Remark about heat diffusion on periodic spaces

Let $M$ be a complete Riemannian manifold with a free cocompact ${\Bbb Z}^k$-action. Let $k(t, m_1, m_2)$ be the heat kernel on $M$. We compute the asymptotics of $k(t, m_1, m_2)$ in the limit in which $t \rightarrow \infty$ and $d(m_1, m_2) \sim \sqrt{t}$. We show that in this limit, the heat diffusion is governed by an effective Euclidean metric on ${\Bbb R}^k$ coming from the Hodge inner product on $\mathrm{H}^1(M/{\Bbb Z}^k; {\Bbb R})$.