251 research outputs found
Invariant Gaussian processes and independent sets on regular graphs of large Girth
We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that |λ|≤2d-1. We then use these approximations for λ=-2d-1 to produce factor of i.i.d. independent sets on regular trees. © 2014 Wiley Periodicals, Inc
Independence ratio and random eigenvectors in transitive graphs
A theorem of Hoffman gives an upper bound on the independence ratio of
regular graphs in terms of the minimum of the spectrum of the
adjacency matrix. To complement this result we use random eigenvectors to gain
lower bounds in the vertex-transitive case. For example, we prove that the
independence ratio of a -regular transitive graph is at least
The same bound holds for infinite transitive graphs: we
construct factor of i.i.d. independent sets for which the probability that any
given vertex is in the set is at least . We also show that the set of
the distributions of factor of i.i.d. processes is not closed w.r.t. the weak
topology provided that the spectrum of the graph is uncountable.Comment: Published at http://dx.doi.org/10.1214/14-AOP952 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Factors of IID on Trees
Classical ergodic theory for integer-group actions uses entropy as a complete
invariant for isomorphism of IID (independent, identically distributed)
processes (a.k.a. product measures). This theory holds for amenable groups as
well. Despite recent spectacular progress of Bowen, the situation for
non-amenable groups, including free groups, is still largely mysterious. We
present some illustrative results and open questions on free groups, which are
particularly interesting in combinatorics, statistical physics, and
probability. Our results include bounds on minimum and maximum bisection for
random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur
Independent sets and cuts in large-girth regular graphs
We present a local algorithm producing an independent set of expected size
on large-girth 3-regular graphs and on large-girth
4-regular graphs. We also construct a cut (or bisection or bipartite subgraph)
with edges on large-girth 3-regular graphs. These decrease the gaps
between the best known upper and lower bounds from to , from
to and from to , respectively. We are using
local algorithms, therefore, the method also provides upper bounds for the
fractional coloring numbers of and and fractional edge coloring number . Our algorithms are applications of the technique introduced by Hoppen
and Wormald
Correlation bound for distant parts of factor of IID processes
We study factor of i.i.d. processes on the -regular tree for .
We show that if such a process is restricted to two distant connected subgraphs
of the tree, then the two parts are basically uncorrelated. More precisely, any
functions of the two parts have correlation at most ,
where denotes the distance of the subgraphs. This result can be considered
as a quantitative version of the fact that factor of i.i.d. processes have
trivial 1-ended tails.Comment: 18 pages, 5 figure
Spectral measures of factor of i.i.d. processes on vertex-transitive graphs
We prove that a measure on is the spectral measure of a factor of
i.i.d. process on a vertex-transitive infinite graph if and only if it is
absolutely continuous with respect to the spectral measure of the graph.
Moreover, we show that the set of spectral measures of factor of i.i.d.
processes and that of -limits of factor of i.i.d. processes are the
same.Comment: 26 pages; proof of Proposition 9 shortene
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