3,422 research outputs found
Computing the Loewner driving process of random curves in the half plane
We simulate several models of random curves in the half plane and numerically
compute their stochastic driving process (as given by the Loewner equation).
Our models include models whose scaling limit is the Schramm-Loewner evolution
(SLE) and models for which it is not. We study several tests of whether the
driving process is Brownian motion. We find that just testing the normality of
the process at a fixed time is not effective at determining if the process is
Brownian motion. Tests that involve the independence of the increments of
Brownian motion are much more effective. We also study the zipper algorithm for
numerically computing the driving function of a simple curve. We give an
implementation of this algorithm which runs in a time O(N^1.35) rather than the
usual O(N^2), where N is the number of points on the curve.Comment: 20 pages, 4 figures. Changes to second version: added new paragraph
to conclusion section; improved figures cosmeticall
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
Interference of nematic quantum critical quasiparticles: a route to the octet model
Repeated observations of inhomogeneity in cuperate superconductors[1-5] make
one immediately question the existance of coherent quasiparticles(qp's) and the
applicability of a momentum space picture. Yet, obversations of interference
effects[6-9] suggest that the qp's maintain a remarkable coherence under
special circumstances. In particular, quasi-particle interference (QPI) imaging
using scanning tunneling spectroscopy revealed a highly unusual form of
coherence: accumulation of coherence only at special points in momentum space
with a particular energy dispersion[5-7]. Here we show that nematic quantum
critical fluctuations[10], combined with the known extreme velocity
anisotropy[11] provide a natural mechanism for the accumulation of coherence at
those special points. Our results raise the intriguing question of whether the
nematic fluctuations provide the unique mechanism for such a phenomenon.Comment: 4 pages, 3 figure
A Fast Algorithm for Simulating the Chordal Schramm-Loewner Evolution
The Schramm-Loewner evolution (SLE) can be simulated by dividing the time
interval into N subintervals and approximating the random conformal map of the
SLE by the composition of N random, but relatively simple, conformal maps. In
the usual implementation the time required to compute a single point on the SLE
curve is O(N). We give an algorithm for which the time to compute a single
point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both give a
value of p of approximately 0.4.Comment: 17 pages, 10 figures. Version 2 revisions: added a paragraph to
introduction, added 5 references and corrected a few typo
Stationarity of SLE
A new method to study a stopped hull of SLE(kappa,rho) is presented. In this
approach, the law of the conformal map associated to the hull is invariant
under a SLE induced flow. The full trace of a chordal SLE(kappa) can be studied
using this approach. Some example calculations are presented.Comment: 14 pages with 1 figur
Note on SLE and logarithmic CFT
It is discussed how stochastic evolutions may be linked to logarithmic
conformal field theory. This introduces an extension of the stochastic Loewner
evolutions. Based on the existence of a logarithmic null vector in an
indecomposable highest-weight module of the Virasoro algebra, the
representation theory of the logarithmic conformal field theory is related to
entities conserved in mean under the stochastic process.Comment: 10 pages, LaTeX, v2: version to be publishe
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
Derivatives of spin dynamics simulations
We report analytical equations for the derivatives of spin dynamics
simulations with respect to pulse sequence and spin system parameters. The
methods described are significantly faster, more accurate and more reliable
than the finite difference approximations typically employed. The resulting
derivatives may be used in fitting, optimization, performance evaluation and
stability analysis of spin dynamics simulations and experiments.
Keywords: NMR, EPR, simulation, analytical derivatives, optimal control, spin
chemistry, radical pair.Comment: Accepted by The Journal of Chemical Physic
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
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