708 research outputs found
Divergence of the correlation length for critical planar FK percolation with via parafermionic observables
Parafermionic observables were introduced by Smirnov for planar FK
percolation in order to study the critical phase . This
article gathers several known properties of these observables. Some of these
properties are used to prove the divergence of the correlation length when
approaching the critical point for FK percolation when . A crucial
step is to consider FK percolation on the universal cover of the punctured
plane. We also mention several conjectures on FK percolation with arbitrary
cluster-weight .Comment: 26 page
Limit of the Wulff Crystal when approaching criticality for site percolation on the triangular lattice
The understanding of site percolation on the triangular lattice progressed
greatly in the last decade. Smirnov proved conformal invariance of critical
percolation, thus paving the way for the construction of its scaling limit.
Recently, the scaling limit of near-critical percolation was also constructed
by Garban, Pete and Schramm. The aim of this very modest contribution is to
explain how these results imply the convergence, as p tends to p_c, of the
Wulff crystal to a Euclidean disk. The main ingredient of the proof is the
rotational invariance of the scaling limit of near-critical percolation proved
by these three mathematicians.Comment: 16 pages, 1 figur
The self-dual point of the two-dimensional random-cluster model is critical for
We prove a long-standing conjecture on random-cluster models, namely that the
critical point for such models with parameter on the square lattice is
equal to the self-dual point . This gives a
proof that the critical temperature of the -state Potts model is equal to
for all . We further prove that the transition is
sharp, meaning that there is exponential decay of correlations in the
sub-critical phase. The techniques of this paper are rigorous and valid for all
, in contrast to earlier methods valid only for certain given . The
proof extends to the triangular and the hexagonal lattices as well.Comment: 27 pages, 10 figure
Smirnov's fermionic observable away from criticality
In a recent and celebrated article, Smirnov [Ann. of Math. (2) 172 (2010)
1435-1467] defines an observable for the self-dual random-cluster model with
cluster weight q = 2 on the square lattice , and uses it to
obtain conformal invariance in the scaling limit. We study this observable away
from the self-dual point. From this, we obtain a new derivation of the fact
that the self-dual and critical points coincide, which implies that the
critical inverse temperature of the Ising model equals .
Moreover, we relate the correlation length of the model to the large deviation
behavior of a certain massive random walk (thus confirming an observation by
Messikh [The surface tension near criticality of the 2d-Ising model (2006)
Preprint]), which allows us to compute it explicitly.Comment: Published in at http://dx.doi.org/10.1214/11-AOP689 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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