A multivariate extension of the Erdős–Taylor theorem

Abstract

AbstractThe Erdős–Taylor theorem (Acta Math Acad Sci Hungar, 1960) states that if $$\textsf{L}_N$$ L N is the local time at zero, up to time 2N, of a two-dimensional simple, symmetric random walk, then $$\tfrac{\pi }{\log N} \,\textsf{L}_N$$ π log N 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 $$\textsf{L}_N^{(1,2)}=\sum _{n=1}^N \mathbb {1}_{\{S_n^{(1)}= S_n^{(2)}\}}$$ L N ( 1 , 2 ) = ∑ n = 1 N 1 { S n ( 1 ) = S n ( 2 ) } , then $$\tfrac{\pi }{\log N}\, \textsf{L}^{(1,2)}_N$$ π log N L N ( 1 , 2 ) converges in distribution to an exponential random variable of parameter one. We prove that for every $$h \geqslant 3$$ h ⩾ 3 , the family $$ \big \{ \frac{\pi }{\log N} \,\textsf{L}_N^{(i,j)} \big \}_{1\leqslant i<j\leqslant h}$$ { π log N L N ( i , j ) } 1 ⩽ i < j ⩽ 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 Erdős–Taylor theorem. We also discuss connections to directed polymers in random environments.