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

Abstract

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 ${P_e}p^α=F({r_e}p^β)$, 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 ${P(t_1)}p^γ = G({t_1}p^δ)$. Where, $F$ and $G$ are two scaling functions and $α$, $β$, $γ$ and $δ$ are some exponents. In both the dimensions, it is found that, $α= 0$, $β=-1/2$, $γ=-1$ and $δ=1$.