On the spectrum of limit models

We study the spectrum of limit models assuming the existence of a nicely behaved independence notion. Under reasonable assumptions, we show that all long' limit models are isomorphic, and all short’ limit models are non-isomorphic. $\mathbf{\text{Theorem.}}$ Let $\mathbf{K}$ be a ${\mathrm{\aleph }}_0$-tame abstract elementary class stable in $\lambda \ge \mathrm{LS}(\mathbf{K})$ with amalgamation, joint embedding and no maximal models. Suppose there is an independence relation on the models of size $\lambda$ that satisfies uniqueness, extension, non-forking amalgamation, universal continuity, and $(\ge \kappa )$-local character in a minimal regular $\kappa <{\lambda }^{+}$. Suppose ${\delta }_1,{\delta }_2<{\lambda }^{+}$ with $\mathrm{cf}({\delta }_1)<\mathrm{cf}({\delta }_2)$. Then for any $N_1,N_2,M\in {\mathbf{K}}_{\lambda }$ where $N_l$ is a $(\lambda ,{\delta }_l)$-limit model over $M$ for $l=1,2$, $$N_1\text{ is isomorphic to }{N}_2\text{ over }M⟺\mathrm{cf}({\delta }_1)\ge \kappa$$ Both implications in the conclusion have improvements. High cofinality limits are isomorphic without the ${\mathrm{\aleph }}_0$-tameness assumption and assuming the independence relation is defined only on high cofinality limit models. Low cofinality limits are non-isomorphic without assuming non-forking amalgamation. We show how our results can be used to study limit models in both abstract settings and in natural examples of abstract elementary classes.