Elephant random walk is a kind of one-dimensional discrete-time random walk with infinite memory: For each step, with probability
α
the walker adopts one of his/her previous steps uniformly chosen at ...random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits a phase transition from diffusive to superdiffusive behavior at the critical value
α
c
=
1
/
2
. For
α
∈
(
α
c
,
1
)
, there is a scaling factor
a
n
of order
n
α
such that the position
S
n
of the walker at time
n
scaled by
a
n
converges to a nondegenerate random variable
W
^
, whose distribution is not Gaussian. Our main result shows that the fluctuation of
S
n
around
W
^
·
a
n
is still Gaussian. We also give a description of a phase transition induced by bias decaying polynomially in time.
We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed Ann. Probab. 37 (2009) 1044—1079 proved the tightness of the minimum centered around its mean value. We show ...that a convergence in law holds, giving the analog of a well-known result of Bramson Mem. Amer. Math. Soc. 44 (1983) iv+190 in the case of the branching Brownian motion.
•Trapping is the predominant sampling method in insect ecology, and 3D elevated traps are used for flying insects.•A 3D random walk model can be used to simulate insect movement in 3D space, and trap ...counts can be computed.•We use numerical simulations to determine the effect of 3D trap shapes and sizes and of movement diffusion on trapping efficiency.•A better understanding of the interplay between the movement pattern, trap geometry and impacts on trapping efficiency, lead to improved trap count interpretations.
Random walks (RWs) have proved to be a powerful modelling tool in ecology, particularly in the study of animal movement. An application of RW concerns trapping which is the predominant sampling method to date in insect ecology and agricultural pest management. A lot of research effort has been directed towards modelling ground-dwelling insects by simulating their movement in 2D, and computing pitfall trap counts, but comparatively very little for flying insects with 3D elevated traps.
We introduce the mathematics behind 3D RWs and present key metrics such as the mean squared displacement (MSD) and path sinuosity, which are already well known in 2D. We develop the mathematical theory behind the 3D correlated random walk (CRW) which involves short-term directional persistence and the 3D Biased random walk (BRW) which introduces a long-term directional bias in the movement so that there is an overall preferred movement direction. In this study, we focus on the geometrical aspects of the 3D trap and thus consider three types of shape; a spheroidal trap, a cylindrical trap and a rectangular cuboidal trap. By simulating movement in 3D space, we investigated the effect of 3D trap shapes and sizes and of movement diffusion on trapping efficiency.
We found that there is a non-linear dependence of trap counts on the trap surface area or volume, but the effect of volume appeared to be a simple consequence of changes in area. Nevertheless, there is a slight but clear hierarchy of trap shapes in terms of capture efficiency, with the spheroidal trap retaining more counts than a cylinder, followed by the cuboidal type for a given area. We also showed that there is no effect of short-term persistence when diffusion is kept constant, but trap counts significantly decrease with increasing diffusion.
Our results provide a better understanding of the interplay between the movement pattern, trap geometry and impacts on trapping efficiency, which leads to improved trap count interpretations, and more broadly, has implications for spatial ecology and population dynamics.
We report on the possibility of controlling quantum random walks (QWs) with a step-dependent coin (SDC). The coin is characterized by a (single) rotation angle. Considering different rotation angles, ...one can find diverse probability distributions for this walk including: complete localization, Gaussian and asymmetric likes. In addition, we explore the entropy of walk in two contexts; for probability density distributions over position space and walker's internal degrees of freedom space (coin space). We show that entropy of position space can decrease for a SDC with the step-number, quite in contrast to a walk with step-independent coin (SIC). For entropy of coin space, a damped oscillation is found for walk with SIC while for a SDC case, the behavior of entropy depends on rotation angle. In general, we demonstrate that quantum walks with simple initiatives may exhibit a quite complex and varying behavior if SDCs are applied. This provides the possibility of controlling QW with a SDC.
We define here a
directed edge reinforced random walk
on a connected locally finite graph. As the name suggests, this walk keeps track of its past, and gives a bias towards directed edges previously ...crossed proportional to the exponential of the number of crossings. The model is inspired by the so called
Ant Mill phenomenon
, in which a group of army ants forms a continuously rotating circle until they die of exhaustion. For that reason we refer to the walk defined in this work as the
Ant RW
. Our main result justifies this name. Namely, we will show that on any finite graph which is not a tree, and on
Z
d
with
d
≥
2
, the Ant RW almost surely gets eventually trapped into some directed circuit which will be followed forever. In the case of
Z
we show that the Ant RW eventually escapes to infinity and satisfies a law of large number with a random limit which we explicitly identify.
PDF
