a
semi-local version of Aldous' skeleton decomposition for size-constrained
Galton--Watson trees.Comment: New version including referee's corrections, accepted for publication
in Electronic Journal of Probabilit
Cutting down trees with a Markov chainsaw
We provide simplified proofs for the asymptotic distribution of the number of
cuts required to cut down a Galton-Watson tree with critical, finite-variance
offspring distribution, conditioned to have total progeny . Our proof is
based on a coupling which yields a precise, nonasymptotic distributional result
for the case of uniformly random rooted labeled trees (or, equivalently,
Poisson Galton-Watson trees conditioned on their size). Our approach also
provides a new, random reversible transformation between Brownian excursion and
Brownian bridge.Comment: Published in at http://dx.doi.org/10.1214/13-AAP978 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The vertical profile of embedded trees
Consider a rooted binary tree with n nodes. Assign with the root the abscissa
0, and with the left (resp. right) child of a node of abscissa i the abscissa
i-1 (resp. i+1). We prove that the number of binary trees of size n having
exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is with n_{l-1}=n_{r+1}=0. The
sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the
tree. The vertical profile of a uniform random tree of size n is known to
converge, in a certain sense and after normalization, to a random mesure called
the integrated superbrownian excursion, which motivates our interest in the
profile. We prove similar looking formulas for other families of trees whose
nodes are embedded in Z. We also refine these formulas by taking into account
the number of nodes at abscissa j whose parent lies at abscissa i, and/or the
number of vertices at abscissa i having a prescribed number of children at
abscissa j, for all i and j. Our proofs are bijective.Comment: 47 page
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