Conditional expectation of the duration of the classical gambler problem with defects

TL;DR

This paper analyzes how a defect site affects the expected crossing time of a 1D random walk, revealing non-monotonic behaviors and conditions where defects can actually reduce residence time, with implications for gambler's ruin scenarios.

Contribution

It introduces a detailed analysis of inhomogeneities in 1D random walks, connecting to gambler's ruin, and provides both numerical and analytical results on residence time behavior.

Findings

Defects can decrease residence time when placed at certain positions.

Residence time exhibits non-monotonic dependence on defect parameters.

Analytical and Monte Carlo methods agree on key behaviors.

Abstract

The effect of space inhomogeneities on a diffusing particle is studied in the framework of the 1D random walk. The typical time needed by a particle to cross a one--dimensional finite lane, the so--called residence time, is computed possibly in presence of a drift. A local inhomogeneity is introduced as a single defect site with jumping probabilities differing from those at all the other regular sites of the system. We find complex behaviors in the sense that the residence time is not monotonic as a function of some parameters of the model, such as the position of the defect site. In particular we show that introducing at suitable positions a defect opposing to the motion of the particles decreases the residence time, i.e., favors the flow of faster particles. The problem we study in this paper is strictly connected to the classical gambler's ruin problem, indeed, it can be thought as that problem in which the rules of the game are changed when the gambler's fortune reaches a particular a priori fixed value. The problem is approached both numerically, via Monte Carlo simulations, and analytically with two different techniques yielding different representations of the exact result.