Grand Canyon, October 2005. 

Uniformly sampled homomorphism and Lipschitz functions in 2 and 3 dimensions Left column: homomorphism and Lipschitz functions on a 100 x 100 square with zero boundary values Right column: middle slice of homomorphism and Lipschitz functions on a 100 x 100 x 100 cube with zero boundary values 
Top: the outermost level sets separating zeros and ones of a uniformly sampled homomorphism on a 40 x 40 and 300 x 300 squares with zero boundary values (pictures produced with the help of Steven M. Heilman) Bottom: The shift transformation applied to the level set of a homomorphism function. This transformation is a major tool in the analysis of homomorphism functions in high dimensions 
Gradient Flow / Gravitational Allocation (pictures based on code by Manjunath Krishnapur) First row: Allocation to the zeros of the planar, hyperbolic and spherical canonical Gaussian Analytic Functions (all cells have equal areas!) Second row: The potential for the allocations to the planar and hyperbolic Gaussian Analytic Functions. 

Kwise independent percolation 
4 coloring of PoissonVoronoi map. 
Rough isometry of 1D percolations 
Brownian motion on a geometric state space and on the Cantor set (pictures courtesy of Peter Ralph) 