5,809 research outputs found
Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models
We study the relation between the minimal spanning tree (MST) on many random
points and the "near-minimal" tree which is optimal subject to the constraint
that a proportion of its edges must be different from those of the
MST. Heuristics suggest that, regardless of details of the probability model,
the ratio of lengths should scale as . We prove this
scaling result in the model of the lattice with random edge-lengths and in the
Euclidean model.Comment: 24 pages, 3 figure
On the Number of Incipient Spanning Clusters
In critical percolation models, in a large cube there will typically be more
than one cluster of comparable diameter. In 2D, the probability of
spanning clusters is of the order . In dimensions d>6, when
the spanning clusters proliferate: for the spanning
probability tends to one, and there typically are spanning
clusters of size comparable to |\C_{max}| \approx L^4. The rigorous results
confirm a generally accepted picture for d>6, but also correct some
misconceptions concerning the uniqueness of the dominant cluster. We
distinguish between two related concepts: the Incipient Infinite Cluster, which
is unique partly due to its construction, and the Incipient Spanning Clusters,
which are not. The scaling limits of the ISC show interesting differences
between low (d=2) and high dimensions. In the latter case (d>6 ?) we find
indication that the double limit: infinite volume and zero lattice spacing,
when properly defined would exhibit both percolation at the critical state and
infinitely many infinite clusters.Comment: Latex(2e), 42 p, 5 figures; to appear in Nucl. Phys. B [FS
A simple renormalization flow for FK-percolation models
We present a setup that enables to define in a concrete way a renormalization
flow for the FK-percolation models from statistical physics (that are closely
related to Ising and Potts models). In this setting that is applicable in any
dimension of space, one can interpret perturbations of the critical
(conjectural) scaling limits in terms of stationary distributions for rather
simple Markov processes on spaces of abstract discrete weighted graphs.Comment: 12 pages, to appear in the Jean-Michel Bismut 65th anniversary volum
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