On the rate of escape or approach to the origin of a random string

TL;DR

This paper investigates the behavior of a random string model in high dimensions, analyzing how quickly it escapes to infinity in dimensions seven and above, and approaches the origin in dimension six.

Contribution

It extends previous results by Mueller and Tribe, providing new bounds on the escape rate in high dimensions and the approach rate in dimension six.

Findings

Established the escape rate in dimensions d ≥ 7.

Provided bounds for the approach rate in dimension d = 6.

Extended the theoretical understanding of Funaki's random string model.

Abstract

In this paper, we extend upon a result by Mueller and Tribe regarding Funaki's model of a random string. Specifically, we examine the rate of escape of this model in dimensions $d \ge 7$. We also provide a bound for the rate of approach to the origin in dimension $d = 6$.