442 research outputs found
Pattern theorems, ratio limit theorems and Gumbel maximal clusters for random fields
We study occurrences of patterns on clusters of size n in random fields on
Z^d. We prove that for a given pattern, there is a constant a>0 such that the
probability that this pattern occurs at most an times on a cluster of size n is
exponentially small. Moreover, for random fields obeying a certain Markov
property, we show that the ratio between the numbers of occurrences of two
distinct patterns on a cluster is concentrated around a constant value. This
leads to an elegant and simple proof of the ratio limit theorem for these
random fields, which states that the ratio of the probabilities that the
cluster of the origin has sizes n+1 and n converges as n tends to infinity.
Implications for the maximal cluster in a finite box are discussed.Comment: 23 pages, 2 figure
Transforming fixed-length self-avoiding walks into radial SLE_8/3
We conjecture a relationship between the scaling limit of the fixed-length
ensemble of self-avoiding walks in the upper half plane and radial SLE with
kappa=8/3 in this half plane from 0 to i. The relationship is that if we take a
curve from the fixed-length scaling limit of the SAW, weight it by a suitable
power of the distance to the endpoint of the curve and then apply the conformal
map of the half plane that takes the endpoint to i, then we get the same
probability measure on curves as radial SLE. In addition to a non-rigorous
derivation of this conjecture, we support it with Monte Carlo simulations of
the SAW. Using the conjectured relationship between the SAW and radial SLE, our
simulations give estimates for both the interior and boundary scaling
exponents. The values we obtain are within a few hundredths of a percent of the
conjectured values
Geometric Exponents, SLE and Logarithmic Minimal Models
In statistical mechanics, observables are usually related to local degrees of
freedom such as the Q < 4 distinct states of the Q-state Potts models or the
heights of the restricted solid-on-solid models. In the continuum scaling
limit, these models are described by rational conformal field theories, namely
the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic
Loewner evolution (SLE_kappa), one can consider observables related to nonlocal
degrees of freedom such as paths or boundaries of clusters. This leads to
fractal dimensions or geometric exponents related to values of conformal
dimensions not found among the finite sets of values allowed by the rational
minimal models. Working in the context of a loop gas with loop fugacity beta =
-2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal
dimensions of various geometric objects such as paths and the generalizations
of cluster mass, cluster hull, external perimeter and red bonds. Specializing
to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we
argue that the geometric exponents are related to conformal dimensions found in
the infinitely extended Kac tables of the logarithmic minimal models LM(p,p').
These theories describe lattice systems with nonlocal degrees of freedom. We
present results for critical dense polymers LM(1,2), critical percolation
LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising
model LM(4,5) as well as LM(3,5). Our results are compared with rigourous
results from SLE_kappa, with predictions from theoretical physics and with
other numerical experiments. Throughout, we emphasize the relationships between
SLE_kappa, geometric exponents and the conformal dimensions of the underlying
CFTs.Comment: Added reference
Two-Dimensional Critical Percolation: The Full Scaling Limit
We use SLE(6) paths to construct a process of continuum nonsimple loops in
the plane and prove that this process coincides with the full continuum scaling
limit of 2D critical site percolation on the triangular lattice -- that is, the
scaling limit of the set of all interfaces between different clusters. Some
properties of the loop process, including conformal invariance, are also
proved.Comment: 45 pages, 12 figures. This is a revised version of math.PR/0504036
without the appendice
The signed loop approach to the Ising model: foundations and critical point
The signed loop method is a beautiful way to rigorously study the
two-dimensional Ising model with no external field. In this paper, we explore
the foundations of the method, including details that have so far been
neglected or overlooked in the literature. We demonstrate how the method can be
applied to the Ising model on the square lattice to derive explicit formal
expressions for the free energy density and two-point functions in terms of
sums over loops, valid all the way up to the self-dual point. As a corollary,
it follows that the self-dual point is critical both for the behaviour of the
free energy density, and for the decay of the two-point functions.Comment: 38 pages, 7 figures, with an improved Introduction. The final
publication is available at link.springer.co
Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback
The onset of pulse propagation is studied in a reaction-diffusion (RD) model
with control by augmented transmission capability that is provided either along
nonlocal spatial coupling or by time-delayed feedback. We show that traveling
pulses occur primarily as solutions to the RD equations while augmented
transmission changes excitability. For certain ranges of the parameter
settings, defined as weak susceptibility and moderate control, respectively,
the hybrid model can be mapped to the original RD model. This results in an
effective change of RD parameters controlled by augmented transmission. Outside
moderate control parameter settings new patterns are obtained, for example
step-wise propagation due to delay-induced oscillations. Augmented transmission
constitutes a signaling system complementary to the classical RD mechanism of
pattern formation. Our hybrid model combines the two major signaling systems in
the brain, namely volume transmission and synaptic transmission. Our results
provide insights into the spread and control of pathological pulses in the
brain
Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke
The stability of cortical function depends critically on proper regulation.
Under conditions of migraine and stroke a breakdown of transmembrane chemical
gradients can spread through cortical tissue. A concomitant component of this
emergent spatio-temporal pattern is a depolarization of cells detected as slow
voltage variations. The velocity of ~3 mm/min indicates a contribution of
diffusion. We propose a mechanism for spreading depolarizations (SD) that rests
upon a nonlocal or non-instantaneous feedback in a reaction-diffusion system.
Depending upon the characteristic space and time scales of the feedback, the
propagation of cortical SD can be suppressed by shifting the bifurcation line,
which separates the parameter regime of pulse propagation from the regime where
a local disturbance dies out. The optimisation of this feedback is elaborated
for different control schemes and ranges of control parameters
LERW as an example of off-critical SLEs
Two dimensional loop erased random walk (LERW) is a random curve, whose
continuum limit is known to be a Schramm-Loewner evolution (SLE) with parameter
kappa=2. In this article we study ``off-critical loop erased random walks'',
loop erasures of random walks penalized by their number of steps. On one hand
we are able to identify counterparts for some LERW observables in terms of
symplectic fermions (c=-2), thus making further steps towards a field theoretic
description of LERWs. On the other hand, we show that it is possible to
understand the Loewner driving function of the continuum limit of off-critical
LERWs, thus providing an example of application of SLE-like techniques to
models near their critical point. Such a description is bound to be quite
complicated because outside the critical point one has a finite correlation
length and therefore no conformal invariance. However, the example here shows
the question need not be intractable. We will present the results with emphasis
on general features that can be expected to be true in other off-critical
models.Comment: 45 pages, 2 figure
Astrocytic Ion Dynamics: Implications for Potassium Buffering and Liquid Flow
We review modeling of astrocyte ion dynamics with a specific focus on the
implications of so-called spatial potassium buffering, where excess potassium
in the extracellular space (ECS) is transported away to prevent pathological
neural spiking. The recently introduced Kirchoff-Nernst-Planck (KNP) scheme for
modeling ion dynamics in astrocytes (and brain tissue in general) is outlined
and used to study such spatial buffering. We next describe how the ion dynamics
of astrocytes may regulate microscopic liquid flow by osmotic effects and how
such microscopic flow can be linked to whole-brain macroscopic flow. We thus
include the key elements in a putative multiscale theory with astrocytes
linking neural activity on a microscopic scale to macroscopic fluid flow.Comment: 27 pages, 7 figure
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