Critical number of walkers for diffusive search processes with resetting

We consider $N$ Brownian motions diffusing independently on a line, starting at $x_0>0$, in the presence of an absorbing target at the origin. The walkers undergo stochastic resetting under two protocols: (A) each walker resets independently to $x_0$ with rate $r$ and (B) all walkers reset simultaneously to $x_0$ with rate $r$. We compute analytically the mean first-passage time to the origin and show that, as a function of $r$ and for fixed $x_0$, it has a minimum at an optimal value $r^{\ast }>0$ as long as $N<N_c$. Thus resetting is beneficial for the search for $N<N_c$. When $N>N_c$, the optimal value occurs at $r^{\ast }=0$ indicating that resetting hinders search processes. Continuing our results analytically to real $N$, we show that $N_c=7.3264773\dots$ for protocol A and $N_c=6.3555864\dots$ for protocol B, independently of $x_0$. Our theoretical predictions are verified in numerical Langevin simulations.