Random walks and the Hagedorn transition

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 single string at finite temperature is the torus amplitude restricted to unit winding around Euclidean time. We use 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 × 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 Barbón and Rabinovici, and by Adams 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.