  We prove a general theorem on cutoffs for symmetric simple exclusion
processes on graphs with open boundaries, under the natural assumption that the
graphs converge geometrically and spectrally to a compact metric measure space
with Dirichlet boundary condition. Our theorem is valid on a variety of
settings including, but not limited to: the $d$-dimensional grid for every
integer dimension $d$; and self-similar fractal graphs and products thereof.
Our method of proof is to identify a rescaled version of the density
fluctuation field---the cutoff martingale---which allows us to prove the mixing
time upper bound that matches the lower bound obtained via Wilson's method.
