Persistent homology of Gaussian fields in Euclidean space

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.

The closest I can find spontaneously would be Matthew Kahle's work on random topology; http://arxiv.org/abs/0910.1649 looks like it would be directly related to your question, and http://arxiv.org/abs/1009.4130 seems related too.

For what it's worth, Laura Balzano and I ran some experiments on this precise question for Gaussian clouds in R^2, trying to understand what kind of barcodes are rare under this "null hypothesis" of data without topological structure. We focused on the question "how long a bar in R^1 constitutes a surprisingly long bar?" and ran some tests on this. No theorems, though. I agree with your implicit assertion that this is an important question for the theory!

http://www.math.wisc.edu/~ellenber/topoData_icassp-3.pdf