5,301 research outputs found
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
Quasi-isometries Between Tubular Groups
We give a method of constructing maps between tubular groups inductively
according to a set of strategies. This map will be a quasi-isometry exactly
when the set of strategies is consistent. Conversely, if there exists a
quasi-isometry between tubular groups, then there is a consistent set of
strategies for them.
There is an algorithm that will in finite time either produce a consistent
set of strategies or decide that such a set does not exist. Consequently, this
algorithm decides whether or not the groups are quasi-isometric.Comment: 44 pages, 11 figures. PDFLaTeX. Improved exposition and added some
auxiliary material to make the paper more self contained, per referee's
comment
The distribution of height and diameter in random non-plane binary trees
This study is dedicated to precise distributional analyses of the height of
non-plane unlabelled binary trees ("Otter trees"), when trees of a given size
are taken with equal likelihood. The height of a rooted tree of size is
proved to admit a limiting theta distribution, both in a central and local
sense, as well as obey moderate as well as large deviations estimates. The
approximations obtained for height also yield the limiting distribution of the
diameter of unrooted trees. The proofs rely on a precise analysis, in the
complex plane and near singularities, of generating functions associated with
trees of bounded height
The continuum random tree is the scaling limit of unlabelled unrooted trees
We prove that the uniform unlabelled unrooted tree with n vertices and vertex
degrees in a fixed set converges in the Gromov-Hausdorff sense after a suitable
rescaling to the Brownian continuum random tree. This proves a conjecture by
Aldous. Moreover, we establish Benjamini-Schramm convergence of this model of
random trees
Geometry and symmetries of multi-particle systems
The quantum dynamical evolution of atomic and molecular aggregates, from
their compact to their fragmented states, is parametrized by a single
collective radial parameter. Treating all the remaining particle coordinates in
d dimensions democratically, as a set of angles orthogonal to this collective
radius or by equivalent variables, bypasses all independent-particle
approximations. The invariance of the total kinetic energy under arbitrary
d-dimensional transformations which preserve the radial parameter gives rise to
novel quantum numbers and ladder operators interconnecting its eigenstates at
each value of the radial parameter.
We develop the systematics and technology of this approach, introducing the
relevant mathematics tutorially, by analogy to the familiar theory of angular
momentum in three dimensions. The angular basis functions so obtained are
treated in a manifestly coordinate-free manner, thus serving as a flexible
generalized basis for carrying out detailed studies of wavefunction evolution
in multi-particle systems.Comment: 37 pages, 2 eps figure
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