2008/06/08 by Itai Benjamini, Itaï Benjamini, Oded Schramm · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #math-ph #math.MP #math.PR #msc:60D05
paper · pdf · doi:10.1007/s00220-009-0752-1
14 pages
arxiv created 2008/06/08 · openalex publication_date 2009/02/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.