This paper deals with different models of random walks with a reinforced memory of preferential attachment type. We consider extensions of the Elephant Random Walk introduced by Schütz and Trimper (Phys Rev E 70:044510(R), 2004) with stronger reinforcement mechanisms, where, roughly speaking, a step from the past is remembered proportional to some weight and then repeated with probability p . With probability $$1-p$$ 1 - p , the random walk performs a step independent of the past. The weight of the remembered step is increased by an additive factor $$b\ge 0$$ b ≥ 0 , making it likelier to repeat the step again in the future. A combination of techniques from the theory of urns, branching processes and $$\alpha $$ α -stable processes enables us to discuss the limit behavior of reinforced versions of both the Elephant Random Walk and its $$\alpha $$ α -stable counterpart, the so-called Shark Random Swim introduced by Businger (J Stat Phys 172(3):701–717, 2004). We establish phase transitions, separating subcritical from supercritical regimes.

中文翻译：

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一类具有强化记忆的随机游走

本文涉及具有优先依恋类型的增强记忆的随机游走的不同模型。我们考虑 Schütz 和 Trimper (Phys Rev E 70:044510(R), 2004) 引入的具有更强强化机制的 Elephant Random Walk 的扩展，其中，粗略地说，记住过去的一步与某些权重成正比，然后重复概率为 p 。随机游走以 $$1-p$$ 1 - p 的概率执行独立于过去的步骤。记住的步骤的权重增加了一个加性因子 $$b\ge 0$$ b ≥ 0 ，使其更有可能在未来再次重复该步骤。结合骨灰盒理论的技术，分支过程和 $$\alpha $$ α -stable 过程使我们能够讨论 Elephant Random Walk 及其 $$\alpha $$ α -stable 对应物的增强版本的极限行为，即所谓的 Shark Random Swim Businger 着（J Stat Phys 172(3):701–717, 2004）。我们建立相变，将亚临界状态与超临界状态分开。