  We study details of the approach to the Hagedorn temperature in string theory
in various static spacetime backgrounds. We show that the partition function
for a {\it single} string at finite temperature is the torus amplitude
restricted to unit winding around Euclidean time. We use the worldsheet path
integral to derive the statement that the the sum over random walks of the
thermal scalar near the Hagedorn transition is precisely the image under a
modular transformation of the sum over spatial configurations of a single
highly excited string. We compute the radius of gyration of thermally excited
strings in $AdS_D\times S^n$. We show that the winding mode indicates an
instability despite the AdS curvature at large radius, and that the negative
mass squared decreases with decreasing AdS radius, much like the type 0
tachyon. We add further arguments to statements by Barbon and Rabinovici, and
by Adams {\it et. al.}, that the Euclidean AdS black hole can thought of as a
condensate of the thermal scalar. We use this to provide circumstantial
evidence that the condensation of the thermal scalar decouples closed string
modes.
