51 research outputs found

### Universality and time-scale invariance for the shape of planar L\'evy processes

For a broad class of planar Markov processes, viz. L\'evy processes
satisfying certain conditions (valid \textit{eg} in the case of Brownian motion
and L\'evy flights), we establish an exact, universal formula describing the
shape of the convex hull of sample paths. We show indeed that the average
number of edges joining paths' points separated by a time-lapse $\Delta \tau
\in \left[\Delta \tau _1, \Delta \tau_2\right]$ is equal to $2\ln \left(\Delta
\tau_2 / \Delta \tau_1 \right)$, regardless of the specific distribution of the
process's increments and regardless of its total duration $T$. The formula also
exhibits invariance when the time scale is multiplied by any constant.
Apart from its theoretical importance, our result provides new insights
regarding the shape of two-dimensional objects modelled by stochastic
processes' sample paths (\textit{eg} polymer chains): in particular for a total
time (or parameter) duration $T$, the average number of edges on the convex
hull ("cut off" to discard edges joining points separated by a time-lapse
shorter than some $\Delta \tau < T$) will be given by $2 \ln
\left(\frac{T}{\Delta \tau}\right)$. Thus it will only grow logarithmically,
rather than at some higher pace.Comment: 8 pages, 3 figures, accepted in PR

### From Markovian to non-Markovian persistence exponents

We establish an exact formula relating the survival probability for certain
L\'evy flights (viz. asymmetric $\alpha$-stable processes where $\alpha = 1/2$)
with the survival probability for the order statistics of the running maxima of
two independent Brownian particles. This formula allows us to show that the
persistence exponent $\delta$ in the latter, non Markovian case is simply
related to the persistence exponent $\theta$ in the former, Markovian case via:
$\delta=\theta/2$. Thus, our formula reveals a link between two recently
explored families of anomalous exponents: one exhibiting continuous deviations
from Sparre-Andersen universality in a Markovian context, and one describing
the slow kinetics of the non Markovian process corresponding to the difference
between two independent Brownian maxima.Comment: Accepted in EP

### Convex hull of n planar Brownian paths: an exact formula for the average number of edges

We establish an exact formula for the average number of edges appearing on
the boundary of the global convex hull of n independent Brownian paths in the
plane. This requires the introduction of a counting criterion which amounts to
"cutting off" edges that are, in a specific sense, small. The main argument
consists in a mapping between planar Brownian convex hulls and configurations
of constrained, independent linear Brownian motions. This new formula is
confirmed by retrieving an existing exact result on the average perimeter of
the boundary of Brownian convex hulls in the plane.Comment: 14 pages, 8 figures, submitted to JPA. (Typo corrected in equation
(14).

### Optimization in First-Passage Resetting

We investigate classic diffusion with the added feature that a diffusing
particle is reset to its starting point each time the particle reaches a
specified threshold. In an infinite domain, this process is non-stationary and
its probability distribution exhibits rich features. In a finite domain, we
define a non-trivial optimization in which a cost is incurred whenever the
particle is reset and a reward is obtained while the particle stays near the
reset point. We derive the condition to optimize the net gain in this system,
namely, the reward minus the cost.Comment: 4 pages, 3 figures, revtex 4-1 format. Version 1 contains changes in
response to referee comments. Version 2: A missing factor of 2 in an inline
formula has been correcte

### Maxima of Two Random Walks: Universal Statistics of Lead Changes

We investigate statistics of lead changes of the maxima of two discrete-time
random walks in one dimension. We show that the average number of lead changes
grows as $\pi^{-1}\ln(t)$ in the long-time limit. We present theoretical and
numerical evidence that this asymptotic behavior is universal. Specifically,
this behavior is independent of the jump distribution: the same asymptotic
underlies standard Brownian motion and symmetric Levy flights. We also show
that the probability to have at most n lead changes behaves as $t^{-1/4}[\ln
t]^n$ for Brownian motion and as $t^{-\beta(\mu)}[\ln t]^n$ for symmetric Levy
flights with index $\mu$. The decay exponent $\beta(\mu)$ varies continuously
with the Levy index when $02$.Comment: 7 pages, 6 figure

### Convex hulls of several multidimensional Gaussian random walks

We derive explicit formulae for the expected volume and the expected number
of facets of the convex hull of several multidimensional Gaussian random walks
in terms of the Gaussian persistence probabilities. Special cases include the
already known results about the convex hull of a single Gaussian random walk
and the $d$-dimensional Gaussian polytope with or without the origin

### Multidimensional Urban Segregation - Toward A Neural Network Measure

We introduce a multidimensional, neural-network approach to reveal and
measure urban segregation phenomena, based on the Self-Organizing Map algorithm
(SOM). The multidimensionality of SOM allows one to apprehend a large number of
variables simultaneously, defined on census or other types of statistical
blocks, and to perform clustering along them. Levels of segregation are then
measured through correlations between distances on the neural network and
distances on the actual geographical map. Further, the stochasticity of SOM
enables one to quantify levels of heterogeneity across census blocks. We
illustrate this new method on data available for the city of Paris.Comment: NCAA S.I. WSOM+ 201

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