KPZ in one dimensional random geometry of multiplicative cascades

TL;DR

This paper establishes a formula linking Hausdorff dimensions of sets in one-dimensional random geometries generated by multiplicative cascades, connecting to the KPZ formula when cascade variables are exponentials of Gaussians.

Contribution

It derives a new relation between Hausdorff dimensions in one-dimensional random geometries and connects it to the KPZ formula in a novel setting.

Findings

Derived a formula relating Hausdorff dimensions in random geometries.

Connected the formula to the KPZ relation for Gaussian exponential cascades.

Facilitated direct determination of Hausdorff dimension in this framework.

Abstract

We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity appears. This note was inspired by the recent work of Duplantier and Sheffield proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension.