66 research outputs found
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
Reconstruction of Random Colourings
Reconstruction problems have been studied in a number of contexts including
biology, information theory and and statistical physics. We consider the
reconstruction problem for random -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
Subgraph densities and scaling limits of random graphs with a prescribed modular decomposition
We consider large uniform labeled random graphs in different classes with
prescribed decorations in their modular decomposition. Our main result is the
estimation of the number of copies of every graph as an induced subgraph. As a
consequence, we obtain the convergence of a uniform random graph in such
classes to a Brownian limit object in the space of graphons.
Our proofs rely on combinatorial arguments, computing generating series using
the symbolic method and deriving asymptotics using singularity analysis.Comment: 32 pages, 11 figures. arXiv admin note: substantial text overlap with
arXiv:2301.1360
The normal distribution is -infinitely divisible
We prove that the classical normal distribution is infinitely divisible with
respect to the free additive convolution. We study the Voiculescu transform
first by giving a survey of its combinatorial implications and then
analytically, including a proof of free infinite divisibility. In fact we prove
that a subfamily Askey-Wimp-Kerov distributions are freely infinitely
divisible, of which the normal distribution is a special case. At the time of
this writing this is only the third example known to us of a nontrivial
distribution that is infinitely divisible with respect to both classical and
free convolution, the others being the Cauchy distribution and the free
1/2-stable distribution.Comment: AMS LaTeX, 29 pages, using tikz and 3 eps figures; new proof
including infinite divisibility of certain Askey-Wilson-Kerov distibution
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