A lower bound for pc in range-R bond percolation in four, five and six dimensions

For the range-$R$ bond percolation in $d=4,5,6$, we obtain a lower bound for the critical probability $p_c$ for $R$ large, agreeing with the conjectured asymptotics and thus complementing the corresponding results of Van der Hofstad and Sakai (2005) for $d>6$, and Frei and Perkins (2016), Hong (2023) for $d\le 3$. The lower bound proof is completed by showing the extinction of the associated SIR epidemic model. To prove the extinction of the SIR epidemics, we introduce a refined model of the branching random walk, called a self-avoiding branching random walk, whose total range dominates that of the SIR epidemic process.