Once-reinforced random walk on $\mathbb{Z}^d$ has range exponent at least $d/(d+1)$

In once-reinforced random walk, introduced by Davis (1990), every edge starts with weight one and is assigned weight $β\geq1$ after its first crossing. At each step the walker chooses an incident edge with probability proportional to its weight. For sufficiently large reinforcement $β$, the expected