5,053 research outputs found
The shattering dimension of sets of linear functionals
We evaluate the shattering dimension of various classes of linear functionals
on various symmetric convex sets. The proofs here relay mostly on methods from
the local theory of normed spaces and include volume estimates, factorization
techniques and tail estimates of norms, viewed as random variables on Euclidean
spheres. The estimates of shattering dimensions can be applied to obtain error
bounds for certain classes of functions, a fact which was the original
motivation of this study. Although this can probably be done in a more
traditional manner, we also use the approach presented here to determine
whether several classes of linear functionals satisfy the uniform law of large
numbers and the uniform central limit theorem.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000038
Random graph states, maximal flow and Fuss-Catalan distributions
For any graph consisting of vertices and edges we construct an
ensemble of random pure quantum states which describe a system composed of
subsystems. Each edge of the graph represents a bi-partite, maximally entangled
state. Each vertex represents a random unitary matrix generated according to
the Haar measure, which describes the coupling between subsystems. Dividing all
subsystems into two parts, one may study entanglement with respect to this
partition. A general technique to derive an expression for the average
entanglement entropy of random pure states associated to a given graph is
presented. Our technique relies on Weingarten calculus and flow problems. We
analyze statistical properties of spectra of such random density matrices and
show for which cases they are described by the free Poissonian
(Marchenko-Pastur) distribution. We derive a discrete family of generalized,
Fuss-Catalan distributions and explicitly construct graphs which lead to
ensembles of random states characterized by these novel distributions of
eigenvalues.Comment: 37 pages, 24 figure
Disorder, Order, and Domain Wall Roughening in the 2d Random Field Ising Model
Ground states and domain walls are investigated with exact combinatorial
optimization in two-dimensional random field Ising magnets. The ground states
break into domains above a length scale that depends exponentially on the
random field strength squared. For weak disorder, this paramagnetic structure
has remnant long-range order of the percolation type. The domain walls are
super-rough in ordered systems with a roughness exponent close to 6/5.
The interfaces exhibit rare fluctuations and multiscaling reminiscent of some
models of kinetic roughening and hydrodynamic turbulence.Comment: to be published in Phys.Rev.E/Rapid.Com
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