76 research outputs found
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
Scaling Limit of the Prudent Walk
We describe the scaling limit of the nearest neighbour prudent walk on the
square lattice, which performs steps uniformly in directions in which it does
not see sites already visited. We show that the scaling limit is given by the
process Z(u) = s_1 theta^+(3u/7) e_1 + s_2 theta^-(3u/7) e_2, where e_1, e_2 is
the canonical basis, theta^+(t), resp. theta^-(t), is the time spent by a
one-dimensional Brownian motion above, resp. below, 0 up to time t, and s_1,
s_2 are two random signs. In particular, the asymptotic speed of the walk is
well-defined in the L^1-norm and equals 3/7.Comment: Better exposition, stronger claim, simpler description of the
limiting process; final version, to appear in Electr. Commun. Probab
Bridge Decomposition of Restriction Measures
Motivated by Kesten's bridge decomposition for two-dimensional self-avoiding
walks in the upper half plane, we show that the conjectured scaling limit of
the half-plane SAW, the SLE(8/3) process, also has an appropriately defined
bridge decomposition. This continuum decomposition turns out to entirely be a
consequence of the restriction property of SLE(8/3), and as a result can be
generalized to the wider class of restriction measures. Specifically we show
that the restriction hulls with index less than one can be decomposed into a
Poisson Point Process of irreducible bridges in a way that is similar to Ito's
excursion decomposition of a Brownian motion according to its zeros.Comment: 24 pages, 2 figures. Final version incorporates minor revisions
suggested by the referee, to appear in Jour. Stat. Phy
Loop-Erasure of Plane Brownian Motion
We use the coupling technique to prove that there exists a loop-erasure of a
plane Brownian motion stopped on exiting a simply connected domain, and the
loop-erased curve is the reversal of a radial SLE curve.Comment: 10 page
The Length of an SLE - Monte Carlo Studies
The scaling limits of a variety of critical two-dimensional lattice models
are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the
parameter kappa. These lattice models have a natural parametrization of their
random curves given by the length of the curve. This parametrization (with
suitable scaling) should provide a natural parametrization for the curves in
the scaling limit. We conjecture that this parametrization is also given by a
type of fractal variation along the curve, and present Monte Carlo simulations
to support this conjecture. Then we show by simulations that if this fractal
variation is used to parametrize the SLE, then the parametrized curves have the
same distribution as the curves in the scaling limit of the lattice models with
their natural parametrization.Comment: 18 pages, 10 figures. Version 2 replaced the use of "nu" for the
"growth exponent" by 1/d_H, where d_H is the Hausdorff dimension. Various
minor errors were also correcte
Abelian Sandpile Model on the Honeycomb Lattice
We check the universality properties of the two-dimensional Abelian sandpile
model by computing some of its properties on the honeycomb lattice. Exact
expressions for unit height correlation functions in presence of boundaries and
for different boundary conditions are derived. Also, we study the statistics of
the boundaries of avalanche waves by using the theory of SLE and suggest that
these curves are conformally invariant and described by SLE2.Comment: 24 pages, 5 figure
Stochastic Loewner evolution driven by Levy processes
Standard stochastic Loewner evolution (SLE) is driven by a continuous
Brownian motion, which then produces a continuous fractal trace. If jumps are
added to the driving function, the trace branches. We consider a generalized
SLE driven by a superposition of a Brownian motion and a stable Levy process.
The situation is defined by the usual SLE parameter, , as well as
which defines the shape of the stable Levy distribution. The resulting
behavior is characterized by two descriptors: , the probability that the
trace self-intersects, and , the probability that it will approach
arbitrarily close to doing so. Using Dynkin's formula, these descriptors are
shown to change qualitatively and singularly at critical values of and
. It is reasonable to call such changes ``phase transitions''. These
transitions occur as passes through four (a well-known result) and as
passes through one (a new result). Numerical simulations are then used
to explore the associated touching and near-touching events.Comment: Published version, minor typos corrected, added reference
Conformal Curves in Potts Model: Numerical Calculation
We calculated numerically the fractal dimension of the boundaries of the
Fortuin-Kasteleyn clusters of the -state Potts model for integer and
non-integer values of on the square lattice.
In addition we calculated with high accuracy the fractal dimension of the
boundary points of the same clusters on the square domain. Our calculation
confirms that this curves can be described by SLE.Comment: 11 Pages, 4 figure
The near-critical planar FK-Ising model
We study the near-critical FK-Ising model. First, a determination of the
correlation length defined via crossing probabilities is provided. Second, a
phenomenon about the near-critical behavior of FK-Ising is highlighted, which
is completely missing from the case of standard percolation: in any monotone
coupling of FK configurations (e.g., in the one introduced in
[Gri95]), as one raises near , the new edges arrive in a
self-organized way, so that the correlation length is not governed anymore by
the number of pivotal edges at criticality.Comment: 34 pages, 8 figures. This is a streamlined version; the previous one
contains more explanations and additional material on exceptional times in FK
models with general . Furthermore, the statement and proof of Theorem 1.2
have slightly change
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
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