49,883 research outputs found
Minimal spanning forests
Minimal spanning forests on infinite graphs are weak limits of minimal
spanning trees from finite subgraphs. These limits can be taken with free or
wired boundary conditions and are denoted FMSF (free minimal spanning forest)
and WMSF (wired minimal spanning forest), respectively. The WMSF is also the
union of the trees that arise from invasion percolation started at all
vertices. We show that on any Cayley graph where critical percolation has no
infinite clusters, all the component trees in the WMSF have one end a.s. In
this was proved by Alexander [Ann. Probab. 23 (1995) 87--104],
but a different method is needed for the nonamenable case. We also prove that
the WMSF components are ``thin'' in a different sense, namely, on any graph,
each component tree in the WMSF has a.s., where
denotes the critical probability for having an infinite
cluster in Bernoulli percolation. On the other hand, the FMSF is shown to be
``thick'': on any connected graph, the union of the FMSF and independent
Bernoulli percolation (with arbitrarily small parameter) is a.s. connected. In
conjunction with a recent result of Gaboriau, this implies that in any Cayley
graph, the expected degree of the FMSF is at least the expected degree of the
FSF (the weak limit of uniform spanning trees). We also show that the number of
infinite clusters for Bernoulli() percolation is at most the
number of components of the FMSF, where denotes the critical
probability for having a unique infinite cluster. Finally, an example is given
to show that the minimal spanning tree measure does not have negative
associations.Comment: Published at http://dx.doi.org/10.1214/009117906000000269 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Spanning trees of 3-uniform hypergraphs
Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting
spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done
in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and
related to the class of Pfaffian graphs. We prove a complexity result for
recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian
3-graphs -- one of these is given by a forbidden subgraph characterization
analogous to Little's for bipartite Pfaffian graphs, and the other consists of
a class of partial Steiner triple systems for which the property of being
3-Pfaffian can be reduced to the property of an associated graph being
Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are
not 3-Pfaffian, none of which can be reduced to any other by deletion or
contraction of triples.
We also find some necessary or sufficient conditions for the existence of a
spanning tree of a 3-graph (much more succinct than can be obtained by the
currently fastest polynomial-time algorithm of Gabow and Stallmann for finding
a spanning tree) and a superexponential lower bound on the number of spanning
trees of a Steiner triple system.Comment: 34 pages, 9 figure
Uniqueness of maximal entropy measure on essential spanning forests
An essential spanning forest of an infinite graph is a spanning forest of
in which all trees have infinitely many vertices. Let be an
increasing sequence of finite connected subgraphs of for which . Pemantle's arguments imply that the uniform measures on spanning trees
of converge weakly to an -invariant measure
on essential spanning forests of . We show that if is a
connected, amenable graph and acts
quasitransitively on , then is the unique -invariant measure
on essential spanning forests of for which the specific entropy is maximal.
This result originated with Burton and Pemantle, who gave a short but incorrect
proof in the case . Lyons discovered the error and
asked about the more general statement that we prove.Comment: Published at http://dx.doi.org/10.1214/009117905000000765 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on
a version of the triangular lattice in the complex plane have unique scaling
limits, which are invariant under rotations, scalings, and, in the case of the
MST, also under translations. However, they are not expected to be conformally
invariant. We also prove some geometric properties of the limiting MST. The
topology of convergence is the space of spanning trees introduced by Aizenman,
Burchard, Newman & Wilson (1999), and the proof relies on the existence and
conformal covariance of the scaling limit of the near-critical percolation
ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio
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