Examples of successful derandomization from BPP to P What are some major examples of successful derandomization or at least progress in showing concrete evidence towards $P=BPP$ goal (not the hardness randomness connection)?

The only example that comes to my mind is AKS deterministic polynomial time primality testing (even for this there was a methodology assuming GRH). So what specific evidence through example do we have for derandomization (again not the hardness or oracle connection)?

Please keep examples to only where time complexity improvement was shown from randomized poly to deterministic poly or something that is very close for specific problems.



Following is more of a comment and I do not know much it will help this query.

Chazelle has a very intriguing statement in http://www.cs.princeton.edu/~chazelle/linernotes.html under 'The Discrepancy Method: Randomness and Complexity   (Cambridge University Press, 2000)'.

'It's been an endless source of fascination for me that a deeper understanding of deterministic computation should require the mastery of randomization. I wrote this book to illustrate this powerful connection. From minimum spanning trees to linear programming to Delaunay triangulations, the most efficient algorithms are often derandomizations of probabilistic solutions. The discrepancy method puts the spotlight on one of the most fruitful questions in all of computer science: if you think you need random bits, please tell us why?'
$SL = L$.

$RL$ stands for randomized logspace and $RL=L$ is a smaller version of the problem $RP=P$. A major stepping stone was the proof of Reingold in '04 ("Undirected S-T Connectivity in Logspace") that $SL = L$, where $S$ stands for "symmetric" and $SL$ is an intermediate class between $RL$ and $L$.

The idea is that you can think of a randomized logspace Turing machine as an polynomial-sized directed graph, where nodes are states of the machine, and an RL algorithm takes a random walk that has good properties. SL corresponds to undirected graphs of this form. Reingold's proof built on work on expander graphs, particularly Reingold, Vadhan, and Wigderson's "zig-zag product", to take any random walk on an undirected graph with good properties and turn it into a psuedorandom walk retaining those properties.

edit this question was posted before the question was explicitly changed to focus exclusively on P vs BPP ... I am leaving it up because it seems to be of interest.