Critical Mandelbrot Cascades

TL;DR

This paper investigates the properties of Mandelbrot's critical multiplicative cascade measures, demonstrating their non-atomic nature, multifractal spectrum, and extending KPZ formula validation to critical and low temperatures.

Contribution

It proves the non-atomicity of critical cascade measures, computes their multifractal spectrum, and extends KPZ formula validation to critical and low temperature regimes.

Findings

Critical measures have no atoms.

Computed the multifractal spectrum of the measures.

Extended KPZ formula validation to critical and low temperatures.

Abstract

We study Mandelbrot's multiplicative cascade measures at the critical temperature. As has been recently shown by Barral, Rhodes and Vargas (arXiv:1203.5445), an appropriately normalized sequence of cascade measures converges weakly in probability to a nontrivial limit measure. We prove that these limit measures have no atoms and give bounds for the modulus of continuity of the cumulative distribution of the measure. Using the earlier work of Barral and Seuret (2007), we compute the multifractal spectrum of the measures. We also extend the result of Benjamini and Schramm (2009), in which the KPZ formula from quantum gravity is validated for the high temperature cascade measures, to the critical and low temperature cases.