  We show that, for a class of critical strings in ${\bf R}\times S^{1}$-target
space, the description of string theory given by its field content (analog
model) breaks down when the radius of $S^{1}$ decreases below
$R_{0}=\sqrt{\alpha^{\prime}}$, the self-dual point of the partition function
$Z(R)$. We find that $Z(R)$ has a soft singularity at $R_{0}$ (a finite jump in
the first derivative of $Z$).
