39 research outputs found
Conformally invariant scaling limits in planar critical percolation
This is an introductory account of the emergence of conformal invariance in
the scaling limit of planar critical percolation. We give an exposition of
Smirnov's theorem (2001) on the conformal invariance of crossing probabilities
in site percolation on the triangular lattice. We also give an introductory
account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of
conformally invariant random curves discovered by Schramm (2000). The article
is organized around the aim of proving the result, due to Smirnov (2001) and to
Camia and Newman (2007), that the percolation exploration path converges in the
scaling limit to chordal SLE(6). No prior knowledge is assumed beyond some
general complex analysis and probability theory.Comment: 55 pages, 10 figure
Strong path convergence from Loewner driving function convergence
We show that, under mild assumptions on the limiting curve, a sequence of
simple chordal planar curves converges uniformly whenever certain Loewner
driving functions converge. We extend this result to random curves. The random
version applies in particular to random lattice paths that have chordal
as a scaling limit, with
(nonspace-filling). Existing convergence proofs often
begin by showing that the Loewner driving functions of these paths (viewed from
) converge to Brownian motion. Unfortunately, this is not sufficient,
and additional arguments are required to complete the proofs. We show that
driving function convergence is sufficient if it can be established for both
parametrization directions and a generic observation point.Comment: Published in at http://dx.doi.org/10.1214/10-AOP627 the Annals of
Probability (http://www.imstat.org/aop/) 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
Factor models on locally tree-like graphs
We consider homogeneous factor models on uniformly sparse graph sequences
converging locally to a (unimodular) random tree , and study the existence
of the free energy density , the limit of the log-partition function
divided by the number of vertices as tends to infinity. We provide a
new interpolation scheme and use it to prove existence of, and to explicitly
compute, the quantity subject to uniqueness of a relevant Gibbs measure
for the factor model on . By way of example we compute for the
independent set (or hard-core) model at low fugacity, for the ferromagnetic
Ising model at all parameter values, and for the ferromagnetic Potts model with
both weak enough and strong enough interactions. Even beyond uniqueness regimes
our interpolation provides useful explicit bounds on . In the regimes in
which we establish existence of the limit, we show that it coincides with the
Bethe free energy functional evaluated at a suitable fixed point of the belief
propagation (Bethe) recursions on . In the special case that has a
Galton-Watson law, this formula coincides with the nonrigorous "Bethe
prediction" obtained by statistical physicists using the "replica" or "cavity"
methods. Thus our work is a rigorous generalization of these heuristic
calculations to the broader class of sparse graph sequences converging locally
to trees. We also provide a variational characterization for the Bethe
prediction in this general setting, which is of independent interest.Comment: Published in at http://dx.doi.org/10.1214/12-AOP828 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The number of solutions for random regular NAE-SAT
Recent work has made substantial progress in understanding the transitions of
random constraint satisfaction problems. In particular, for several of these
models, the exact satisfiability threshold has been rigorously determined,
confirming predictions of statistical physics. Here we revisit one of these
models, random regular k-NAE-SAT: knowing the satisfiability threshold, it is
natural to study, in the satisfiable regime, the number of solutions in a
typical instance. We prove here that these solutions have a well-defined free
energy (limiting exponential growth rate), with explicit value matching the
one-step replica symmetry breaking prediction. The proof develops new
techniques for analyzing a certain "survey propagation model" associated to
this problem. We believe that these methods may be applicable in a wide class
of related problems
Satisfiability threshold for random regular NAE-SAT
We consider the random regular -NAE-SAT problem with variables each
appearing in exactly clauses. For all exceeding an absolute constant
, we establish explicitly the satisfiability threshold . We
prove that for the problem is satisfiable with high probability while
for the problem is unsatisfiable with high probability. If the
threshold lands exactly on an integer, we show that the problem is
satisfiable with probability bounded away from both zero and one. This is the
first result to locate the exact satisfiability threshold in a random
constraint satisfaction problem exhibiting the condensation phenomenon
identified by Krzakala et al. (2007). Our proof verifies the one-step replica
symmetry breaking formalism for this model. We expect our methods to be
applicable to a broad range of random constraint satisfaction problems and
combinatorial problems on random graphs