Abstract: If lengths 1 and 2 are assigned randomly to each edge in the planar grid, what are the fluctuations of distances between far away points?
This problem is open, yet we know, in great detail, what to expect.
The directed landscape, a universal random plane geometry, provides the answer to such questions.
In some models, such as directed polymers, the stochastic heat equation, or the KPZ equation, random plane geometry hides in the background.
Principal component analysis, a fundamental statistical method, comes to the rescue: BBP statistics can be used to show that these models converge to the directed landscape.