569 research outputs found
A note on topological amenability
We point out a simple characterisation of topological amenability in terms of
bounded cohomology, following Johnson's reformulation of amenability
Large Deviations for Brownian Intersection Measures
We consider independent Brownian motions in . We assume that and . Let denote the intersection measure of the
paths by time , i.e., the random measure on that assigns to any
measurable set the amount of intersection local time of the
motions spent in by time . Earlier results of Chen \cite{Ch09} derived
the logarithmic asymptotics of the upper tails of the total mass
as . In this paper, we derive a large-deviation principle for the
normalised intersection measure on the set of positive measures
on some open bounded set as before exiting . The
rate function is explicit and gives some rigorous meaning, in this asymptotic
regime, to the understanding that the intersection measure is the pointwise
product of the densities of the normalised occupation times measures of the
motions. Our proof makes the classical Donsker-Varadhan principle for the
latter applicable to the intersection measure.
A second version of our principle is proved for the motions observed until
the individual exit times from , conditional on a large total mass in some
compact set . This extends earlier studies on the intersection
measure by K\"onig and M\"orters \cite{KM01,KM05}.Comment: To appear in "Communications on Pure and Applied Mathematics
Scaling of human behavior during portal browsing
We investigate transitions of portals users between different subpages. A
weighted network of portals subpages is reconstructed where edge weights are
numbers of corresponding transitions. Distributions of link weights and node
strengths follow power laws over several decades. Node strength increases
faster than linearly with node degree. The distribution of time spent by the
user at one subpage decays as power law with exponent around 1.3. Distribution
of numbers P(z) of unique subpages during one visit is exponential. We find a
square root dependence between the average z and the total number of
transitions n during a single visit. Individual path of portal user resembles
of self-attracting walk on the weighted network. Analytical model is developed
to recover in part the collected data.Comment: 6 pages, 7 figure
Non-additivity of Renyi entropy and Dvoretzky's Theorem
The goal of this note is to show that the analysis of the minimum output
p-Renyi entropy of a typical quantum channel essentially amounts to applying
Milman's version of Dvoretzky's Theorem about almost Euclidean sections of
high-dimensional convex bodies. This conceptually simplifies the
(nonconstructive) argument by Hayden-Winter disproving the additivity
conjecture for the minimal output p-Renyi entropy (for p>1).Comment: 8 pages, LaTeX; v2: added and updated references, minor editorial
changes, no content change
A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space
We consider the task of computing an approximate minimizer of the sum of a
smooth and non-smooth convex functional, respectively, in Banach space.
Motivated by the classical forward-backward splitting method for the
subgradients in Hilbert space, we propose a generalization which involves the
iterative solution of simpler subproblems. Descent and convergence properties
of this new algorithm are studied. Furthermore, the results are applied to the
minimization of Tikhonov-functionals associated with linear inverse problems
and semi-norm penalization in Banach spaces. With the help of
Bregman-Taylor-distance estimates, rates of convergence for the
forward-backward splitting procedure are obtained. Examples which demonstrate
the applicability are given, in particular, a generalization of the iterative
soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as
well as total-variation based image restoration in higher dimensions are
presented
Locally Perturbed Random Walks with Unbounded Jumps
In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively
scaled, simple symmetric random walk, weak convergence to the Brownian motion
holds even in the case of local impurities if . The extension of their
result to finite range random walks is straightforward. Here, however, we are
interested in the situation when the random walk has unbounded range.
Concretely we generalize the statement of \cite{SzT} to unbounded random walks
whose jump distribution belongs to the domain of attraction of the normal law.
We do this first: for diffusively scaled random walks on having finite variance; and second: for random walks with distribution
belonging to the non-normal domain of attraction of the normal law. This result
can be applied to random walks with tail behavior analogous to that of the
infinite horizon Lorentz-process; these, in particular, have infinite variance,
and convergence to Brownian motion holds with the superdiffusive scaling.Comment: 16 page
From Low-Distortion Norm Embeddings to Explicit Uncertainty Relations and Efficient Information Locking
The existence of quantum uncertainty relations is the essential reason that
some classically impossible cryptographic primitives become possible when
quantum communication is allowed. One direct operational manifestation of these
uncertainty relations is a purely quantum effect referred to as information
locking. A locking scheme can be viewed as a cryptographic protocol in which a
uniformly random n-bit message is encoded in a quantum system using a classical
key of size much smaller than n. Without the key, no measurement of this
quantum state can extract more than a negligible amount of information about
the message, in which case the message is said to be "locked". Furthermore,
knowing the key, it is possible to recover, that is "unlock", the message. In
this paper, we make the following contributions by exploiting a connection
between uncertainty relations and low-distortion embeddings of L2 into L1. We
introduce the notion of metric uncertainty relations and connect it to
low-distortion embeddings of L2 into L1. A metric uncertainty relation also
implies an entropic uncertainty relation. We prove that random bases satisfy
uncertainty relations with a stronger definition and better parameters than
previously known. Our proof is also considerably simpler than earlier proofs.
We apply this result to show the existence of locking schemes with key size
independent of the message length. We give efficient constructions of metric
uncertainty relations. The bases defining these metric uncertainty relations
are computable by quantum circuits of almost linear size. This leads to the
first explicit construction of a strong information locking scheme. Moreover,
we present a locking scheme that is close to being implementable with current
technology. We apply our metric uncertainty relations to exhibit communication
protocols that perform quantum equality testing.Comment: 60 pages, 5 figures. v4: published versio
Asymptotics in Knuth's parking problem for caravans
18 pages, 2 figuresWe consider a generalized version of Knuth's parking problem, in which caravans consisting of a number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free space to park. Extending a recent work by Chassaing and Louchard, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent. According to the behavior of the caravan's size tail distribution, several qualitatively different versions of eternal additive coalescent are involved
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