Unique asymptotics of ancient convex mean curvature flow solutions

We study compact noncollapsed ancient convex solutions to Mean Curvature Flow in Rⁿ⁺¹ with O(1) × O(n) symmetry. We show they all have unique asymptotics as t → -∞ and we give a precise asymptotic description of these solutions. The asymptotics apply, in particular, to the solutions constructed by White, and Haslhofer and Hershkovits (in the case of those particular solutions the asymptotics were predicted and formally computed by Angenent).