2,091 research outputs found
Asymptotic direction of random walks in Dirichlet environment
In this paper we generalize the result of directional transience from
[SabotTournier10]. This enables us, by means of [Simenhaus07], [ZernerMerkl01]
and [Bouchet12] to conclude that, on Z^d (for any dimension d), random walks in
i.i.d. Dirichlet environment, or equivalently oriented-edge reinforced random
walks, have almost-surely an asymptotic direction equal to the direction of the
initial drift, unless this drift is zero. In addition, we identify the exact
value or distribution of certain probabilities, answering and generalizing a
conjecture of [SaTo10].Comment: This version includes a second part, proving and generalizing
identities conjectured in a previous paper by C.Sabot and the autho
Biased random walks on random graphs
These notes cover one of the topics programmed for the St Petersburg School
in Probability and Statistical Physics of June 2012.
The aim is to review recent mathematical developments in the field of random
walks in random environment. Our main focus will be on directionally transient
and reversible random walks on different types of underlying graph structures,
such as , trees and for .Comment: Survey based one of the topics programmed for the St Petersburg
School in Probability and Statistical Physics of June 2012. 64 pages, 16
figure
Rocking Subdiffusive Ratchets: Origin, Optimization and Efficiency
We study origin, parameter optimization, and thermodynamic efficiency of
isothermal rocking ratchets based on fractional subdiffusion within a
generalized non-Markovian Langevin equation approach. A corresponding
multi-dimensional Markovian embedding dynamics is realized using a set of
auxiliary Brownian particles elastically coupled to the central Brownian
particle (see video on the journal web site). We show that anomalous
subdiffusive transport emerges due to an interplay of nonlinear response and
viscoelastic effects for fractional Brownian motion in periodic potentials with
broken space-inversion symmetry and driven by a time-periodic field. The
anomalous transport becomes optimal for a subthreshold driving when the driving
period matches a characteristic time scale of interwell transitions. It can
also be optimized by varying temperature, amplitude of periodic potential and
driving strength. The useful work done against a load shows a parabolic
dependence on the load strength. It grows sublinearly with time and the
corresponding thermodynamic efficiency decays algebraically in time because the
energy supplied by the driving field scales with time linearly. However, it
compares well with the efficiency of normal diffusion rocking ratchets on an
appreciably long time scale
Explicit formulae in probability and in statistical physics
We consider two aspects of Marc Yor's work that have had an impact in
statistical physics: firstly, his results on the windings of planar Brownian
motion and their implications for the study of polymers; secondly, his theory
of exponential functionals of Levy processes and its connections with
disordered systems. Particular emphasis is placed on techniques leading to
explicit calculations.Comment: 14 pages, 2 figures. To appear in Seminaire de Probabilites, Special
Issue Marc Yo
Anomalous 1D fluctuations of a simple 2D random walk in a large deviation regime
The following question is the subject of our work: could a two-dimensional
random path pushed by some constraints to an improbable "large deviation
regime", possess extreme statistics with one-dimensional Kardar-Parisi-Zhang
(KPZ) fluctuations? The answer is positive, though non-universal, since the
fluctuations depend on the underlying geometry. We consider in details two
examples of 2D systems for which imposed external constraints force the
underlying stationary stochastic process to stay in an atypical regime with
anomalous statistics. The first example deals with the fluctuations of a
stretched 2D random walk above a semicircle or a triangle. In the second
example we consider a 2D biased random walk along a channel with forbidden
voids of circular and triangular shapes. In both cases we are interested in the
dependence of a typical span \left \sim t^{\gamma} of the
trajectory of steps above the top of the semicircle or the triangle. We
show that , i.e. \left shares the KPZ
statistics for the semicircle, while for the triangle. We propose
heuristic derivations of scaling exponents for different geometries,
justify them by explicit analytic computations and compare with numeric
simulations. For practical purposes, our results demonstrate that the geometry
of voids in a channel might have a crucial impact on the width of the boundary
layer and, thus, on the heat transfer in the channel.Comment: 17 pages, 8 figures, some parts of the paper are rewritte
Zero-one law for directional transience of one-dimensional random walks in dynamic random environments
We prove the trichotomy between transience to the right, transience to the
left and recurrence of one-dimensional nearest-neighbour random walks in
dynamic random environments under fairly general assumptions, namely:
stationarity under space-time translations, ergodicity under spatial
translations, and a mild ellipticity condition. In particular, the result
applies to general uniformly elliptic models and also to a large class of
non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An
immediate consequence is the recurrence of models that are symmetric with
respect to reflection through the origin.Comment: 14 pages, 1 figure. Added Corollary 2.
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