293,763 research outputs found
The spectral dimension of generic trees
We define generic ensembles of infinite trees. These are limits as
of ensembles of finite trees of fixed size , defined in terms
of a set of branching weights. Among these ensembles are those supported on
trees with vertices of a uniformly bounded order. The associated probability
measures are supported on trees with a single spine and Hausdorff dimension
. Our main result is that their spectral dimension is , and
that the critical exponent of the mass, defined as the exponential decay rate
of the two-point function along the spine, is 1/3
Invariant Measures, Hausdorff Dimension and Dimension Drop of some Harmonic Measures on Galton-Watson Trees
We consider infinite Galton-Watson trees without leaves together with
i.i.d.~random variables called marks on each of their vertices. We define a
class of flow rules on marked Galton-Watson trees for which we are able, under
some algebraic assumptions, to build explicit invariant measures. We apply this
result, together with the ergodic theory on Galton-Watson trees developed in
\cite{LPP95}, to the computation of Hausdorff dimensions of harmonic measures
in two cases. The first one is the harmonic measure of the (transient)
-biased random walk on Galton-Watson trees, for which the invariant
measure and the dimension were not explicitly known. The second case is a model
of random walk on a Galton-Watson trees with random lengths for which we
compute the dimensions of the harmonic measure and show dimension drop
phenomenon for the natural metric on the boundary and another metric that
depends on the random lengths.Comment: 37 pages, 5 figure
Perturbative Quantum Field Theory on Random Trees
In this paper we start a systematic study of quantum field theory on random
trees. Using precise probability estimates on their Galton-Watson branches and
a multiscale analysis, we establish the general power counting of averaged
Feynman amplitudes and check that they behave indeed as living on an effective
space of dimension 4/3, the spectral dimension of random trees. In the `just
renormalizable' case we prove convergence of the averaged amplitude of any
completely convergent graph, and establish the basic localization and
subtraction estimates required for perturbative renormalization. Possible
consequences for an SYK-like model on random trees are briefly discussed.Comment: 44 page
On the partition dimension of trees
Given an ordered partition of the vertex set
of a connected graph , the \emph{partition representation} of a vertex
with respect to the partition is the vector
, where represents the
distance between the vertex and the set . A partition of is
a \emph{resolving partition} of if different vertices of have different
partition representations, i.e., for every pair of vertices ,
. The \emph{partition dimension} of is the minimum
number of sets in any resolving partition of . In this paper we obtain
several tight bounds on the partition dimension of trees
Spanning trees on the Sierpinski gasket
We obtain the numbers of spanning trees on the Sierpinski gasket
with dimension equal to two, three and four. The general expression for the
number of spanning trees on with arbitrary is conjectured. The
numbers of spanning trees on the generalized Sierpinski gasket
with and are also obtained.Comment: 20 pages, 8 figures, 1 tabl
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