4,176 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
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Improved Log(gf) Values for Lines of Ti I and Abundance Determinations in the Photospheres of the Sun and Metal-Poor Star HD 84937 (Accurate Transition Probabilities for Ti I)
New atomic transition probability measurements for 948 lines of Ti I are reported. Branching fractions from Fourier transform spectra and from spectra recorded using a 3 m echelle spectrometer are combined with published radiative lifetimes from laser-induced fluorescence measurements to determine these transition probabilities. Generally good agreement is found in comparisons to the NIST Atomic Spectra Database. The new Ti I data are applied to re-determine the Ti abundance in the photospheres of the Sun and metal-poor star HD 84937 using many lines covering a range of wavelength and excitation potential to explore possible non-local thermal equilibrium effects. The variation of relative Ti/Fe abundance with metallicity in metal-poor stars observed in earlier studies is supported in this study.NSF AST-1211055, AST-0908978, AST-1211585NSF REU grant AST-1004881ESO Science Archive Facility 073.D-0024, 266.D-5655NASA NAS 5-26555Astronom
Field theory conjecture for loop-erased random walks
We give evidence that the functional renormalization group (FRG), developed
to study disordered systems, may provide a field theoretic description for the
loop-erased random walk (LERW), allowing to compute its fractal dimension in a
systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with
rigorous bounds, correctly reproduces the leading logarithmic corrections at
the upper critical dimension d=4, and compares well with numerical studies. We
obtain the universal subleading logarithmic correction in d=4, which can be
used as a further test of the conjecture.Comment: 5 page
The dimension of loop-erased random walk in 3D
We measure the fractal dimension of loop-erased random walk (LERW) in 3
dimensions, and estimate that it is 1.62400 +- 0.00005. LERW is closely related
to the uniform spanning tree and the abelian sandpile model. We simulated LERW
on both the cubic and face-centered cubic lattices; the corrections to scaling
are slightly smaller for the face-centered cubic lattice.Comment: 4 pages, 4 figures. v2 has more data, minor additional change
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in
is supported on a set of Hausdorff dimension strictly less than
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of , . By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any , and any | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where denotes the
probability that is the first entrance point of the simple random walk
starting at into . Furthermore, must converge to as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne
Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile
The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We prove
that the asymptotic shape of this model is a Euclidean ball, in a sense which
is stronger than our earlier work. For the shape consisting of
sites, where is the volume of the unit ball in , we show that
the inradius of the set of occupied sites is at least , while the
outradius is at most for any . For a related
model, the divisible sandpile, we show that the domain of occupied sites is a
Euclidean ball with error in the radius a constant independent of the total
mass. For the classical abelian sandpile model in two dimensions, with particles, we show that the inradius is at least , and the
outradius is at most . This improves on bounds of Le Borgne
and Rossin. Similar bounds apply in higher dimensions.Comment: [v3] Added Theorem 4.1, which generalizes Theorem 1.4 for the abelian
sandpile. [v4] Added references and improved exposition in sections 2 and 4.
[v5] Final version, to appear in Potential Analysi
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
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.
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