KPZ formula for log-infinitely divisible multifractal random measures
By Rémi Rhodes, Vincent Vargas. 2008
157 citations recorded by the source corpus.
We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in [E. Bacry et al. Comm. Math. Phys. 236 (2003) 449–475]. If M is a non degenerate multifractal measure with associated metric ρ ( x,y ) = M ([ x,y ]) and structure function ζ, we show that we have the following relation between the (Euclidian) Hausdorff dimension dim H of a measurable set K and the Hausdorff dimension dim H ρ with respect to ρ of the same set: ζ(dim H ρ ( K )) = dim H ( K ). Our results can be extended to all dimensions: inspired by quantum gravity in dimension 2 , we focus on the log normal case in dimension 2 .
Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/b25de23489b9f0562826494bc52cd05e96aeb730
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In Geometry and Topology
In Mathematical Physics
- Rémi Rhodes
- Vincent Vargas
- A. Kupiainen
- F. David
- Martin Hairer
- Scott Sheffield
- Eero Saksman
- Thomas Madaule
In Statistical and Nonlinear Physics
Citations in the corpus
- Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012, unknown): “In this context, the KPZ formula has been proved rigorously [30], as well as in the general context of Gaussian multiplicative chaos [69]; see also [13] in the context of Mandelbrot’s multiplicative cascades.”
- Liouville Quantum Gravity on the Riemann Sphere (2014, background): “Up to now such studies have exclusively focused on the incarnation of the Liouville theory as a free field theory (the Gaussian Free Field or GFF), where the parameter μ is set to zero2: see for instance [21, 46] and the review [25].”
- Gaussian multiplicative chaos and applications: A review (2013, methodology): “We will explain the geometrical KPZ formulae rigorously proved in [56, 128] (see also [27, 17]).”