3,777 research outputs found
SLE-type growth processes and the Yang-Lee singularity
The recently introduced SLE growth processes are based on conformal maps from
an open and simply-connected subset of the upper half-plane to the half-plane
itself. We generalize this by considering a hierarchy of stochastic evolutions
mapping open and simply-connected subsets of smaller and smaller fractions of
the upper half-plane to these fractions themselves. The evolutions are all
driven by one-dimensional Brownian motion. Ordinary SLE appears at grade one in
the hierarchy. At grade two we find a direct correspondence to conformal field
theory through the explicit construction of a level-four null vector in a
highest-weight module of the Virasoro algebra. This conformal field theory has
central charge c=-22/5 and is associated to the Yang-Lee singularity. Our
construction may thus offer a novel description of this statistical model.Comment: 12 pages, LaTeX, v2: thorough revision with corrections, v3: version
to be publishe
Random walk on the range of random walk
We study the random walk X on the range of a simple random walk on ℤ d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin
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
Conformal invariance in two-dimensional turbulence
Simplicity of fundamental physical laws manifests itself in fundamental
symmetries. While systems with an infinity of strongly interacting degrees of
freedom (in particle physics and critical phenomena) are hard to describe, they
often demonstrate symmetries, in particular scale invariance. In two dimensions
(2d) locality often promotes scale invariance to a wider class of conformal
transformations which allow for nonuniform re-scaling. Conformal invariance
allows a thorough classification of universality classes of critical phenomena
in 2d. Is there conformal invariance in 2d turbulence, a paradigmatic example
of strongly-interacting non-equilibrium system? Here, using numerical
experiment, we show that some features of 2d inverse turbulent cascade display
conformal invariance. We observe that the statistics of vorticity clusters is
remarkably close to that of critical percolation, one of the simplest
universality classes of critical phenomena. These results represent a new step
in the unification of 2d physics within the framework of conformal symmetry.Comment: 10 pages, 5 figures, 1 tabl
Limiting shapes for deterministic centrally seeded growth models
We study the rotor router model and two deterministic sandpile models. For
the rotor router model in , Levine and Peres proved that the
limiting shape of the growth cluster is a sphere. For the other two models,
only bounds in dimension 2 are known. A unified approach for these models with
a new parameter (the initial number of particles at each site), allows to
prove a number of new limiting shape results in any dimension .
For the rotor router model, the limiting shape is a sphere for all values of
. For one of the sandpile models, and (the maximal value), the
limiting shape is a cube. For both sandpile models, the limiting shape is a
sphere in the limit . Finally, we prove that the rotor router
shape contains a diamond.Comment: 18 pages, 3 figures, some errors corrected and more explanation
added, to appear in Journal of Statistical Physic
Families of Vicious Walkers
We consider a generalisation of the vicious walker problem in which N random
walkers in R^d are grouped into p families. Using field-theoretic
renormalisation group methods we calculate the asymptotic behaviour of the
probability that no pairs of walkers from different families have met up to
time t. For d>2, this is constant, but for d<2 it decays as a power t^(-alpha),
which we compute to O(epsilon^2) in an expansion in epsilon=2-d. The second
order term depends on the ratios of the diffusivities of the different
families. In two dimensions, we find a logarithmic decay (ln t)^(-alpha'), and
compute alpha' exactly.Comment: 20 pages, 5 figures. v.2: minor additions and correction
Proposal for a CFT interpretation of Watts' differential equation for percolation
G. M. T. Watts derived that in two dimensional critical percolation the
crossing probability Pi_hv satisfies a fifth order differential equation which
includes another one of third order whose independent solutions describe the
physically relevant quantities 1, Pi_h, Pi_hv.
We will show that this differential equation can be derived from a level
three null vector condition of a rational c=-24 CFT and motivate how this
solution may be fitted into known properties of percolation.Comment: LaTeX, 20p, added references, corrected typos and additional content
Analysis of a fully packed loop model arising in a magnetic Coulomb phase
The Coulomb phase of spin ice, and indeed the Ic phase of water ice,
naturally realise a fully-packed two-colour loop model in three dimensions. We
present a detailed analysis of the statistics of these loops, which avoid
themselves and other loops of the same colour, and contrast their behaviour to
an analogous two-dimensional model. The properties of another extended degree
of freedom are also addressed, flux lines of the emergent gauge field of the
Coulomb phase, which appear as "Dirac strings" in spin ice. We mention
implications of these results for related models, and experiments.Comment: 5 pages, 4 figure
Determinantal Correlations of Brownian Paths in the Plane with Nonintersection Condition on their Loop-Erased Parts
As an image of the many-to-one map of loop-erasing operation \LE of random
walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk
(LERW) model is the statistical ensemble of SAWs such that the weight of each
SAW is given by the total weight of all random walks which are
inverse images of , \{\pi: \LE(\pi)=\zeta \}. We regard the Brownian
paths as the continuum limits of random walks and consider the statistical
ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the
LERW model. Following the theory of Fomin on nonintersecting LERWs, we
introduce a nonintersecting system of -tuples of LEBPs in a domain in
the complex plane, where the total weight of nonintersecting LEBPs is given by
Fomin's determinant of an matrix whose entries are boundary
Poisson kernels in . We set a sequence of chambers in a planar domain and
observe the first passage points at which Brownian paths first enter each chamber, under the condition that the loop-erased
parts (\LE(\gamma_1),..., \LE(\gamma_N)) make a system of nonintersecting
LEBPs in the domain in the sense of Fomin. We prove that the correlation
functions of first passage points of the Brownian paths of the present system
are generally given by determinants specified by a continuous function called
the correlation kernel. The correlation kernel is of Eynard-Mehta type, which
has appeared in two-matrix models and time-dependent matrix models studied in
random matrix theory. Conformal covariance of correlation functions is
demonstrated.Comment: v3: REVTeX4, 27 pages, 10 figures, corrections made for publication
in Phys.Rev.
Temporal Correlations and Persistence in the Kinetic Ising Model: the Role of Temperature
We study the statistical properties of the sum , that is the difference of time spent positive or negative by the
spin , located at a given site of a -dimensional Ising model
evolving under Glauber dynamics from a random initial configuration. We
investigate the distribution of and the first-passage statistics
(persistence) of this quantity. We discuss successively the three regimes of
high temperature (), criticality (), and low temperature
(). We discuss in particular the question of the temperature
dependence of the persistence exponent , as well as that of the
spectrum of exponents , in the low temperature phase. The
probability that the temporal mean was always larger than the
equilibrium magnetization is found to decay as . This
yields a numerical determination of the persistence exponent in the
whole low temperature phase, in two dimensions, and above the roughening
transition, in the low-temperature phase of the three-dimensional Ising model.Comment: 21 pages, 11 PostScript figures included (1 color figure
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