190 research outputs found
Ising models on locally tree-like graphs
We consider ferromagnetic Ising models on graphs that converge locally to
trees. Examples include random regular graphs with bounded degree and uniformly
random graphs with bounded average degree. We prove that the "cavity"
prediction for the limiting free energy per spin is correct for any positive
temperature and external field. Further, local marginals can be approximated by
iterating a set of mean field (cavity) equations. Both results are achieved by
proving the local convergence of the Boltzmann distribution on the original
graph to the Boltzmann distribution on the appropriate infinite random tree.Comment: Published in at http://dx.doi.org/10.1214/09-AAP627 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Central limit theorem for biased random walk on multi-type Galton-Watson trees
Let T be a rooted supercritical multi-type Galton-Watson (MGW) tree with
types coming from a finite alphabet, conditioned to non-extinction. The
lambda-biased random walk (X_t, t>=0) on T is the nearest-neighbor random walk
which, when at a vertex v with d(v) offspring, moves closer to the root with
probability lambda/[lambda+d(v)], and to each of the offspring with probability
1/[lambda+d(v)]. This walk is recurrent for lambda>=rho and transient for
0<lambda<rho, with rho the Perron-Frobenius eigenvalue for the (assumed)
irreducible matrix of expected offspring numbers. Subject to finite moments of
order p>4 for the offspring distributions, we prove the following quenched CLT
for lambda-biased random walk at the critical value lambda=rho: for almost
every T, the process |X_{floor(nt)}|/sqrt{n} converges in law as n tends to
infinity to a reflected Brownian motion rescaled by an explicit constant. This
result was proved under some stronger assumptions by Peres-Zeitouni (2008) for
single-type Galton-Watson trees. Following their approach, our proof is based
on a new explicit description of a reversing measure for the walk from the
point of view of the particle (generalizing the measure constructed in the
single-type setting by Peres-Zeitouni), and the construction of appropriate
harmonic coordinates. In carrying out this program we prove moment and
conductance estimates for MGW trees, which may be of independent interest. In
addition, we extend our construction of the reversing measure to a biased
random walk with random environment (RWRE) on MGW trees, again at a critical
value of the bias. We compare this result against a transience-recurrence
criterion for the RWRE generalizing a result of Faraud (2011) for Galton-Watson
trees.Comment: 44 pages, 1 figur
No zero-crossings for random polynomials and the heat equation
Consider random polynomial of independent mean-zero
normal coefficients , whose variance is a regularly varying function (in
) of order . We derive general criteria for continuity of
persistence exponents for centered Gaussian processes, and use these to show
that such polynomial has no roots in with probability
, and no roots in with probability
, hence for even, it has no real roots with probability
. Here, when and
otherwise is independent of the detailed regularly
varying variance function and corresponds to persistence probabilities for an
explicit stationary Gaussian process of smooth sample path. Further, making
precise the solution to the -dimensional heat
equation initiated by a Gaussian white noise , we
confirm that the probability of for all
, is , for .Comment: Published in at http://dx.doi.org/10.1214/13-AOP852 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Finite size scaling for the core of large random hypergraphs
The (two) core of a hypergraph is the maximal collection of hyperedges within
which no vertex appears only once. It is of importance in tasks such as
efficiently solving a large linear system over GF[2], or iterative decoding of
low-density parity-check codes used over the binary erasure channel. Similar
structures emerge in a variety of NP-hard combinatorial optimization and
decision problems, from vertex cover to satisfiability. For a uniformly chosen
random hypergraph of vertices and hyperedges, each consisting of
the same fixed number of vertices, the size of the core exhibits for
large a first-order phase transition, changing from for to a positive fraction of for , with
a transition window size around .
Analyzing the corresponding ``leaf removal'' algorithm, we determine the
associated finite-size scaling behavior. In particular, if is inside the
scaling window (more precisely, ), the
probability of having a core of size has a limit strictly between 0
and 1, and a leading correction of order . The correction
admits a sharp characterization in terms of the distribution of a Brownian
motion with quadratic shift, from which it inherits the scaling with . This
behavior is expected to be universal for a wide collection of combinatorial
problems.Comment: Published in at http://dx.doi.org/10.1214/07-AAP514 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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