The impact of a random metric upon a diffusing particle

TL;DR

This paper investigates how a specific singularity in a quantized gravity propagator affects the short-time behavior of a diffusing particle's displacement, revealing a power-law relationship influenced by the singularity parameter.

Contribution

It introduces a novel analysis linking the singularity of a quantum gravity propagator to the diffusive behavior of particles over short times.

Findings

Mean fourth power of displacement scales as t^{2(1-γ)} for small t

The singularity in the propagator directly influences diffusive scaling

Provides a mathematical relationship between quantum gravity features and diffusion dynamics

Abstract

We show that if the singularity of the quantized gravity propagator is $\vert {\bf x}\vert^{-2\gamma}$ then the mean value of the fourth power of the distance achieved in time $t$ by a diffusing particle behaves as $t^{2(1-\gamma)}$ for a small $t$.