147,678 research outputs found
Random walk on the range of random walk
We study the random walk X on the range of a simple random walk on ℤ d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin
Excited Random Walk
A random walk on Z^d is excited if the first time it visits a vertex there is
a bias in one direction, but on subsequent visits to that vertex the walker
picks a neighbor uniformly at random. We show that excited random walk on Z^d,
is transient iff d>1.Comment: 7 pages, v2 is journal versio
Slow movement of a random walk on the range of a random walk in the presence of an external field
In this article, a localisation result is proved for the biased random walk
on the range of a simple random walk in high dimensions (d \geq 5). This
demonstrates that, unlike in the supercritical percolation setting, a slowdown
effect occurs as soon a non-trivial bias is introduced. The proof applies a
decomposition of the underlying simple random walk path at its cut-times to
relate the associated biased random walk to a one-dimensional random walk in a
random environment in Sinai's regime
A multifractal random walk
We introduce a class of multifractal processes, referred to as Multifractal
Random Walks (MRWs). To our knowledge, it is the first multifractal processes
with continuous dilation invariance properties and stationary increments. MRWs
are very attractive alternative processes to classical cascade-like
multifractal models since they do not involve any particular scale ratio. The
MRWs are indexed by few parameters that are shown to control in a very direct
way the multifractal spectrum and the correlation structure of the increments.
We briefly explain how, in the same way, one can build stationary multifractal
processes or positive random measures.Comment: 5 pages, 4 figures, uses RevTe
On the Speed of an Excited Asymmetric Random Walk
An excited random walk is a non-Markovian extension of the simple random
walk, in which the walk's behavior at time is impacted by the path it has
taken up to time . The properties of an excited random walk are more
difficult to investigate than those of a simple random walk. For example, the
limiting speed of an excited random walk is either zero or unknown depending on
its initial conditions. While its limiting speed is unknown in most cases, the
qualitative behavior of an excited random walk is largely determined by a
parameter which can be computed explicitly. Despite this, it is known
that the limiting speed cannot be written as a function of . We offer a
new proof of this fact, and use techniques from this proof to further
investigate the relationship between and speed. We also generalize the
standard excited random walk by introducing a "bias" to the right, and call
this generalization an excited asymmetric random walk. Under certain initial
conditions we are able to compute an explicit formula for the limiting speed of
an excited asymmetric random walk.Comment: 22 pages, 4 figures, presented at 2017 MAA MathFes
- …