Zero error randomized communication complexity vs deterministic communication complexity
Indeed, for distjointness of sets of size $\log(n)$ out of $n$ items, it is known that 
the $0$-error randomized communication complexity is $\Theta(\log n)$, while the deterministic
complexity is $\Theta(\log^2 n)$.  

Recall that there can be at most a quadratic gap since the $0$-error randomized complexity is bounded from below by the non-deterministic and co-non-deterministic complexities.

See: http://mirror.theoryofcomputing.org/articles/v003a011/v003a011.pdf