Critical Mandelbrot Cascades

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.