Infinite collision property for the three-dimensional uniform spanning tree

Let $\mathcal{U}$ be the uniform spanning tree on ${\mathbb{Z}}^3$, whose probability law is denoted by $\mathbf{P}$. For $\mathbf{P}$-a.s. realization of $\mathcal{U}$, the recurrence of the the simple random walk on $\mathcal{U}$ is proved in [5] and it is also demonstrated in [8] that two independent simple random walks on $\mathcal{U}$ collide infinitely often. In this article, we will give a quantitative estimate on the number of collisions of two independent simple random walks on $\mathcal{U}$, which provides another proof of the infinite collision property of $\mathcal{U}$.