1,761 research outputs found
A glimpse of the conformal structure of random planar maps
We present a way to study the conformal structure of random planar maps. The
main idea is to explore the map along an SLE (Schramm--Loewner evolution)
process of parameter and to combine the locality property of the
SLE_{6} together with the spatial Markov property of the underlying lattice in
order to get a non-trivial geometric information. We follow this path in the
case of the conformal structure of random triangulations with a boundary. Under
a reasonable assumption called (*) that we have unfortunately not been able to
verify, we prove that the limit of uniformized random planar triangulations has
a fractal boundary measure of Hausdorff dimension almost surely.
This agrees with the physics KPZ predictions and represents a first step
towards a rigorous understanding of the links between random planar maps and
the Gaussian free field (GFF).Comment: To appear in Commun. Math. Phy
Introduction to linear logic and ludics, part II
This paper is the second part of an introduction to linear logic and ludics,
both due to Girard. It is devoted to proof nets, in the limited, yet central,
framework of multiplicative linear logic and to ludics, which has been recently
developped in an aim of further unveiling the fundamental interactive nature of
computation and logic. We hope to offer a few computer science insights into
this new theory
Random laminations and multitype branching processes
We consider multitype branching processes arising in the study of random
laminations of the disk. We classify these processes according to their
subcritical or supercritical behavior and provide Kolmogorov-type estimates in
the critical case corresponding to the random recursive lamination process of
[1]. The proofs use the infinite dimensional Perron-Frobenius theory and
quasi-stationary distributions
Random non-crossing plane configurations: A conditioned Galton-Watson tree approach
We study various models of random non-crossing configurations consisting of
diagonals of convex polygons, and focus in particular on uniform dissections
and non-crossing trees. For both these models, we prove convergence in
distribution towards Aldous' Brownian triangulation of the disk. In the case of
dissections, we also refine the study of the maximal vertex degree and validate
a conjecture of Bernasconi, Panagiotou and Steger. Our main tool is the use of
an underlying Galton-Watson tree structure.Comment: 24 pages, 9 figure
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