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 $P_e{p}^{\alpha }=F(r_e{p}^{\beta })$, 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 $\delta =1$.

10 Pages Latex with 4 figures. arXiv admin note: text overlap with arXiv:1506.00096

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