  We study the Asymptotic Cone of Teichm\"uller space equipped with the
Weil-Petersson metric. In particular, we provide a characterization of the
canonical finest pieces in the tree-graded structure of the asymptotic cone of
Teichm\"uller space along the same lines as a similar characterization for
right angled Artin groups by Behrstock-Charney and for mapping class groups by
Behrstock-Kleiner-Minksy-Mosher. As a corollary of the characterization, we
complete the thickness classification of Teichm\"uller spaces for all surfaces
of finite type, thereby answering questions of Behrstock-Drutu,
Behrstock-Drutu-Mosher, and Brock-Masur. In particular, we prove that
Teichm\"uller space of the genus two surface with one boundary component (or
puncture) can be uniquely characterized in the following two senses: it is
thick of order two, and it has superquadratic yet at most cubic divergence. In
addition, we characterize strongly contracting quasi-geodesics in Teichm\"uller
space, generalizing results of Brock-Masur-Minsky. As a tool, we develop a
complex of separating multicurves, which may be of independent interest.
