  We show that random walk on the incipient infinite cluster (IIC) of
two-dimensional critical percolation is subdiffusive in the chemical distance
(i.e., in the intrinsic graph metric). Kesten (1986) famously showed that this
is true for the Euclidean distance, but it is known that the chemical distance
is typically asymptotically larger. More generally, we show that subdiffusivity
in the chemical distance holds for stationary random graphs of polynomial
volume growth, as long as there is a multi-scale way of covering the graph so
that "deep patches" have "thin backbones". Our estimates are quantitative and
give explicit bounds in terms of the one and two-arm exponents $\eta_2 > \eta_1
> 0$: For $d$-dimensional models, the mean chemical displacement after $T$
steps of random walk scales asymptotically slower than $T^{1/\beta}$, whenever
\[
  \beta < 2 + \frac{\eta_2-\eta_1}{d-\eta_1}\,.
  \] Using the conjectured values of $\eta_2 = \eta_1 + 1/4$ and $\eta_1 =
5/48$ for 2D lattices, the latter quantity is $2+12/91$.
