n^{-1} C^2).$$

This allows to define the *non-microstates free entropy dimension* $\delta^\ast(X_1,\dots,X_n)$ by $$\delta^\ast(X_1,\dots,X_n) := n - \liminf_{{\varepsilon}\searrow 0} \frac{\chi^\ast(X_1+\sqrt{{\varepsilon}}S_1,\dots,X_n+\sqrt{{\varepsilon}}S_n)}{\log(\sqrt{{\varepsilon}})}.$$

We note that there is actually a variant of $\delta^\ast(X_1,\dots,X_n)$, given by $$\hat\delta^\star(X_1,\dots,X_n) := n - \liminf_{t\searrow0} t \Phi^\ast(X_1+\sqrt{t}S_1,\dots,X_n+\sqrt{t}S_n),$$ which is formally obtained by applying L’Hospital’s rule to the $\liminf$ appearing in the definition of $\delta^\ast(X_1,\dots,X_n)$. In [@Connes-Shlyakhtenko], where $\hat\delta^\star$ was introduced, it was denoted by $\delta^\star$; we have slightly changed the notation for better legibility. Due to , we have that $0 \leq \hat\delta^\star(X_1,\dots,X_n) \leq n$.

It was already mentioned in [@Shly2014] that $\delta^\ast(X_1,\dots,X_n)=n$ or even $\hat\delta^\star(X_1,\dots,X_n)=n$ are the weakest possible assumptions where we can expect a version of Theorem \[main2\] to hold true. Accordingly, it sits at the end of a longer chain of general implications, namely $$\begin{aligned}
\Phi^\ast(X_1,\dots,X_n) < \infty & \quad \Longrightarrow \quad \chi^\ast(X_1,\dots,X_n) > - \infty\\
                                  & \quad \Longrightarrow \quad \delta^\ast(X_1,\dots,X_n)=n\\
                                  & \quad \Longrightarrow \quad \hat\delta^\star(X_1,\dots,X_n)=n.\end{aligned}$$ The first implication follows by definition of $\chi^\ast(X_1,\dots,X_n)$, the second implication is a direct consequence of the definition of $\delta^\ast(X_1,\dots,X_n)$, and the last implication is justified by $\hat\delta^\star(X_1,\dots,X_n) \geq \delta^\ast(X_1,\dots,X_n),$ which was shown in [@Connes-Shlyakhtenko Lemma 4.1] by a straightforward computation.

The case of finite non-microstates free Fisher information 