If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be happening, with the barcodes tending towards something like a "wing" shape, fat in lower dimensions, thinning out towards dimension $n$.
Has anyone proven any theorems that describe the asymptotic "shape" of the barcodes?
Ideally I'd like a test so that I can look at some barcodes and say "that's typical of a Gaussian normal distribution".
The closest thing I've been able to find is experiments and results on the expected Euler characteristic of the persistent homology, in the following two references (arXiv links): Persistent homology for random fields and complexes, Euler integration of Gaussian random fields and persistent homology.
Edit:
I did a very rough computation to try and get some kind of guess as to what the distribution of barcodes should look like. So I made a very coarse estimate based on a distribution of points that is roughly `locally cubical' and approximately respecting a normal distribution.
The density is given by:
$$\mu = N e^{-r^2}$$
where $r$ is the distance from the origin. Then if $\epsilon$ is the parameter for persistent homology, it appears that $H_0$ is rank approximately
$$N \int_{\sqrt{\ln(N\epsilon^{1/n})}}^\infty r^{n-1}e^{-r^2} dr$$
and $H_k$ for $k \in \{1,2,\cdots,n-1\}$ has rank approximately
$$ {n \choose k+1}\frac{(\sqrt{\ln(N\epsilon^{1/n}/\sqrt{k}))}^{n-2}}{4\sqrt{k}\epsilon^{1/n}} $$
These are fairly coarse estimates, and in no way rigorous. But if something like this is actually true it seems to be saying that for $N$ large and $n \geq 3$, the $H_0$ betti number tends to some asymptote (dependent on $\epsilon$), and $H_1, \cdots, H_{n-1}$ are non-trivial but small. So most of the points in the distribution are in a giant homology `black hole' at the centre and persitent homology sees the thin crust around the outside.
I'd be curious if people have done other similar guestimates (or better) and if they had similar-looking results.
Best Answer
Adler, Bobrowski and Weinberber's "Crackle: The Persistent Homology of Noise" is an answer to my question. I have not read it closely yet but it appears to confirm the guess in the question, and provide answers for other distributions as well.
Although this paper does not target my question directly it gives a more quantitative answer to a nearby question, that of the length of the largest barcode for certain types of random point clouds. Maximally persistent cycles in random geometric complexes.