A multivariate extension of the Erdös-Taylor theorem

Abstract

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.

Figures (5)

• Figure 1: This is a graphical representation of expansion \ref{['def:expansion']} corresponding to the collisions of four random walks, each starting from the origin. Each solid line will be marked with the label of the walk that it corresponds to throughout the diagram. Each solid dot, which marks a collision among a subset $A$ of the random walks, is given a weight $\prod_{ i,j\in A} \sigma_N^{i,j}$. Any solid line between points $(m,x), (n,y)$ is assigned the weight of the simple random walk transition kernel $q_{m-n}(y-x)$. The hollow dots are assigned weight $1$ and they mark the places where we simply apply the Chapman-Kolmogorov formula.
• Figure 2: This is the simplified version of Figure's \ref{['fig:expansion']} graphical representation of the expansion \ref{['laplace_expansion']}, where we have grouped together the blocks of consecutive collisions between the same pair of random walks. These are now represented by the wiggle lines ( replicas) and we call the evolution in strips that contain only one replica as replica evolution (although strip seven is the beginning of another wiggle line, we have not represented it as such since we have not completed the picture beyond that point). The wiggle lines (replicas) between points $(n,x), (m,y)$, corresponding to collisions of a single pair of walks $S^{(k)}, S^{(\ell)}$, are assigned weight $U_N^{\beta_{k,\ell}}(m-n, y-x)$. A solid line between points $(m,x), (n,y)$ is assigned the weight of the simple random walk transition kernel $q_{m-n}(y-x)$.
• Figure 3: A diagramatic representation of a configuration of collisions between $4$ random walks in $H^{(2)}_{4,N}$ with $I_1=\{2,3\}$, $I_2=\{1,2\}$, $I_3=\{3,4\}$ and $I_4=\{2,3\}$. Wiggly lines represent replica evolution, see \ref{['replica_op']}.
• Figure 4: Figure \ref{['fig:scales']} after rewiring. We use blue lines to represent the new kernels produced by rewiring. The dashed lines represent remaining free kernels from the rewiring procedure as well as kernels coming from using the Chapman-Kolmogorov formula for the simple random walk.
• Figure 5: Graphical representation of a term of the chaos representation of $\prod_{1\leq i<j \leq h}\mathrm E[e^{\frac{\pi\beta_{i,j}}{\log N}\, \mathsf L_N^{(i,j)}}]$ for $h=3$. This diagram captures the main contribution, which is when all $a$'s and $b$'s are distinct. Configurations where more than one pair of walks collide at a same time $n\leq N$ have lower order contributions.

Theorems & Definitions (17)

• Theorem A: ET60
• Theorem B: LZ21
• Theorem 1.1
• Lemma 2.1
• Proposition 2.2
• Lemma 2.3
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Lemma 3.4