First passage on disordered intervals

We derive unexpected first-passage properties for nearest-neighbor hopping on finite intervals with disordered hopping rates, including (a) a highly variable spatial dependence of the first-passage time, (b) huge disparities in first-passage times for different realizations of hopping rates, (c) significant discrepancies between the first moment and the square root of the second moment of the first-passage time, and (d) bimodal first-passage time distributions. Our approach relies on the backward equation, in conjunction with probability generating functions, to obtain all moments, as well as the distribution of first-passage times. Our approach is simpler than previous approaches based on the forward equation, in which computing the $m\mathrm{th}$ moment of the first-passage time requires all preceding moments.

Figure 1

Hopping rates for the interval with (a) two absorbing boundaries, and (b) one reflecting and one absorbing boundary.

Figure 2

(a) Mean first-passage times on a disordered interval of length $N=100$ for three realizations of the hopping rates as a function of starting position $i$ (solid). The dashed curves show the square root of the second moment of the first-passage time for these same realizations. The red solid curve shows the mean first-passage time of the homogeneous interval with $b_i=d_i=1/3$ for all $i$. (b) The local bias $ln(b_i/d_i)$ [smoothed over a 10-point range (green) and by a Bezier curve (black)] for the realization in black in (a). The arrows show the direction of the local average bias.

Figure 3

(a) The average deviation in Eq. (12) when a finite fraction $f$ of all hopping-rate realizations is sampled for an interval length $N=12$. (b) The average deviation as a function of $f$ for various starting positions along the interval.

Figure 4

(a) The first-passage probability $f_2\left(t\right)$ on the interval [0,10]. The schematic shows the local bias $ln(b_i/d_i)$ at each site. The inset shows the explicit values of $ln(b_i/d_i)$. (b) The first-passage probability $f_6\left(t\right)$ for an interval of length $N=20$ in which the synthetically generated bias changes sign at $i=5$ with $\left|v\right|=|b_i-d_i|=0.3$.