353 research outputs found
Exact Occupation Time Distribution in a Non-Markovian Sequence and Its Relation to Spin Glass Models
We compute exactly the distribution of the occupation time in a discrete {\em
non-Markovian} toy sequence which appears in various physical contexts such as
the diffusion processes and Ising spin glass chains. The non-Markovian property
makes the results nontrivial even for this toy sequence. The distribution is
shown to have non-Gaussian tails characterized by a nontrivial large deviation
function which is computed explicitly. An exact mapping of this sequence to an
Ising spin glass chain via a gauge transformation raises an interesting new
question for a generic finite sized spin glass model: at a given temperature,
what is the distribution (over disorder) of the thermally averaged number of
spins that are aligned to their local fields? We show that this distribution
remains nontrivial even at infinite temperature and can be computed explicitly
in few cases such as in the Sherrington-Kirkpatrick model with Gaussian
disorder.Comment: 10 pages Revtex (two-column), 1 eps figure (included
Persistence in higher dimensions : a finite size scaling study
We show that the persistence probability , in a coarsening system of
linear size at a time , has the finite size scaling form where is the persistence exponent and
is the coarsening exponent. The scaling function for
and is constant for large . The scaling form implies a fractal
distribution of persistent sites with power-law spatial correlations. We study
the scaling numerically for Glauber-Ising model at dimension to 4 and
extend the study to the diffusion problem. Our finite size scaling ansatz is
satisfied in all these cases providing a good estimate of the exponent
.Comment: 4 pages in RevTeX with 6 figures. To appear in Phys. Rev.
Large-Deviation Functions for Nonlinear Functionals of a Gaussian Stationary Markov Process
We introduce a general method, based on a mapping onto quantum mechanics, for
investigating the large-T limit of the distribution P(r,T) of the nonlinear
functional r[V] = (1/T)\int_0^T dT' V[X(T')], where V(X) is an arbitrary
function of the stationary Gaussian Markov process X(T). For T tending to
infinity at fixed r we find that P(r,T) behaves as exp[-theta(r) T], where
theta(r) is a large deviation function. We present explicit results for a
number of special cases, including the case V(X) = X \theta(X) which is related
to the cooling and the heating degree days relevant to weather derivatives.Comment: 8 page
Exact Phase Diagram of a model with Aggregation and Chipping
We revisit a simple lattice model of aggregation in which masses diffuse and
coalesce upon contact with rate 1 and every nonzero mass chips off a single
unit of mass to a randomly chosen neighbour with rate . The dynamics
conserves the average mass density and in the stationary state the
system undergoes a nonequilibrium phase transition in the plane
across a critical line . In this paper, we show analytically that in
arbitrary spatial dimensions, exactly and hence,
remarkably, independent of dimension. We also provide direct and indirect
numerical evidence that strongly suggest that the mean field asymptotic answer
for the single site mass distribution function and the associated critical
exponents are super-universal, i.e., independent of dimension.Comment: 11 pages, RevTex, 3 figure
Persistence in a Stationary Time-series
We study the persistence in a class of continuous stochastic processes that
are stationary only under integer shifts of time. We show that under certain
conditions, the persistence of such a continuous process reduces to the
persistence of a corresponding discrete sequence obtained from the measurement
of the process only at integer times. We then construct a specific sequence for
which the persistence can be computed even though the sequence is
non-Markovian. We show that this may be considered as a limiting case of
persistence in the diffusion process on a hierarchical lattice.Comment: 8 pages revte
Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
The Langevin equation for a particle (`random walker') moving in
d-dimensional space under an attractive central force, and driven by a Gaussian
white noise, is considered for the case of a power-law force, F(r) = -
Ar^{-sigma}. The `persistence probability', P_0(t), that the particle has not
visited the origin up to time t, is calculated. For sigma > 1, the force is
asymptotically irrelevant (with respect to the noise), and the asymptotics of
P_0(t) are those of a free random walker. For sigma < 1, the noise is
(dangerously) irrelevant and the asymptotics of P_0(t) can be extracted from a
weak noise limit within a path-integral formalism. For the case sigma=1,
corresponding to a logarithmic potential, the noise is exactly marginal. In
this case, P_0(t) decays as a power-law, P_0(t) \sim t^{-theta}, with an
exponent theta that depends continuously on the ratio of the strength of the
potential to the strength of the noise. This case, with d=2, is relevant to the
annihilation dynamics of a vortex-antivortex pair in the two-dimensional XY
model. Although the noise is multiplicative in the latter case, the relevant
Langevin equation can be transformed to the standard form discussed in the
first part of the paper. The mean annihilation time for a pair initially
separated by r is given by t(r) \sim r^2 ln(r/a) where a is a microscopic
cut-off (the vortex core size). Implications for the nonequilibrium critical
dynamics of the system are discussed and compared to numerical simulation
results.Comment: 10 pages, 1 figur
A new family of matrix product states with Dzyaloshinski-Moriya interactions
We define a new family of matrix product states which are exact ground states
of spin 1/2 Hamiltonians on one dimensional lattices. This class of
Hamiltonians contain both Heisenberg and Dzyaloshinskii-Moriya interactions but
at specified and not arbitrary couplings. We also compute in closed forms the
one and two-point functions and the explicit form of the ground state. The
degeneracy structure of the ground state is also discussed.Comment: 15 pages, 1 figur
Neutrino Decays over Cosmological Distances and the Implications for Neutrino Telescopes
We discuss decays of ultra-relativistic neutrinos over cosmological distances
by solving the decay equation in terms of its redshift dependence. We
demonstrate that there are significant conceptual differences compared to more
simplified treatments of neutrino decay. For instance, the maximum distance the
neutrinos have traveled is limited by the Hubble length, which means that the
common belief that longer neutrino lifetimes can be probed by longer distances
does not apply. As a consequence, the neutrino lifetime limit from supernova
1987A cannot be exceeded by high-energy astrophysical neutrinos. We discuss the
implications for neutrino spectra and flavor ratios from gamma-ray bursts as
one example of extragalactic sources, using up-to-date neutrino flux
predictions. If the observation of SN 1987A implies that \nu_1 is stable and
the other mass eigenstates decay with rates much smaller than their current
bounds, the muon track rate can be substantially suppressed compared to the
cascade rate in the region IceCube is most sensitive to. In this scenario, no
gamma-ray burst neutrinos may be found using muon tracks even with the full
scale experiment, whereas reliable information on high-energy astrophysical
sources can only be obtained from cascade measurements. As another consequence,
the recently observed two cascade event candidates at PeV energies will not be
accompanied by corresponding muon tracks.Comment: 20 pages, 6 figures, 1 table. Matches published versio
Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation
We study various statistical properties of real roots of three different
classes of random polynomials which recently attracted a vivid interest in the
context of probability theory and quantum chaos. We first focus on gap
probabilities on the real axis, i.e. the probability that these polynomials
have no real root in a given interval. For generalized Kac polynomials, indexed
by an integer d, of large degree n, one finds that the probability of no real
root in the interval [0,1] decays as a power law n^{-\theta(d)} where \theta(d)
> 0 is the persistence exponent of the diffusion equation with random initial
conditions in spatial dimension d. For n \gg 1 even, the probability that they
have no real root on the full real axis decays like
n^{-2(\theta(2)+\theta(d))}. For Weyl polynomials and Binomial polynomials,
this probability decays respectively like \exp{(-2\theta_{\infty}} \sqrt{n})
and \exp{(-\pi \theta_{\infty} \sqrt{n})} where \theta_{\infty} is such that
\theta(d) = 2^{-3/2} \theta_{\infty} \sqrt{d} in large dimension d. We also
show that the probability that such polynomials have exactly k roots on a given
interval [a,b] has a scaling form given by \exp{(-N_{ab} \tilde
\phi(k/N_{ab}))} where N_{ab} is the mean number of real roots in [a,b] and
\tilde \phi(x) a universal scaling function. We develop a simple Mean Field
(MF) theory reproducing qualitatively these scaling behaviors, and improve
systematically this MF approach using the method of persistence with partial
survival, which in some cases yields exact results. Finally, we show that the
probability density function of the largest absolute value of the real roots
has a universal algebraic tail with exponent {-2}. These analytical results are
confirmed by detailed numerical computations.Comment: 32 pages, 16 figure
Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
We provide a comprehensive report on scale-invariant fluctuations of growing
interfaces in liquid-crystal turbulence, for which we recently found evidence
that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1
dimensions [Phys. Rev. Lett. 104, 230601 (2010); Sci. Rep. 1, 34 (2011)]. Here
we investigate both circular and flat interfaces and report their statistics in
detail. First we demonstrate that their fluctuations show not only the KPZ
scaling exponents but beyond: they asymptotically share even the precise forms
of the distribution function and the spatial correlation function in common
with solvable models of the KPZ class, demonstrating also an intimate relation
to random matrix theory. We then determine other statistical properties for
which no exact theoretical predictions were made, in particular the temporal
correlation function and the persistence probabilities. Experimental results on
finite-time effects and extreme-value statistics are also presented. Throughout
the paper, emphasis is put on how the universal statistical properties depend
on the global geometry of the interfaces, i.e., whether the interfaces are
circular or flat. We thereby corroborate the powerful yet geometry-dependent
universality of the KPZ class, which governs growing interfaces driven out of
equilibrium.Comment: 31 pages, 21 figures, 1 table; references updated (v2,v3); Fig.19
updated & minor changes in text (v3); final version (v4); J. Stat. Phys.
Online First (2012
- …