KPZ formula for log-infinitely divisible multifractal random measures

TL;DR

This paper establishes a KPZ-type relation connecting Euclidean and multifractal Hausdorff dimensions for log-infinitely divisible measures, extending results to higher dimensions in the log-normal case, with implications for quantum gravity.

Contribution

It introduces a KPZ formula for log-infinitely divisible multifractal measures and extends the relation to higher dimensions in the log-normal case.

Findings

Derived a relation between Euclidean and measure-based Hausdorff dimensions.

Extended the KPZ relation to 2D in the log-normal case.

Applicable to multifractal measures in quantum gravity contexts.

Abstract

We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in \cite{bacry} . If M is a non degenerate multifractal measure with associated metric $\rho(x,y)=M([x,y])$ and structure function $\zet a$, we show that we have the following relation between the (Euclidian) Hausdorff dimension ${\rm dim}_H$ of a measurable set K and the Hausdorff dimension ${\rm dim}_H^{\rho}$ with respect to \rho of the same set: $\zeta({\rm dim}_H^{\rho}(K))={\r m dim}_H(K)$. Our results can be extended to higher dimensions in the log normal case: inspired by quantum gravity in dime nsion 2, we consider the 2 dimensional case.