## Fine structure of Hagedorn transitions

We study non-perturbative aspects of the Hagedorn transition for IIB
string theory in an anti-de Sitter spacetime in the limit that the
string length goes to infinity. The theory has a holographic dual in
terms of free $\NN=4$ super-Yang-Mills theory on a three-dimensional
sphere. We define a double scaling limit in which the width of the
transition region around the Hagedorn temperature scales with the
effective string coupling with a critical exponent. We show that in this
limit the transition is smoothed out by quantum effects. In particular,
the Hagedorn singularity of perturbative string theory is removed by
summing over two different string geometries: one from the thermal AdS
background, the other from a noncritical string background. The
associated noncritical string has the scaling of the unconventional
branch of super-Liouville theory or a branched polymer.