[Paper Review] Projections of Mandelbrot percolation in higher dimensions

Károly Simon, Lajos Vágó|arXiv (Cornell University)|Jul 8, 2014

Stochastic processes and statistical mechanics5 references4 citations

TL;DR

This paper extends results on projections of Mandelbrot percolation sets from the plane to higher dimensions, proving that under suitable conditions on the retention probabilities, orthogonal, radial, and co-radial projections of the random fractal set almost surely contain non-empty interiors in all directions when conditioned on non-emptiness. The key contribution is a generalization of earlier planar results to d-dimensional fractal percolation using a modified almost linear family of projections and concentration inequalities.

ABSTRACT

We consider fractal percolation (or Mandelbrot percolation) which is one of the most well studied example of random Cantor sets. Rams and the first author studied the projections (orthogonal, radial and co-radial) of fractal percolation sets on the plane. We extend their results to higher dimension.

Motivation & Objective

• To generalize the results of Rams and Simon (2013) on projections of planar fractal percolation to higher-dimensional settings.
• To establish that for d-dimensional Mandelbrot percolation with Hausdorff dimension greater than k, projections onto k-dimensional subspaces contain intervals almost surely when conditioned on non-emptiness.
• To handle the increased technical complexity in higher dimensions, particularly in controlling the variation of projection directions across dyadic cubes.
• To extend Falconer and Grimmett's result on coordinate-plane projections to all projections simultaneously in higher dimensions.
• To verify that radial and co-radial projections in R^d (e.g., shadows of a 3D set) contain disks when the set has dimension > d−1.

Proposed method

• Introduces an 'almost linear family of projections' defined by a Lipschitz selection of projection directions α_t(x), ensuring that projections of dyadic cubes remain connected.
• Uses a hierarchical dyadic decomposition of the unit cube, labeling level-n cubes via d×n matrices with entries in {0,…,M−1}, and defines random sets E_n via independent retention with given probabilities.
• Applies a modified version of the Azuma-Hoeffding inequality to control the growth of the number of cubes projecting into a fixed target region, using the random variable V_n(x,t) counting such cubes.
• Imposes Condition A(α) on projection families, ensuring sufficient geometric spread and uniformity in the projection behavior across scales and locations.
• Reduces radial and co-radial projections to linear-like projections via geometric equivalence, allowing application of the main theorem to these cases.
• Uses statistical self-similarity of Mandelbrot percolation to reduce the analysis to projections from distant centers, ensuring the almost linear condition holds uniformly.

Experimental results

Research questions

• RQ1 Under what conditions do orthogonal projections of d-dimensional Mandelbrot percolation sets onto k-dimensional subspaces contain non-empty interiors almost surely, given non-emptiness?
• RQ2 Can the planar results of Rams and Simon (2013) on interval-valued projections be extended to all directions in higher dimensions?
• RQ3 How can the technical challenges of non-uniform projection directions in higher dimensions be overcome to ensure uniform control over projection geometry?
• RQ4 Do radial and co-radial projections of 3D Mandelbrot percolation sets contain disks in their shadows when the set has dimension greater than 2?
• RQ5 Is it possible to simultaneously guarantee interval-valued projections in all directions for d-dimensional fractal percolation under a natural extension of the 2D condition?

Key findings

• For d-dimensional Mandelbrot percolation with retention probabilities satisfying a generalized condition (analogous to p > 1/M in the homogeneous case), projections onto any k-dimensional subspace contain non-empty interiors almost surely when conditioned on non-emptiness.
• The result holds for orthogonal, radial, and co-radial projections, extending the planar result to R^d for d ≥ 3.
• In particular, if the 3D Mandelbrot percolation set has Hausdorff dimension greater than 2, then for almost every realization, the shadow of the set under radial projection contains a disk regardless of the sun's position.
• The proof relies on constructing an almost linear family of projections where the direction map α_t(x) is Lipschitz and varies slowly across dyadic cubes, ensuring uniformity.
• The number of level-n cubes projecting into a fixed target region grows at least as (3/2)^n with positive probability, implying the existence of a dense set of points in the projection.
• The method successfully handles the increased complexity of higher-dimensional projections by subdividing parameter spaces and using concentration inequalities to control fluctuations in cube counts.