Dimension result and KPZ formula for two-dimensional multiplicative cascade processes

TL;DR

This paper establishes a Hausdorff dimension result for two-dimensional multiplicative cascade processes and derives a KPZ-type formula with a phase transition point, advancing understanding of fractal geometry in stochastic processes.

Contribution

It provides a new Hausdorff dimension theorem and a KPZ formula for two-dimensional multiplicative cascades, linking fractal geometry with probabilistic models.

Findings

Hausdorff dimension result for the image of 2D multiplicative cascades

Derivation of a KPZ-type formula with a phase transition

Identification of a phase transition point in the KPZ formula

Abstract

We prove a Hausdorff dimension result for the image of two-dimensional multiplicative cascade processes, and we obtain from this result a KPZ-type formula which normally has one point of phase transition.