一维整数格上的篡改记忆大象随机游走
Tampered Memory Elephant Random Walk on One-Dimensional Integer Lattice
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中文总结 AI 辅助
该研究针对一维整数格提出篡改记忆大象随机游走，确定其相变的尖锐记忆阈值为1/2，推测该阈值也适用于随机记忆划分的情形，为大象随机游走的记忆相变问题提供了关键理论结果。
中文摘要 AI 辅助
Gut与Stadtmüller（2023）提出的大象随机游走理论中的一个突出问题是：确定在扩散、临界和超扩散 regime 之间发生相变需要多少记忆。为研究这一记忆临界点，我们引入篡改记忆大象随机游走，其中记忆被划分为两个不相交的集合 $D_n$ 和 $D_n^c$，二者可为确定性或随机集合。在 $D_n^c$ 上，动力学与大象随机游走相同；而在 $D_n$ 上，增量被替换为独立的新息。由此产生的游走由两个相互竞争的分量驱动：大象随机游走和对应新息的独立简单随机游走。我们首先证明，当递增集合族 $\{D_n\}_{n \ge 1}$ 和 $\{D^c_n\}_{n \ge 1}$ 具有带指数矩的更新结构时，该游走满足大数定律。随后，我们确定了确定性记忆划分下支配相变持续存在的尖锐阈值。我们证明，若 $\{D_n\}_{n \ge 1}$ 是具有递增补集 $\{D^c_n\}_{n \ge 1}$ 的非随机递增集合，且满足 $\lim_{n \to \infty} \frac{|D^c_n|}{n} >1/2$，则扩散、临界和超扩散 regime 之间的相变会持续存在；而当 $\lim_{n \to \infty} \frac{|D^c_n|}{n}<1/2$ 时，仅存在阶为 $\mathcal{O}(\sqrt{n})$ 的扩散 regime。我们也刻画了 $\lim_{n \to \infty}\frac{|D^c_n|}{n}=1/2$ 的情形。因此，在这种竞争环境中，1/2 成为异常扩散持续存在的尖锐临界点。我们推测，当 $\{D_n\}_{n \ge 1}$ 为随机集合时，该阈值同样适用。我们的证明依赖于将随机近似方法应用于游走的两个相互依赖的竞争分量，这两个分量分别代表保留的记忆和新息。
英文摘要
One of the outstanding questions in the theory of elephant random walks as observed by Gut and Stadtmüller (2023), is to determine how much memory is needed for a phase transition between the diffusive, critical and superdiffusive regimes to persist. To investigate this memory breakpoint, we introduce the tampered memory elephant random walk, in which the memory is partitioned into two disjoint sets $D_n$ and $D_n^c$, which may be deterministic or random. On $D_n^c$ the dynamics is the same as an elephant random walk, while on $D_n$ the increments are replaced by independent innovations. The resulting walk is thus driven by two competing components: elephant random walk and an independent simple random walk corresponding to the innovations. We first establish a law of large numbers when the increasing collections $\{D_n\}_{n \ge 1}$ and $\{D^c_n\}_{n \ge 1}$ have a renewal structure with exponential moments. We then identify a sharp threshold that governs the persistence of the phase transition for deterministic memory partitions. We show that if $\{D_n\}_{n \ge 1}$ is non-random increasing collection with increasing complement $\{D^c_n\}_{n \ge 1}$ such that $\lim_{n \to \infty} \frac{\lvert D^c_n\rvert}{n} >1/2$, then a phase transition into diffusive, critical and superdiffusive regimes persists, whereas for $\lim_{n \to \infty} \frac{\lvert D^c_n \rvert}{n}<1/2$, there is only the diffusive regime with $\mathcal{O}(\sqrt{n})$. The case of $\lim_{n\to \infty}\frac{\lvert D^c_n \rvert}{n}=1/2$ is also characterised. Thus, one-half emerges as the sharp breakpoint for the persistence of anomalous diffusion in this competitive setting. We conjecture that the same threshold governs the case when $\{D_n\}_{n \ge 1}$ is random. Our proofs rely on stochastic approximation applied to the two dependent competing components of the walk, representing the retained memory and the innovations.
