183 research outputs found
Extremal Values of Ratios: Distance Problems vs. Subtree Problems in Trees
The authors discovered a dual behaviour of two tree indices, the Wiener index and the number of subtrees, for a number of extremal problems [Discrete Appl. Math. 155 (3) 2006, 374-385; Adv. Appl. Math. 34 (2005), 138-155]. Barefoot, Entringer and Székely [Discrete Appl. Math. 80 (1997), 37-56] determined extremal values of σT(w)/σT(u), σT(w)/σT(v), σ(T)/σT(v), and σ(T)/σT(w), where T is a tree on n vertices, v is in the centroid of the tree T, and u,w are leaves in T.
In this paper we test how far the negative correlation between distances and subtrees go if we look for the extremal values of FT(w)/FT(u), FT(w)/FT(v), F(T)/FT(v), and F(T)/FT(w), where T is a tree on n vertices, v is in the subtree core of the tree T, and u,w are leaves in T-the complete analogue of [Discrete Appl. Math. 80 (1997), 37-56], changing distances to the number of subtrees. We include a number of open problems, shifting the interest towards the number of subtrees in graphs
Random subtrees of complete graphs
We study the asymptotic behavior of four statistics associated with subtrees
of complete graphs: the uniform probability that a random subtree is a
spanning tree of , the weighted probability (where the probability a
subtree is chosen is proportional to the number of edges in the subtree) that a
random subtree spans and the two expectations associated with these two
probabilities. We find and both approach ,
while both expectations approach the size of a spanning tree, i.e., a random
subtree of has approximately edges
Trickle-down processes and their boundaries
It is possible to represent each of a number of Markov chains as an evolving
sequence of connected subsets of a directed acyclic graph that grow in the
following way: initially, all vertices of the graph are unoccupied, particles
are fed in one-by-one at a distinguished source vertex, successive particles
proceed along directed edges according to an appropriate stochastic mechanism,
and each particle comes to rest once it encounters an unoccupied vertex.
Examples include the binary and digital search tree processes, the random
recursive tree process and generalizations of it arising from nested instances
of Pitman's two-parameter Chinese restaurant process, tree-growth models
associated with Mallows' phi model of random permutations and with
Schuetzenberger's non-commutative q-binomial theorem, and a construction due to
Luczak and Winkler that grows uniform random binary trees in a Markovian
manner. We introduce a framework that encompasses such Markov chains, and we
characterize their asymptotic behavior by analyzing in detail their Doob-Martin
compactifications, Poisson boundaries and tail sigma-fields.Comment: 62 pages, 8 figures, revised to address referee's comment
Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree
Consider a low temperature stochastic Ising model in the phase coexistence
regime with Markov semigroup . A fundamental and still largely open
problem is the understanding of the long time behavior of \d_\h P_t when the
initial configuration \h is sampled from a highly disordered state
(e.g. a product Bernoulli measure or a high temperature Gibbs measure).
Exploiting recent progresses in the analysis of the mixing time of Monte Carlo
Markov chains for discrete spin models on a regular -ary tree \Tree^b, we
tackle the above problem for the Ising and hard core gas (independent sets)
models on \Tree^b. If is a biased product Bernoulli law then, under
various assumptions on the bias and on the thermodynamic parameters, we prove
-almost sure weak convergence of \d_\h P_t to an extremal Gibbs measure
(pure phase) and show that the limit is approached at least as fast as a
stretched exponential of the time . In the context of randomized algorithms
and if one considers the Glauber dynamics on a large, finite tree, our results
prove fast local relaxation to equilibrium on time scales much smaller than the
true mixing time, provided that the starting point of the chain is not taken as
the worst one but it is rather sampled from a suitable distribution.Comment: 35 page
Diameter of the thick part of moduli space and simultaneous Whitehead moves
Let S be a surface of genus g with p punctures with negative Euler
characteristic. We study the diameter of the -thick part of moduli
space of S equipped with the Teichm\"uller or Thurston's Lipschitz metric. We
show that the asymptotic behaviors in both metrics are of order . The same result also holds for the -thick part
of the moduli space of metric graphs of rank n equipped with the Lipschitz
metric. The proof involves a sorting algorithm that sorts an arbitrary labeled
tree with n labels with simultaneous Whitehead moves, where the number of steps
is of order log(n).Comment: 34 pages, 10 figures. Referee's comments incorporated. An appendix
section is added to discuss the growth rate of the diameter of the space of
graphs equipped with the metric of (non-simultaneous) Whitehead moves. The
final version will appear in Duke Mathematical Journa
- …