371 research outputs found
The peeling process of infinite Boltzmann planar maps
We start by studying a peeling process on finite random planar maps with
faces of arbitrary degrees determined by a general weight sequence, which
satisfies an admissibility criterion. The corresponding perimeter process is
identified as a biased random walk, in terms of which the admissibility
criterion has a very simple interpretation. The finite random planar maps under
consideration were recently proved to possess a well-defined local limit known
as the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien
and Le Gall, we show that the peeling process on the IBPM can be obtained from
the peeling process of finite random maps by conditioning the perimeter process
to stay positive. The simplicity of the resulting description of the peeling
process allows us to obtain the scaling limit of the associated perimeter and
volume process for arbitrary regular critical weight sequences.Comment: 29 pages, 5 figures, several improvement
Martingales in self-similar growth-fragmentations and their connections with random planar maps
The purpose of the present work is twofold. First, we develop the theory of
general self-similar growth-fragmentation processes by focusing on martingales
which appear naturally in this setting and by recasting classical results for
branching random walks in this framework. In particular, we establish
many-to-one formulas for growth-fragmentations and define the notion of
intrinsic area of a growth-fragmentation. Second, we identify a distinguished
family of growth-fragmentations closely related to stable L\'evy processes,
which are then shown to arise as the scaling limit of the perimeter process in
Markovian explorations of certain random planar maps with large degrees (which
are, roughly speaking, the dual maps of the stable maps of Le Gall & Miermont.
As a consequence of this result, we are able to identify the law of the
intrinsic area of these distinguished growth-fragmentations. This generalizes a
geometric connection between large Boltzmann triangulations and a certain
growth-fragmentation process, which was established in arXiv:1507.02265 .Comment: 50 pages, 5 figures. Final version: to appear in Probab. Theory
Related Field
Local limits of uniform triangulations in high genus
We prove a conjecture of Benjamini and Curien stating that the local limits
of uniform random triangulations whose genus is proportional to the number of
faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in
arXiv:1401.3297. The proof relies on a combinatorial argument and the
Goulden--Jackson recurrence relation to obtain tightness, and probabilistic
arguments showing the uniqueness of the limit. As a consequence, we obtain
asymptotics up to subexponential factors on the number of triangulations when
both the size and the genus go to infinity.
As a part of our proof, we also obtain the following result of independent
interest: if a random triangulation of the plane is weakly Markovian in the
sense that the probability to observe a finite triangulation around the
root only depends on the perimeter and volume of , then is a mixture of
PSHT.Comment: 36 pages, 10 figure
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