Characteristics of a random walk on a self-inflating support

TL;DR

This paper investigates how a random walk behaves on a dynamically inflating space, revealing that inflation type affects self-similarity and alters the dynamical exponent, with implications for understanding processes on evolving backgrounds.

Contribution

It introduces a model of a random walk on an inflating space and analyzes how different inflation rates impact self-similarity and the dynamical exponent.

Findings

Exponential inflation destroys self-similarity of the random walk.

Power-law inflation preserves self-similarity if the growth exponent is small.

The dynamical exponent z varies continuously with the inflation control exponent.

Abstract

Self-similar dynamical processes are characterized by a growing length scale $\xi$ which increases with time as $\xi \sim t^{1/z}$, where z is the dynamical exponent. The best known example is a simple random walk with z=2. Usually such processes are assumed to take place on a static background. In this paper we address the question what changes if the background itself evolves dynamically. As an example we consider a random walk on an isotropically and homogeneously inflating space. For an exponentially fast expansion it turns out that the self-similar properties of the random walk are destroyed. For an inflation with power-law characteristics, however, self-similarity is preserved provided that the exponent controlling the growth is small enough. The resulting probability distribution is analyzed in terms of cumulant ratios. Moreover, the dynamical exponent z is found to change continuously with the control exponent.