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.