59,971 research outputs found
Structure and Generation of Crossing-Critical Graphs
We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For c=1 there are only two such graphs without degree-2 vertices, K_5 and K_{3,3}, but for any fixed c>1 there exist infinitely many c-crossing-critical graphs. It has been previously shown that c-crossing-critical graphs have bounded path-width and contain only a bounded number of internally disjoint paths between any two vertices. We expand on these results, providing a more detailed description of the structure of crossing-critical graphs. On the way towards this description, we prove a new structural characterisation of plane graphs of bounded path-width. Then we show that every c-crossing-critical graph can be obtained from a c-crossing-critical graph of bounded size by replicating bounded-size parts that already appear in narrow "bands" or "fans" in the graph. This also gives an algorithm to generate all the c-crossing-critical graphs of at most given order n in polynomial time per each generated graph
Ising Spins on Thin Graphs
The Ising model on ``thin'' graphs (standard Feynman diagrams) displays
several interesting properties. For ferromagnetic couplings there is a mean
field phase transition at the corresponding Bethe lattice transition point. For
antiferromagnetic couplings the replica trick gives some evidence for a spin
glass phase. In this paper we investigate both the ferromagnetic and
antiferromagnetic models with the aid of simulations. We confirm the Bethe
lattice values of the critical points for the ferromagnetic model on
and graphs and examine the putative spin glass phase in the
antiferromagnetic model by looking at the overlap between replicas in a
quenched ensemble of graphs. We also compare the Ising results with those for
higher state Potts models and Ising models on ``fat'' graphs, such as those
used in 2D gravity simulations.Comment: LaTeX 13 pages + 9 postscript figures, COLO-HEP-340,
LPTHE-Orsay-94-6
Percolation model for nodal domains of chaotic wave functions
Nodal domains are regions where a function has definite sign. In recent paper
[nlin.CD/0109029] it is conjectured that the distribution of nodal domains for
quantum eigenfunctions of chaotic systems is universal. We propose a
percolation-like model for description of these nodal domains which permits to
calculate all interesting quantities analytically, agrees well with numerical
simulations, and due to the relation to percolation theory opens the way of
deeper understanding of the structure of chaotic wave functions.Comment: 4 pages, 6 figures, Late
Branching random walks and multi-type contact-processes on the percolation cluster of
In this paper we prove that, under the assumption of quasi-transitivity, if a
branching random walk on survives locally (at arbitrarily
large times there are individuals alive at the origin), then so does the same
process when restricted to the infinite percolation cluster
of a supercritical Bernoulli percolation. When no
more than individuals per site are allowed, we obtain the -type contact
process, which can be derived from the branching random walk by killing all
particles that are born at a site where already individuals are present. We
prove that local survival of the branching random walk on
also implies that for sufficiently large the associated -type contact
process survives on . This implies that the strong
critical parameters of the branching random walk on and on
coincide and that their common value is the limit of
the sequence of strong critical parameters of the associated -type contact
processes. These results are extended to a family of restrained branching
random walks, that is, branching random walks where the success of the
reproduction trials decreases with the size of the population in the target
site.Comment: Published at http://dx.doi.org/10.1214/14-AAP1040 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Phase diagram of an Ising model with competitive interactions on a Husimi tree and its disordered counterpart
We consider an Ising competitive model defined over a triangular Husimi tree
where loops, responsible for an explicit frustration, are even allowed. After a
critical analysis of the phase diagram, in which a ``gas of non interacting
dimers (or spin liquid) - ferro or antiferromagnetic ordered state'' transition
is recognized in the frustrated regions, we introduce the disorder for studying
the spin glass version of the model: the triangular +/- J model. We find out
that, for any finite value of the averaged couplings, the model exhibits always
a phase transition, even in the frustrated regions, where the transition turns
out to be a glassy transition. The analysis of the random model is done by
applying a recently proposed method which allows to derive the upper phase
boundary of a random model through a mapping with a corresponding non random
one.Comment: 19 pages, 11 figures; content change
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