Improved lower bound on monotone circuit complexity of perfect matching? Razborov proved that every monotone circuit that computes the perfect matching function for bipartite graphs must have at least $n^{\Omega(\log n)}$ gates (he called it "logical permanent"). Has a better lower bound for the same problem been proved since then? (say $2^{n^\epsilon}$?) As far as I remember this problem was open in the mid 1990's.

I am aware that the clique function requires exponential-size monotone circuits and so on, but I am interested in perfect matching specifically.
Eva Tardos proved that the gap is truly exponential by showing that there is a monotone boolean function that has poly size circuits but requires exponential size monotone circuits. Nothing better than super-polynomial is known for matching.

Raz has a result that monotone circuits for matching have linear depth. (Thanks Klauck, for pointing the typo.) 

AFAIK, we know nothing better.

Ref: (1) http://www.springerlink.com/index/P25X5838624J0352.pdf

(2) http://www.wisdom.weizmann.ac.il/~ranraz/publications/Pmatching.ps