38 research outputs found
Level number variance and spectral compressibility in a critical two-dimensional random matrix model
We study level number variance in a two-dimensional random matrix model
characterized by a power-law decay of the matrix elements. The amplitude of the
decay is controlled by the parameter b. We find analytically that at small
values of b the level number variance behaves linearly, with the
compressibility chi between 0 and 1, which is typical for critical systems. For
large values of b, we derive that chi=0, as one would normally expect in the
metallic phase. Using numerical simulations we determine the critical value of
b at which the transition between these two phases occurs.Comment: 6 page
Universal and non-universal features of the multifractality exponents of critical wave-functions
We calculate perturbatively the multifractality spectrum of wave-functions in
critical random matrix ensembles in the regime of weak multifractality. We show
that in the leading order the spectrum is universal, while the higher order
corrections are model-specific. Explicit results for the anomalous dimensions
are derived in the power-law and ultrametric random matrix ensembles.Comment: 9 page
Global properties of Stochastic Loewner evolution driven by Levy processes
Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian
motion which then produces a trace, a continuous fractal curve connecting the
singular points of the motion. If jumps are added to the driving function, the
trace branches. In a recent publication [1] we introduced a generalized SLE
driven by a superposition of a Brownian motion and a fractal set of jumps
(technically a stable L\'evy process). We then discussed the small-scale
properties of the resulting L\'evy-SLE growth process. Here we discuss the same
model, but focus on the global scaling behavior which ensues as time goes to
infinity. This limiting behavior is independent of the Brownian forcing and
depends upon only a single parameter, , which defines the shape of the
stable L\'evy distribution. We learn about this behavior by studying a
Fokker-Planck equation which gives the probability distribution for endpoints
of the trace as a function of time. As in the short-time case previously
studied, we observe that the properties of this growth process change
qualitatively and singularly at . We show both analytically and
numerically that the growth continues indefinitely in the vertical direction
for , goes as for , and saturates for . The probability density has two different scales corresponding to
directions along and perpendicular to the boundary. In the former case, the
characteristic scale is . In the latter case the scale
is for , and
for . Scaling functions for the probability density are given for
various limiting cases.Comment: Published versio
On harmonic measure of critical curves
Fractal geometry of critical curves appearing in 2D critical systems is
characterized by their harmonic measure. For systems described by conformal
field theories with central charge , scaling exponents of
harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. {\bf
84}, 1363 (2000)] by relating the problem to boundary two-dimensional gravity.
We present a simple argument that allows us to connect harmonic measure of
critical curves to operators obtained by fusion of primary fields, and compute
characteristics of fractal geometry by means of regular methods of conformal
field theory. The method is not limited to theories with .Comment: Some more correction
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
Critical curves in conformally invariant statistical systems
We consider critical curves -- conformally invariant curves that appear at
critical points of two-dimensional statistical mechanical systems. We show how
to describe these curves in terms of the Coulomb gas formalism of conformal
field theory (CFT). We also provide links between this description and the
stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the
long-time limit of stochastic evolution of various SLE observables related to
CFT primary fields. We show how the multifractal spectrum of harmonic measure
and other fractal characteristics of critical curves can be obtained.Comment: Published versio
Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal noise
To understand the sample-to-sample fluctuations in disorder-generated
multifractal patterns we investigate analytically as well as numerically the
statistics of high values of the simplest model - the ideal periodic
Gaussian noise. By employing the thermodynamic formalism we predict the
characteristic scale and the precise scaling form of the distribution of number
of points above a given level. We demonstrate that the powerlaw forward tail of
the probability density, with exponent controlled by the level, results in an
important difference between the mean and the typical values of the counting
function. This can be further used to determine the typical threshold of
extreme values in the pattern which turns out to be given by
with . Such observation provides a
rather compelling explanation of the mechanism behind universality of .
Revealed mechanisms are conjectured to retain their qualitative validity for a
broad class of disorder-generated multifractal fields. In particular, we
predict that the typical value of the maximum of intensity is to be
given by , where is the
corresponding singularity spectrum vanishing at . For the
noise we also derive exact as well as well-controlled approximate
formulas for the mean and the variance of the counting function without
recourse to the thermodynamic formalism.Comment: 28 pages; 7 figures, published version with a few misprints
corrected, editing done and references adde
Cluster pinch-point densities in polygons
In a statistical cluster or loop model such as percolation, or more generally
the Potts models or O(n) models, a pinch point is a single bulk point where
several distinct clusters or loops touch. In a polygon P harboring such a model
in its interior and with 2N sides exhibiting free/fixed side-alternating
boundary conditions, "boundary" clusters anchor to the fixed sides of P. At the
critical point and in the continuum limit, the density (i.e., frequency of
occurrence) of pinch-point events between s distinct boundary clusters at a
bulk point w in P is proportional to
_P. The
w_i are the vertices of P, psi_1^c is a conformal field theory (CFT) corner
one-leg operator, and Psi_s is a CFT bulk 2s-leg operator. In this article, we
use the Coulomb gas formalism to construct explicit contour integral formulas
for these correlation functions and thereby calculate the density of various
pinch-point configurations at arbitrary points in the rectangle, in the
hexagon, and for the case s=N, in the 2N-sided polygon at the system's critical
point. Explicit formulas for these results are given in terms of algebraic
functions or integrals of algebraic functions, particularly Lauricella
functions. In critical percolation, the result for s=N=2 gives the density of
red bonds between boundary clusters (in the continuum limit) inside a
rectangle. We compare our results with high-precision simulations of critical
percolation and Ising FK clusters in a rectangle of aspect ratio two and in a
regular hexagon and find very good agreement.Comment: 31 pages, 1 appendix, 21 figures. In the second version of this
article, we have improved the organization, figures, and references that
appeared in the first versio
Geometrical properties of parafermionic spin models
We present measurements of the fractal dimensions associated to the
geometrical clusters for Z_4 and Z_5 spin models. We also attempted to measure
similar fractal dimensions for the generalised Fortuyin Kastelyn (FK) clusters
in these models but we discovered that these clusters do not percolate at the
critical point of the model under consideration. These results clearly mark a
difference in the behaviour of these non local objects compared to the Ising
model or the 3-state Potts model which corresponds to the simplest cases of Z_N
spin models with N=2 and N=3 respectively. We compare these fractal dimensions
with the ones obtained for SLE interfaces.Comment: 18 pages, 10 figures. v2: published versio