Intuitively speaking, a point pattern is said to be strongly rigid if the boundary conditions fix the positions of all the points. As the name indicates, it is a strong property. For some time, it was believed that any rigid point pattern was hyperuniform. Together with Günter Last, I disproved this conjecture by providing a counter example that is stationary and ergodic.
Michael Andreas Klatt and Günter Last, On strongly rigid hyperfluctuating random measures, Journal of Applied Probability 59, 948–961 (2022).
Therefore, we studied hyperplane intersection processes (HIPs). These point patterns are formed by the vertices of random hyperplanes. These HIPs are known to be hyperfluctuating, that is, the variance of the number of points in an observation window grows faster than the size of the window, which is the opposite of hyperuniformity.
Nevertheless, these point patterns exhibit a particularly strong form of rigidity. Even exponentially small set around the observation window suffices to reconstruct the position of all points inside the window.
Michael Andreas Klatt 