AI 中文总结
该论文证明了在 $\mathbb{Z}^d$ 上，对于任意维度 $d\geq2$ 和任意强化参数 $\beta$，一次强化随机游走的前 $n$ 步期望范围的下界为 $n^{d/(d+1)}$，验证了 Ordemann 等人和 Beffara 的预测。
AI 中文摘要
在一次强化随机游走中（由 Davis (1990) 引入），每条边初始权重为 1，并在首次被跨越后被赋予权重 $\beta\geq1$。每一步，游走者以与权重成正比的概率选择一条关联边。对于足够大的强化参数 $\beta$，Ordemann 等人 (2000) 和 Beffara (2011) 独立预测，在 $\mathbb{Z}^d$ 上，前 $n$ 步的期望范围的数量级为 $n^{d/(d+1)}$。我们在每个维度 $d\geq2$ 以及每个强化参数 $\beta$ 下证明了相应的下界。
英文摘要
In once-reinforced random walk, introduced by Davis (1990), every edge starts with weight one and is assigned weight $β\geq1$ after its first crossing. At each step the walker chooses an incident edge with probability proportional to its weight. For sufficiently large reinforcement $β$, the expected range in the first $n$ steps was predicted by Ordemann et al. (2000) and independently by Beffara (2011) to have order $n^{d/(d+1)}$ on $\mathbb{Z}^d$. We prove the corresponding lower bound in every dimension $d\geq2$ and for every reinforcement parameter $β$.
Comments15 pages, 2 figures. Lean 4 formalization: https://github.com/nitromannitol/ORRW-Lower-Bound
