4,805 research outputs found
The Brownian continuum random tree as the unique solution to a fixed point equation
In this note, we provide a new characterization of Aldous' Brownian continuum
random tree as the unique fixed point of a certain natural operation on
continuum trees (which gives rise to a recursive distributional equation). We
also show that this fixed point is attractive.Comment: 15 pages, 3 figure
A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components
Tutte has described in the book "Connectivity in graphs" a canonical
decomposition of any graph into 3-connected components. In this article we
translate (using the language of symbolic combinatorics)
Tutte's decomposition into a general grammar expressing any family of graphs
(with some stability conditions) in terms of the 3-connected subfamily. A key
ingredient we use is an extension of the so-called dissymmetry theorem, which
yields negative signs in the grammar.
As a main application we recover in a purely combinatorial way the analytic
expression found by Gim\'enez and Noy for the series counting labelled planar
graphs (such an expression is crucial to do asymptotic enumeration and to
obtain limit laws of various parameters on random planar graphs). Besides the
grammar, an important ingredient of our method is a recent bijective
construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page
Ising spin glass models versus Ising models: an effective mapping at high temperature III. Rigorous formulation and detailed proof for general graphs
Recently, it has been shown that, when the dimension of a graph turns out to
be infinite dimensional in a broad sense, the upper critical surface and the
corresponding critical behavior of an arbitrary Ising spin glass model defined
over such a graph, can be exactly mapped on the critical surface and behavior
of a non random Ising model. A graph can be infinite dimensional in a strict
sense, like the fully connected graph, or in a broad sense, as happens on a
Bethe lattice and in many random graphs. In this paper, we firstly introduce
our definition of dimensionality which is compared to the standard definition
and readily applied to test the infinite dimensionality of a large class of
graphs which, remarkably enough, includes even graphs where the tree-like
approximation (or, in other words, the Bethe-Peierls approach), in general, may
be wrong. Then, we derive a detailed proof of the mapping for all the graphs
satisfying this condition. As a byproduct, the mapping provides immediately a
very general Nishimori law.Comment: 25 pages, 5 figures, made statements in Sec. 10 cleare
Random Forests and Networks Analysis
D. Wilson~\cite{[Wi]} in the 1990's described a simple and efficient
algorithm based on loop-erased random walks to sample uniform spanning trees
and more generally weighted trees or forests spanning a given graph. This
algorithm provides a powerful tool in analyzing structures on networks and
along this line of thinking, in recent works~\cite{AG1,AG2,ACGM1,ACGM2} we
focused on applications of spanning rooted forests on finite graphs. The
resulting main conclusions are reviewed in this paper by collecting related
theorems, algorithms, heuristics and numerical experiments. A first
foundational part on determinantal structures and efficient sampling procedures
is followed by four main applications: 1) a random-walk-based notion of
well-distributed points in a graph 2) how to describe metastable dynamics in
finite settings by means of Markov intertwining dualities 3) coarse graining
schemes for networks and associated processes 4) wavelets-like pyramidal
algorithms for graph signals.Comment: Survey pape
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