54 research outputs found
Optimal estimates of the diffusion coefficient of a single Brownian trajectory
Modern developments in microscopy and image processing are revolutionizing
areas of physics, chemistry and biology as nanoscale objects can be tracked
with unprecedented accuracy. The goal of single particle tracking is to
determine the interaction between the particle and its environment. The price
paid for having a direct visualization of a single particle is a consequent
lack of statistics. Here we address the optimal way of extracting diffusion
constants from single trajectories for pure Brownian motion. It is shown that
the maximum likelihood estimator is much more efficient than the commonly used
least squares estimate. Furthermore we investigate the effect of disorder on
the distribution of estimated diffusion constants and show that it increases
the probability of observing estimates much smaller than the true (average)
value.Comment: 8 pages, 5 figure
Chemical Distance in Geometric Random Graphs with Long Edges and Scale-Free Degree Distribution
We study geometric random graphs defined on the points of a Poisson process in d-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance between points in a flexible way. Our framework includes the soft Boolean model (where marks play the role of radii of balls centered in the vertices), a version of spatial preferential attachment (where marks play the role of birth times), and a whole range of other graph models with scale-free degree distributions and edges spanning large distances. In this versatile framework we give sharp criteria for absence of ultrasmallness of the graphs and in the ultrasmall regime establish a limit theorem for the chemical distance of two points. Other than in the mean-field scale-free network models the boundary of the ultrasmall regime depends not only on the power-law exponent of the degree distribution but also on the spatial embedding of the graph, quantified by the rate of decay of the probability of an edge connecting typical points in terms of their spatial distance
A Probabilistic proof of the breakdown of Besov regularity in -shaped domains
{We provide a probabilistic approach in order to investigate the smoothness
of the solution to the Poisson and Dirichlet problems in -shaped domains. In
particular, we obtain (probabilistic) integral representations for the
solution. We also recover Grisvard's classic result on the angle-dependent
breakdown of the regularity of the solution measured in a Besov scale
Superprocesses as models for information dissemination in the Future Internet
Future Internet will be composed by a tremendous number of potentially
interconnected people and devices, offering a variety of services, applications
and communication opportunities. In particular, short-range wireless
communications, which are available on almost all portable devices, will enable
the formation of the largest cloud of interconnected, smart computing devices
mankind has ever dreamed about: the Proximate Internet. In this paper, we
consider superprocesses, more specifically super Brownian motion, as a suitable
mathematical model to analyse a basic problem of information dissemination
arising in the context of Proximate Internet. The proposed model provides a
promising analytical framework to both study theoretical properties related to
the information dissemination process and to devise efficient and reliable
simulation schemes for very large systems
A recursive approach to the O(n) model on random maps via nested loops
We consider the O(n) loop model on tetravalent maps and show how to rephrase
it into a model of bipartite maps without loops. This follows from a
combinatorial decomposition that consists in cutting the O(n) model
configurations along their loops so that each elementary piece is a map that
may have arbitrary even face degrees. In the induced statistics, these maps are
drawn according to a Boltzmann distribution whose parameters (the face weights)
are determined by a fixed point condition. In particular, we show that the
dense and dilute critical points of the O(n) model correspond to bipartite maps
with large faces (i.e. whose degree distribution has a fat tail). The
re-expression of the fixed point condition in terms of linear integral
equations allows us to explore the phase diagram of the model. In particular,
we determine this phase diagram exactly for the simplest version of the model
where the loops are "rigid". Several generalizations of the model are
discussed.Comment: 47 pages, 13 figures, final version (minor changes with v2 after
proof corrections
The universality classes in the parabolic Anderson model
We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on. We consider general i.i.d. potentials and show that exactly four qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at 8 that are thicker than the double-exponential tails, (2) double-exponential tails at 8 studied by Gärtner and Molchanov, (3) a new class called almost bounded potentials, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities. Our analysis of class (3) relies on two large deviation results for the local times of continuous-time simple random walk. One of these results is proved by Brydges and the first two authors in [BHK04], and is also used here to correct a proof in [BK01]
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