The skew Brownian permuton

The skew Brownian permuton is a universal family of limiting permutons introduced in this work.

Here we collect some simulations of the skew Brownian permuton \(\mu_{\rho,q} \) for different values of the parameters \(\rho\in(-1,1]\) and \(q\in[0,1]\).

In every row there are five simulations of \(\mu_{\rho,q} \) and at the end there is the corresponding two-dimensional Brownian excursion of correlation \(\rho \) in the non-negative quadrant (the specific value of \(\rho \)  is indicated at the beginning of every row). In each row, moving from left to right,  there are increasing values for the parameter \(q \) (roughly \(q\approx 0.1,0.4,0.5,0.6,0.9 \) ).

We highlight that permutons in the same row are driven by the same Brownian excursion plotted at the end of the row and so they are coupled. Note that when \(\rho=1 \)  (this is the case of the last row) the corresponding two-dimensional Brownian excursion is simply a one-dimensional Brownian excursion and it is plotted using the standard diagram for real-valued functions.

\(\rho=-0.995 \)
\(\rho=-0.5 \)
\(\rho=0 \)
\(\rho=0.5 \)
\(\rho=0.995 \)
\(\rho=1 \)

Here \(q= 1/2\) and \(\rho= \)-0.999999999,-0.9999999,-0.99999, -0.99994,-0.9999,-0.9994,-0.999,-0.994,-0.99,-0.98,-0.97,-0.96,-0.95-0.94,-0.92,-0.91,-0.9,-0.8,-0.7,-0.6,-0.5,-0.4,-0.3,-0.2,-0.1,0,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.995,0.99999