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

TL;DR

This paper investigates the scaling behavior of exit probability and first passage time for a lazy Pearson walker in two and three dimensions, revealing universal scaling laws and exponents.

Contribution

It introduces a scaling framework for exit probability and first passage time of lazy Pearson walkers, with derived universal exponents.

Findings

Exit probability scales with radius and jump probability with specific exponents.

First passage time distribution exhibits a scaling form with universal exponents.

Scaling functions collapse data across different parameters, indicating universal behavior.

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 $\delta=1$.