181 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
Approximation of dynamical systems using S-systems theory : application to biological systems
In this paper we propose a new symbolic-numeric algorithm to find positive
equilibria of a n-dimensional dynamical system. This algorithm implies a
symbolic manipulation of ODE in order to give a local approximation of
differential equations with power-law dynamics (S-systems). A numerical
calculus is then needed to converge towards an equilibrium, giving at the same
time a S-system approximating the initial system around this equilibrium. This
algorithm is applied to a real biological example in 14 dimensions which is a
subsystem of a metabolic pathway in Arabidopsis Thaliana
Random walks in Dirichlet environment: an overview
Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in
Random Environment (RWRE) on where the transition probabilities are
i.i.d. at each site with a Dirichlet distribution. Hence, the model is
parametrized by a family of positive weights ,
one for each direction of . In this case, the annealed law is that
of a reinforced random walk, with linear reinforcement on directed edges. RWDE
have a remarkable property of statistical invariance by time reversal from
which can be inferred several properties that are still inaccessible for
general environments, such as the equivalence of static and dynamic points of
view and a description of the directionally transient and ballistic regimes. In
this paper we give a state of the art on this model and several sketches of
proofs presenting the core of the arguments. We also present new computation of
the large deviation rate function for one dimensional RWDE.Comment: 35 page
Non-fixation for Biased Activated Random Walks
We prove that the model of Activated Random Walks on Z^d with biased jump
distribution does not fixate for any positive density, if the sleep rate is
small enough, as well as for any finite sleep rate, if the density is close
enough to 1. The proof uses a new criterion for non-fixation. We provide a
pathwise construction of the process, of independent interest, used in the
proof of this non-fixation criterion
A deterministic walk on the randomly oriented Manhattan lattice
Consider a randomly-oriented two dimensional Manhattan lattice where each
horizontal line and each vertical line is assigned, once and for all, a random
direction by flipping independent and identically distributed coins. A
deterministic walk is then started at the origin and at each step moves
diagonally to the nearest vertex in the direction of the horizontal and
vertical lines of the present location. This definition can be generalized, in
a natural way, to larger dimensions, but we mainly focus on the two dimensional
case. In this context the process localizes on two vertices at all large times,
almost surely. We also provide estimates for the tail of the length of paths,
when the walk is defined on the two dimensional lattice. In particular, the
probability of the path to be larger than decays sub-exponentially in .
It is easy to show that higher dimensional paths may not localize on two
vertices but will still eventually become periodic, and are therefore bounded.Comment: 18 pages, 12 figure
A note on the recurrence of edge reinforced random walks
2 pagesWe give a short proof of Theorem 2.1 from [MR07], stating that the linearly edge reinforced random walk (ERRW) on a locally finite graph is recurrent if and only if it returns to its starting point almost surely. This result was proved in [MR07] by means of the much stronger property that the law of the ERRW is a mixture of Markov chains. Our proof only uses this latter property on finite graphs, in which case it is a consequence of De Finetti's theorem on exchangeability
Stable fluctuations for ballistic random walks in random environment on Z
We consider transient random walks in random environment on Z in the positive
speed (ballistic) and critical zero speed regimes. A classical result of
Kesten, Kozlov and Spitzer proves that the hitting time of level , after
proper centering and normalization, converges to a completely asymmetric stable
distribution, but does not describe its scale parameter. Following a previous
article by three of the authors, where the (non-critical) zero speed case was
dealt with, we give a new proof of this result in the subdiffusive case that
provides a complete description of the limit law. The case of Dirichlet
environment turns out to be remarkably explicit
Uncovering operational interactions in genetic networks using asynchronous boolean dynamics
To analyze and gain intuition on the mechanisms of complex systems of large dimensions, one strategy is to simplify the model by identifying a reduced system, in the form of a smaller set of variables and interactions that still capture specific properties of the system. For large models of biological networks, the diagram of interactions is often well represented by a Boolean model with a family of logical rules. The state space of a Boolean model is finite, and its asynchronous dynamics are fully described by a transition graph in the state space. In this context, a method will be developed for identifying the active or operational interactions responsible for a given dynamic behaviour. The first step in this procedure is the decomposition of the asynchronous transition graph into its strongly connected components, to obtain a ``reduced'' and hierarchically organized graph of transitions. The second step consists of the identification of a partial graph of interactions and a sub-family of logical rules that remain operational in a given region of the state space. This model reduction method and its usefulness are illustrated by an application to a model of programmed cell death. The method identifies two mechanisms used by the cell to respond to death-receptor stimulation and decide between the survival or apoptotic pathways
Integrability of exit times and ballisticity for random walks in Dirichlet environment
We consider random walks in Dirichlet environment, introduced by Enriquez and
Sabot in 2006. As this distribution on environments is not uniformly elliptic,
the annealed integrability of exit times out of a given finite subset is a
non-trivial property. We provide here an explicit equivalent condition for this
integrability to happen, on general directed graphs. Such integrability
problems arise for instance from the definition of Kalikow auxiliary random
walk. Using our condition, we prove a refined version of the ballisticity
criterion given by Enriquez and Sabot
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