The simulations in this page are for this work on the high-dimensional theory for permutons.
The Schnyder wood permuton & the Brownian separable 3-permuton
Simulations for two 3-dimensional permutons. Each simulation is spinning along the Z-axis (w.r.t. the notation used in our paper).
The 3-dimensional permuton associated with a permutation of size 10000 sampled from the Schnyder wood permuton.
The 3-dimensional permuton associated with a permutation of size 10000 sampled from the Brownian separable 3-permuton with parameters (1/2,1/2).
Some additional simulations for Schnyder wood permutations:
In this simulation we repeatedly sampled 10000 steps of the unconditioned 2D random walk W* and considered a sample paths started at zero driven according to the rules of the “red process” (e.g. the right side of Figure 8 of our paper). Under rescaling (so by sqrt(10000) above the x-axis and sqrt(20000) below the x-axis), the height of the endpoint should converge to a skew Brownian motion of parameter q. The line chart shows that simulations (in blue) do match this curve (in green) and that the number of walks that end up positive is quite close to 200000 * 1 / (1 + sqrt(2)) ~ 82843.
Remark 1.16 of our paper claims that the expected frequency of inversions for the marginals of uniform SW permutations is 2/3. The histogram shows the inversion counts of the marginal sigma_M^g for 250 uniform Schnyder wood permutations of size 50, and indeed the average is quite close to 2/3 * (50 choose 2) ~ 816.7.