The images on this page are taken from this work on the scaling and local limits of random Young tableaux.
The heart example
Figures for the heart example. Top-left: The Young diagram \(\lambda^0\) considered in the heart example. Top-right: The frozen boundary curve of the corresponding liquid region. Bottom (from left to right): a uniform random tableau of shape \(\lambda_N\) with \(N=130000\) boxes displayed as a discrete surface in a 3D space (brown is used for small values of the surface and blue for large ones); the corresponding bead processes \(M_{\lambda_N}\); the corresponding height function \(H_{\lambda_N}\).
The pipe example
Figures for the pipe example. Top-left: The Young diagram \(\widetilde \lambda^0\) considered in the pipe example. Top-right: The frozen boundary curve of the corresponding liquid region. Bottom (from left to right): a uniform random tableau of shape \(\lambda_N\) with \(N=59400\) boxes displayed as a discrete surface in a 3D space (brown is used for small values of the surface and blue for large ones); the corresponding bead process \(M_{\lambda_N}\); the corresponding height function \(H_{\lambda_N}\). The red circle in the picture in the left-hand side highlights the discontinuous location of the limiting surface.
See also here the nice 3D pictures made by Istvan Prause.