2,618 research outputs found
An Elementary Proof of Hawkes's Conjecture on Galton-Watson Trees
In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function
for the boundary of supercritical Galton-Watson trees under a certain
assumption on the tail at the infinity of the total mass of the branching
measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as
other other precise results on fractal properties of the boundary of
Galton-Watson trees. The goal of this paper is to provide an elementary proof
of Hawkes's conjecture under a less restrictive assumption than in T.
Watanabe's paper, by use of size-biased Galton-Watson trees introduced by
Lyons, Pemantle and Peres in 1995
Continuum random trees and branching processes with immigration
We study a genealogical model for continuous-state branching processes with
immigration with a (sub)critical branching mechanism. This model allows the
immigrants to be on the same line of descent. The corresponding family tree is
an ordered rooted continuum random tree with a single infinite end defined
thanks to two continuous processes denoted by
and that code the parts at resp. the left and
the right hand of the infinite line of descent of the tree. These processes are
called the left and the right height processes. We define their local time
processes via an approximation procedure and we prove that they enjoy a
Ray-Knight property. We also discuss the important special case corresponding
to the size-biased Galton-Watson tree in the continuous setting. In the last
part of the paper we give a convergence result under general assumptions for
rescaled discrete left and right contour processes of sequences of
Galton-Watson trees with immigration. We also provide a strong invariance
principle for a sequence of rescaled Galton-Watson processes with immigration
that also holds in the supercritical case.Comment: 35 page
Thermal conductivity of semiconductor superlattices: Experimental study of interface scattering
We present thermal conductivity measurements performed in three short-period
(GaAs)_9(AlAs)_5 superlattices. The samples were grown at different
temperatures, leading to different small scale roughness and broadening of the
interfaces. The cross-plane conductivity is measured with a differential 3w
method, at room temperature. The order of magnitude of the overall thermal
conductivity variation is consistent with existing theoretical models, although
the actual variation is smaller than expected
Decomposition of Levy trees along their diameter
We study the diameter of L{\'e}vy trees that are random compact metric spaces
obtained as the scaling limits of Galton-Watson trees. L{\'e}vy trees have been
introduced by Le Gall and Le Jan (1998) and they generalise Aldous' Continuum
Random Tree (1991) that corresponds to the Brownian case. We first characterize
the law of the diameter of L{\'e}vy trees and we prove that it is realized by a
unique pair of points. We prove that the law of L{\'e}vy trees conditioned to
have a fixed diameter r (0, ) is obtained by glueing at their
respective roots two independent size-biased L{\'e}vy trees conditioned to have
height r/2 and then by uniformly re-rooting the resulting tree; we also
describe by a Poisson point measure the law of the subtrees that are grafted on
the diameter. As an application of this decomposition of L{\'e}vy trees
according to their diameter, we characterize the joint law of the height and
the diameter of stable L{\'e}vy trees conditioned by their total mass; we also
provide asymptotic expansions of the law of the height and of the diameter of
such normalised stable trees, which generalises the identity due to Szekeres
(1983) in the Brownian case
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