First-passage times to a fractal boundary: local persistence exponent and its log-periodic oscillations

TL;DR

This paper studies the first-passage times to a fractal boundary, revealing power-law decay, log-periodic oscillations, and analyzing the effects of boundary self-similarity and starting point on survival probabilities.

Contribution

It introduces a detailed analysis of first-passage times to fractal boundaries, highlighting log-periodic oscillations and providing theoretical bounds based on diffusion models.

Findings

FPT distribution shows power-law decay with exponential cutoff.

Local persistence exponent exhibits log-periodic oscillations.

Boundary self-similarity influences survival probability behavior.

Abstract

We investigate the statistics of the first-passage time (FPT) to a fractal self-similar boundary of the Koch snowflake. When the starting position is fixed near the absorbing boundary, the FPT distribution exhibits an apparent power-law decay over a broad range of timescales, culminated by an exponential cut-off. By extensive Monte Carlo simulations, we compute the local persistence exponent of the survival probability and reveal its log-periodic oscillations in time due to self-similarity of the boundary. The effect of the starting point onto this behavior is analyzed in depth. Theoretical bounds on the survival probability are derived from the analysis of diffusion in a circular sector. Physical rationales for the refined structure of the survival probability are presented.