多平面随机游走碰撞时间的独立阈值
Independence Threshold for Collision Times of Many Planar Random Walks
AI总结：
本文研究了二维格点上多个独立随机游走碰撞时间的独立性，确定了游走数量增长时独立性失效的阈值约为$(\log N)^{1/3}$，并利用混沌展开和局部极限定理证明了这一转变。
AI中文摘要：
我们通过成对碰撞局部时间的联合矩生成函数，研究了在$\mathbb Z^2$上多个独立简单随机游走的碰撞时间。对于固定数量的游走，已知这些碰撞时间在适当的对数归一化后渐近独立。我们探讨了当游走数量增长时，这种独立性在多大程度上得以保持。以$N$为游走长度，我们的结果确定了一个阈值$\asymp (\log N)^{\frac{1}{3}}$，在此阈值上发生从独立到依赖的转变。证明结合了混沌展开技术和相关性不等式，后者是局部极限定理的结果。
英文摘要：
We study collision times of many independent simple random walks on $\mathbb Z^2$ through the joint moment generating function of their pairwise collision local times. For a fixed number of walks, these collision times are known to be asymptotically independent after a suitable logarithmic normalisation. We investigate the extent to which this independence persists when the number of walks grows. For $N$ being the walk length, our results identify a threshold $\asymp (\log N)^{\frac{1}{3}}$, on which the transition from independence to dependence happens. The proofs combine chaos expansion techniques and a correlation inequality, which is the result of a local limit theorem.
