论文标题
Erdös-Taylor定理的多元扩展
A multivariate extension of the Erdös-Taylor theorem
论文作者
论文摘要
Erdös-Taylor定理[Acta Math。学院。科学。 Hungar, 1960] states that if $\mathsf{L}_N$ is the local time at zero, up to time $2N$, of a two-dimensional simple, symmetric random walk, then $\tfracπ{\log N} \,\mathsf{L}_N$ converges in distribution to an exponential random variable with parameter one.可以根据两个独立的简单随机步行在平面上的总碰撞时间等等。更准确地说，如果$ \ m var {l} _n^{（1,2）} = \ sum_ {n = 1}^n 1 _ {\ {\ {s_n^{（1）= s_n^{（1）} = s_n^{（（2）}}}}} $ \ Mathsf {l}^{（1,2）} _ n $在分发中收敛到参数一个的指数随机变量。我们证明，在每$ h \ geq 3 $中，家庭$ \ big \ {\fracπ{\ log n} \，\ Mathsf {l} _n^{（i，j）} \ big \ \} _ {1 \ big \} _ {1 \ leq i <j \ j \ leq h}对称随机步行在平面上共同收敛到具有参数1的独立指数随机变量的向量，从而提供了Erdös-Taylor定理的多元版本。我们还讨论了与随机环境中定向聚合物的连接。
The Erdös-Taylor theorem [Acta Math. Acad. Sci. Hungar, 1960] states that if $\mathsf{L}_N$ is the local time at zero, up to time $2N$, of a two-dimensional simple, symmetric random walk, then $\tfracπ{\log N} \,\mathsf{L}_N$ converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if $\mathsf{L}_N^{(1,2)}=\sum_{n=1}^N 1_{\{S_n^{(1)}= S_n^{(2)}\}}$, then $\tfracπ{\log N}\, \mathsf{L}^{(1,2)}_N$ converges in distribution to an exponential random variable of parameter one. We prove that for every $h \geq 3$, the family $ \big\{ \fracπ{\log N} \,\mathsf{L}_N^{(i,j)} \big\}_{1\leq i<j\leq h}$, of logarithmically rescaled, two-body collision local times between $h$ independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdös-Taylor theorem. We also discuss connections to directed polymers in random environments.
