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Phase transitions in Phylogeny
We apply the theory of markov random fields on trees to derive a phase
transition in the number of samples needed in order to reconstruct phylogenies.
We consider the Cavender-Farris-Neyman model of evolution on trees, where all
the inner nodes have degree at least 3, and the net transition on each edge is
bounded by e. Motivated by a conjecture by M. Steel, we show that if 2 (1 - 2
e) (1 - 2e) > 1, then for balanced trees, the topology of the underlying tree,
having n leaves, can be reconstructed from O(log n) samples (characters) at the
leaves. On the other hand, we show that if 2 (1 - 2 e) (1 - 2 e) < 1, then
there exist topologies which require at least poly(n) samples for
reconstruction.
Our results are the first rigorous results to establish the role of phase
transitions for markov random fields on trees as studied in probability,
statistical physics and information theory to the study of phylogenies in
mathematical biology.Comment: To appear in Transactions of the AM
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The role of local space charge concentrations in producing branched tree structures
Electrical trees are branched damage structures produced in polymeric insulation subject to high divergent fields. The density of branching ranges from a sparse form like a tree in winter to a dense compact form like a bush. This variation in form is significant as the bush structure occurs at higher voltages but grows slower. We present here a deterministic model for the formation of electrical trees based on damage produced by charges injected into the polymer from discharges taking place within the gas-filled tubules of the tree. A number of processes within the mechanism cause the space charge fields to fluctuate chaotically, and this is held to be responsible for the branching that is observed. Different tree shapes are found depending on whether or not injected/extracted charges reach a kinetic energy high enough for damage only at a few tree tips or everywhere around the tree periphery
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