Limit profile for high-temperature Ising Glauber dynamics beyond the complete graph

Determine the limit profile of the continuous-time Glauber (single-site heat-bath) dynamics for the high-temperature Ising model on finite graphs other than the complete graph, including lattices and bounded-degree graphs. Specifically, establish the asymptotic profile of the total variation distance around the cutoff window in these geometries, as existing information-percolation methods prove cutoff but do not yield a limit profile.

Background

The paper determines the limit profile for the high-temperature Curie–Weiss (complete graph) Ising model by analyzing a two-coordinate chain and proving convergence to a two-dimensional diffusion near the cutoff window. This provides a precise description of the total variation distance around the cutoff time in the mean-field geometry.

For more general geometries, Lubetzky and Sly used information percolation to prove cutoff for the high-temperature Ising model on lattices and, more generally, bounded-degree graphs. However, their method does not provide limit profiles. Consequently, while cutoff is known in these settings, the detailed form of the sharp transition (i.e., the limit profile) remains undetermined outside the mean-field case.