Minimum dilatations of pseudo-Anosov braids

We determine the minimum dilatation ${\delta }_n$ among pseudo-Anosov braids with $n$ strands, for large enough values of $n$. These are the dilatations attained by the examples of Hironaka-Kin and Venzke, and they satisfy ${\lim }_{n\to \mathrm{\infty }}{\delta }_n^n=(2+\sqrt{3}{)}^2\approx 13.928$. Together with previous work, this result confirms conjectures by Kin-Takasawa and Venzke, and solves the minimum dilatation problem on the $n$-punctured sphere, for all but $6$ values of $n$.