Exit probability and first passage time of a lazy Pearson walker: Scaling behaviour

The motion of a lazy Pearson walker is studied with different probability ($p$) of jump in two and three dimensions. The probability of exit ($P_e$) from a zone of radius $r_e$, is studied as a function of $r_e$ with different values of jump probability $p$. The exit probability $P_e$ is found to scale as ${P_e}p^{\alpha}=F({r_e}p^{\beta})$, which is obtained by method of data collapse. The first passage time ($t_1$) i.e., the time required for first exit from a zone is studied. The probability distribution ($P(t_1)$) of first passage time was studied for different values of jump probability ($p$). The probability distribution of first passage time was found to scale as ${P(t_1)}p^{\gamma} = G({t_1}p^{\delta})$. Where, $F$ and $G$ are two scaling functions and $\alpha$, $\beta$, $\gamma$ and $\delta$ are some exponents. In both the dimensions, it is found that, $\alpha = 0$, $\beta=-1/2$, $\gamma=-1$ and $\delta=1$.