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

The Asymptotic Cone of Teichm\"uller Space: Thickness and Divergence