A very rough sketch of a proof that random walk excited to the center is recurrent in all dimensions can be found here. The main lemma is only explained by examples, not proved. The proof dates from 2006, but this sketch was only written in 2013.
A sketch of a proof that Gromov random groups do not have low-rank quotients is here. Dates to approximately 2011. My proof of what happens at density half was included in Duchin et al., A sharper threshold for random groups at density one-half, see here.
Here is a construction of a measure mu on [0,1] such that if you take random samples from mu X mu and then take their convex hull, then the number of sides of the resulting polygon is tight as the number of samples goes to infinity. Dates to 2010.
Cover time for branching random walk on graphs, with Itai Benjamini, circa 2007. Download here. Why was this written but never published (even on the arxiv)? לעת עתה זה סוד
The "cccp" draft which circulated is now published (still with no proofs). See Itai Benjamini's book, Coarse geometry and Randomness, section 10.2.
The discrete curvature draft with Bo'az Klartag which circulated and was even referenced was included in a paper with Ralli and Tetali, see here.
A construction of a stationary sequence of orthonormal functions which does not satisfy Carleson's theorem can be found here. The proof is complete. Dates to about 2006.
Strong uniform distribution — the case of infinitely many primes, circa 2001. Download here. The opposite result was published in Lacroix, 2004. I do not know who is correct.