399 research outputs found
A Hypergraph Dictatorship Test with Perfect Completeness
A hypergraph dictatorship test is first introduced by Samorodnitsky and
Trevisan and serves as a key component in their unique games based \PCP
construction. Such a test has oracle access to a collection of functions and
determines whether all the functions are the same dictatorship, or all their
low degree influences are Their test makes queries and has
amortized query complexity but has an inherent loss of
perfect completeness. In this paper we give an adaptive hypergraph dictatorship
test that achieves both perfect completeness and amortized query complexity
.Comment: Some minor correction
Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
We find a general formula for the distribution of time-averaged observables
for systems modeled according to the sub-diffusive continuous time random walk.
For Gaussian random walks coupled to a thermal bath we recover ergodicity and
Boltzmann's statistics, while for the anomalous subdiffusive case a weakly
non-ergodic statistical mechanical framework is constructed, which is based on
L\'evy's generalized central limit theorem. As an example we calculate the
distribution of : the time average of the position of the particle,
for unbiased and uniformly biased particles, and show that exhibits
large fluctuations compared with the ensemble average .Comment: 5 pages, 2 figure
Kernel PCA for multivariate extremes
We propose kernel PCA as a method for analyzing the dependence structure of
multivariate extremes and demonstrate that it can be a powerful tool for
clustering and dimension reduction. Our work provides some theoretical insight
into the preimages obtained by kernel PCA, demonstrating that under certain
conditions they can effectively identify clusters in the data. We build on
these new insights to characterize rigorously the performance of kernel PCA
based on an extremal sample, i.e., the angular part of random vectors for which
the radius exceeds a large threshold. More specifically, we focus on the
asymptotic dependence of multivariate extremes characterized by the angular or
spectral measure in extreme value theory and provide a careful analysis in the
case where the extremes are generated from a linear factor model. We give
theoretical guarantees on the performance of kernel PCA preimages of such
extremes by leveraging their asymptotic distribution together with Davis-Kahan
perturbation bounds. Our theoretical findings are complemented with numerical
experiments illustrating the finite sample performance of our methods
Levy ratchets with dichotomic random flashing
Additive symmetric L\'evy noise can induce directed transport of overdamped
particles in a static asymmetric potential. We study, numerically and
analytically, the effect of an additional dichotomous random flashing in such
L\'evy ratchet system. For this purpose we analyze and solve the corresponding
fractional Fokker-Planck equations and we check the results with Langevin
simulations. We study the behavior of the current as function of the stability
index of the L\'evy noise, the noise intensity and the flashing parameters. We
find that flashing allows both to enhance and diminish in a broad range the
static L\'evy ratchet current, depending on the frequencies and asymmetry of
the multiplicative dichotomous noise, and on the additive L\'evy noise
parameters. Our results thus extend those for dichotomous flashing ratchets
with Gaussian noise to the case of broadly distributed noises.Comment: 15 pages, 6 figure
A Stochastic Description for Extremal Dynamics
We show that extremal dynamics is very well modelled by the "Linear
Fractional Stable Motion" (LFSM), a stochastic process entirely defined by two
exponents that take into account spatio-temporal correlations in the
distribution of active sites. We demonstrate this numerically and analytically
using well-known properties of the LFSM. Further, we use this correspondence to
write an exact expressions for an n-point correlation function as well as an
equation of fractional order for interface growth in extremal dynamics.Comment: 4 pages LaTex, 3 figures .ep
How Heavy Are the Tails of a Stationary HARCH(k) Process? A Study of the Moments
How Heavy Are the Tails of a Stationary HARCH(k) Process? A Study of the Moment
On the high density behavior of Hamming codes with fixed minimum distance
We discuss the high density behavior of a system of hard spheres of diameter
d on the hypercubic lattice of dimension n, in the limit n -> oo, d -> oo,
d/n=delta. The problem is relevant for coding theory. We find a solution to the
equations describing the liquid up to very large values of the density, but we
show that this solution gives a negative entropy for the liquid phase when the
density is large enough. We then conjecture that a phase transition towards a
different phase might take place, and we discuss possible scenarios for this
transition. Finally we discuss the relation between our results and known
rigorous bounds on the maximal density of the system.Comment: 15 pages, 6 figure
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