Consider Varadhan's famous formula for the kernel of the heat equation on a manifold:

$$ \lim_{t \rightarrow 0} t \log h(t,x,y) = - \frac{d(x,y)^2}{4} .$$

I do not have access to his 1967 two papers, but from what I can understand Varadhan obtained this as a byproduct of a much more general probabilistic result (something about stochastic processes, apparently) and by using probabilistic methods and concepts, hence my question: has anybody found in the meantime a more straightforward, purely geometric (Riemannian) proof of the above formula? (I would be happy even with a slightly weaker result than the above, but not much weaker.)