First Order Transition for the Optimal Search Time of Lévy Flights with Resetting

We study analytically an intermittent search process in one dimension. There is an immobile target at the origin and a searcher undergoes a discrete time jump process starting at $x_0\ge 0$, where successive jumps are drawn independently from an arbitrary jump distribution $f(\eta )$. In addition, with a probability $0\le r<1$, the position of the searcher is reset to its initial position $x_0$. The efficiency of the search strategy is characterized by the mean time to find the target, i.e., the mean first passage time (MFPT) to the origin. For arbitrary jump distribution $f(\eta )$, initial position $x_0$ and resetting probability $r$, we compute analytically the MFPT. For the heavy-tailed Lévy stable jump distribution characterized by the Lévy index $0<\mu <2$, we show that, for any given $x_0$, the MFPT has a global minimum in the $(\mu ,r)$ plane at $\mathbf{(}{\mu }^{*}(x_0),r^{*}(x_0)\mathbf{)}$. We find a remarkable first-order phase transition as $x_0$ crosses a critical value $x_0^{*}$ at which the optimal parameters change discontinuously. Our analytical results are in good agreement with numerical simulations.

Figure 1

Illustration of the search strategy which combines long jumps (Lévy flights) and random resettings, with probability $r$, at the initial position $x_0$. Here, the search time, i.e., the first passage time in 0 where the target is located is $T_{x_0}(\mu ,r)=9$ while there have been two resettings, at step 4 and 7. The integers $n$ and $m$ with $n=6$ and $m=2$ here illustrate the notations in the renewal equation in Eq. (4).

Figure 2

2D plots of the average search time $⟨T_{x_0}(\mu ,r)⟩$, computed using numerical simulations, in the $(\mu ,r)$ plane for different values of the initial position $x_0$: (a) $x_0=0.56<x_0^{*}$, (b) $x_0=x_0^{*}\simeq 0.58$, and (c) $x_0=0.65>x_0^{*}$.

Figure 3

Plot of the optimal parameters ${\mu }^{*}(x_0)$ and $r^{*}(x_0)$, obtained from numerical simulations, as a function of $x_0$. Both of them exhibit a clear discontinuity for $x_0=x_0^{*}\simeq 0.58$, reminiscent of a first order phase transition.

Figure 4

$⟨T_{x_0}(\mu ,r)⟩$ vs $r$—comparison between numerical results for small $\mu$ and analytical prediction for $\mu \to 0$.