Are you using LWJGL 2.8.3 or the nightly build? There is an updated Display.sync which may fix your problem.
Im using the nightly build.
The spike could also be caused by Java's garbage collector. Are you creating any garbage?
I don't know. I don't i'm creating enough for it to have a noticable effect. Im actually starting to believe that theres something wrong with my drivers or something. I let a few of my friends test the game and none of them could notice the lag.
Edit: Nope! Updating my drivers did not change anything.