I have been given a list of eigenvalues and, sorting them in decreasing order, I notice that their values decay very nicely with a power law. Actually, these eigenvalues have been obtained from a POD (proper orthogonal decomposition) analysis; so sorting according to the eigenvalue order number and value is the same (principle of optimality, if I got it right).
For me this regular decaying of the values is just a naive observation, yet fascinating. Is there a special meaning attached to this type of decay and to its parameter? Can this behaviour be anticipated based on overarching properties of something else?
Disclaimer: I am no mathematician, please accompany formalism with commentary if possible
What you are seeing is an instance of universality. Just like primes and energy levels, eigenvalues of nearby matrices don't like to be close to each other. The results in power-law decay. This gives more general background.
Specifically for your case of a single matrix, a useful formula is Equation 12 in this paper, which gives distribution of gaps between nearby eigenvalues for a large random matrix.