scholarly journals Variations of the elephant random walk

2021 ◽  
Vol 58 (3) ◽  
pp. 805-829
Author(s):  
Allan Gut ◽  
Ulrich Stadtmüller
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
General Step

AbstractIn the classical simple random walk the steps are independent, that is, the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schütz and Trimper [19] in 2004, the next step always depends on the whole path so far. Our main aim is to prove analogous results when the elephant has only a restricted memory, for example remembering only the most remote step(s), the most recent step(s), or both. We also extend the models to cover more general step sizes.

The size of the region occupied by one type in an invasion process

1976 ◽  
Vol 13 (02) ◽  
pp. 355-356 ◽  
Author(s):  
Aidan Sudbury
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Rectangular Lattice ◽  
Invasion Process

Particles are situated on a rectangular lattice and proceed to invade each other's territory. When they are equally competitive this creates larger and larger blocks of one type as time goes by. It is shown that the expected size of such blocks is equal to the expected range of a simple random walk.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

A Simple Random Walk Model for Dairy Cow Calving Time Prediction

2021 ◽  
Author(s):  
Thi Thi Zin ◽  
Pyke Tin ◽  
Pann Thinzar Seint ◽  
Kosuke Sumi ◽  
Ikuo Kobayashi ◽  
...  
Keyword(s):  
Random Walk ◽  
Dairy Cow ◽  
Simple Random Walk ◽  
Random Walk Model ◽  
Time Prediction

scholarly journals Diamond aggregation

2010 ◽  
Vol 149 (2) ◽  
pp. 351-372
Author(s):  
WOUTER KAGER ◽  
LIONEL LEVINE
Keyword(s):  
Random Walk ◽  
Growth Model ◽  
Power Law ◽  
Simple Random Walk ◽  
Internal Diffusion ◽  
Diffusion Limited Aggregation ◽  
Directional Bias ◽  
Model Based ◽  
Diffusion Limited ◽  
Logarithmic Fluctuations

AbstractInternal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

On the number of times where a simple random walk reaches its maximum

1992 ◽  
Vol 29 (02) ◽  
pp. 305-312 ◽  
Author(s):  
W. Katzenbeisser ◽  
W. Panny
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Probability And Statistics

Let Qn denote the number of times where a simple random walk reaches its maximum, where the random walk starts at the origin and returns to the origin after 2n steps. Such random walks play an important role in probability and statistics. In this paper the distribution and the moments of Qn , are considered and their asymptotic behavior is studied.

scholarly journals Capturing the Drunk Robber on a Graph

10.37236/3398 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Natasha Komarov ◽  
Peter Winkler
Keyword(s):  
Random Walk ◽  
Undirected Graph ◽  
Simple Random Walk ◽  
Adjacent Vertex ◽  
Expected Time ◽  
Capture Time ◽  
Vertex Graph

We show that the expected time for a smart "cop"' to catch a drunk "robber" on an $n$-vertex graph is at most $n + {\rm o}(n)$. More precisely, let $G$ be a simple, connected, undirected graph with distinguished points $u$ and $v$ among its $n$ vertices. A cop begins at $u$ and a robber at $v$; they move alternately from vertex to adjacent vertex. The robber moves randomly, according to a simple random walk on $G$; the cop sees all and moves as she wishes, with the object of "capturing" the robber—that is, occupying the same vertex—in least expected time. We show that the cop succeeds in expected time no more than $n {+} {\rm o}(n)$. Since there are graphs in which capture time is at least $n {-} o(n)$, this is roughly best possible. We note also that no function of the diameter can be a bound on capture time.

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