My question is rather general - what is known about derandomization of results in random matrix theory, high-dimensional geometry, Banach spaces etc. using probabilistic constructions (like estimates of eigenvalues, Dvoretzky's theorem, metric embeddings)? Here I'm interested both in fully explicit counterparts of random constructions as well as "pseudorandom" (in some sense) examples, using "less" randomness than, say, filling every entry of a matrix with a random variable etc. For example - suppose we know that for a fixed norm an n x n matrix with IID standard gaussian entries has "large" norm with high probability. How to find an explicit infinite family of such matrices?

My question is rather vague, of course I have a specific application of this kind of results in mind, but at this moment I am more interested in general methodology of constructing "derandomized" examples, where to start looking for such objects etc. My only contact so far with pseudorandomness has been in the context of spectral graph theory, expander graphs, property (T) etc., I'm not sure if this perspective is relevant for high-dimensional geometry.

I'd be grateful for any hints, references or advice on who may know this kind of things.