  We introduce a new framework for analyzing Glauber dynamics for the Ising
model. The traditional approach for obtaining sharp mixing results has been to
appeal to estimates on spatial properties of the stationary measure from within
a multi-scale analysis of the dynamics. Here we propose to study these
simultaneously by examining "information percolation" clusters in the
space-time slab.
  Using this framework, we obtain new results for the Ising model on
$(\mathbb{Z}/n\mathbb{Z})^d$ throughout the high temperature regime:
total-variation mixing exhibits cutoff with an $O(1)$-window around the time at
which the magnetization is the square-root of the volume. (Previously, cutoff
in the full high temperature regime was only known for $d\leq 2$, and only with
an $O(\log\log n)$-window.)
  Furthermore, the new framework opens the door to understanding the effect of
the initial state on the mixing time. We demonstrate this on the 1D Ising
model, showing that starting from the uniform ("disordered") initial
distribution asymptotically halves the mixing time, whereas almost every
deterministic starting state is asymptotically as bad as starting from the
("ordered") all-plus state.
