I was reading A Gerald's "Integral, Probability and Fractal Measure" and I found the concept of $\bf{random}$ ${\bf homeomorphism}$ and I was wondering about the following:
What kind of applications do random homeomorphisms have in other areas of mathematics, especially those regarding Topological Data Analysis?.
Someone once told me that random homeomorphisms can be used to prove the existence of surfaces of a "special kind", but since I don't remember that "special kind" of surfaces, specifically, I've been unable to find any articles about it.
I would be really thankful if anyone could provide orientation to find articles about the above points.