A random walk on a non-intersecting two-sided random walk trace is subdiffusive in low dimensions

Let $(\overline {S}^{1}, \overline {S}^{2})$ be the two-sided random walks in $\mathbb{Z}^{d} \ (d=2,3)$ conditioned so that $\overline {S}^{1}[0,\infty ) \cap \overline {S}^{2}[1, \infty ) = \emptyset$, which was constructed by the author in 2012. We prove that the number of global cut times up to $n$ grows like $n^{\frac {3}{8}}$ for $d=2$. In particular, we show that each $\overline {S}^{i}$ has infinitely many global cut times with probability one. Using this property, we prove that the simple random walk on $\overline {S}^{1}[0,\infty ) \cup \overline {S}^{2}[0,\infty )$ is subdiffusive for $d=2$. We show the same result for $d=3$.