Transient superdiffusion in random walks with a<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-exponentially decaying memory profile

2021

Preprint

In the simple random walk the steps are independent, viz., the walker has no memory. In contrast, in the elephant random walk (ERW), which was introduced by Schütz and Trimper [13] in 2004, the next step always depends on the whole path so far. Various authors have studied further properties of the ERW. In [8] we studied the case when the Elephant remembers only a finite part of the first or last steps. In both cases there was no separation into two different regimes as in the classical ERW. We also posed the question about what happens if she remembers a gradually increasing past. This paper will give some answers to that question. We also discuss related questions for ERW:s with delays.

Section: Introductionmentioning

confidence: 93%

The Elephant random walk with gradually increasing memory

Gut¹,

2021

Preprint

“…An obvious, as well as interesting, question would be to find the breaking point. There exist some papers on this topic using simulations, see e.g., [12,2,10] and further papers cited therein, but we are not aware of any theoretical results concerning this matter.…”

Section: Phase Transitionmentioning

confidence: 99%

“…Our main interest is the situation in which the elephant has only a limited memory, either that he or she remembers only some distant past, only a recent past or a mixture of both. No paper on exact results and proofs seems to exist, only simulations, in which case a given fraction of the distant/recent past is remembered; [12,2,10].…”

Section: Introductionmentioning

confidence: 99%

Variations of the elephant random walk

Gut¹,

2018

Preprint

In the classical simple random walk the steps are independent, viz., the walker has no memory. In contrast, in the elephant random walk, which was introduced by Schütz and Trimper [11] in 2004, the walker remembers the whole past, and 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.

“…There exist some papers on this topic using simulations (see e.g. [20], [6], and [18], and further papers cited therein), but we are not aware of any theoretical results concerning this matter.…”

Section: Phase Transitionmentioning

confidence: 99%

“…Our main interest is the situation in which the elephant has only a limited memory, remembering only some distant past, only a recent past, or a mixture of both. We were motivated by simulations studying the case in which a given fraction of the distant/recent past is remembered [6,18,20] but where no theoretical results are provided. This is a first step towards a better understanding of situations with a memory increasing in time, and for finding the breaking point for the phase transitions.…”

Section: Introductionmentioning

confidence: 99%

Variations of the elephant random walk

Gut

2021

J. Appl. Probab.

In 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.