Random walk on a linear chain with a quenched distribution of jump lengths

We study the random walk of a particle on a linear chain, where a jump length 1 or 2 is assigned randomly to each lattice site with probability ${\mathrm{p}}_1$ and ${\mathrm{p}}_2$=1-${\mathrm{p}}_1$, respectively. We find that the probability ${\mathrm{p}}_1^{\mathrm{eff}}$ for the particle to be at a site with jump length 1 is different from ${\mathrm{p}}_1$, which causes the diffusion coefficient D to differ from the mean-field result. A theory is developed that allows us to calculate ${\mathrm{p}}_1^{\mathrm{eff}}$ and D for all values of ${\mathrm{p}}_1$. In the limit ${\mathrm{p}}_1$→0, the theory yields a nonanalytic dependence of ${\mathrm{p}}_1^{\mathrm{eff}}$ on ${\mathrm{p}}_1$,${\mathrm{p}}_1^{\mathrm{eff}}$∼-${\mathrm{p}}_1^2$ln ${\mathrm{p}}_1$.