Near-Hagedorn thermodynamics and random walks — extensions and examples

Abstract

In this paper, we discuss several explicit examples of the results obtained in [1]. We elaborate on the random walk picture in these spacetimes and how it is modified. Firstly we discuss the linear dilaton background. Then we analyze a previously studied toroidally compactified background where we determine the Hagedorn temperature and study the random walk picture. We continue with flat space orbifold models where we discuss boundary conditions for the thermal scalar. Finally, we study the general link between the quantum numbers in the fundamental domain and the strip and their role in thermodynamics.

Perturbative string thermodynamics near black hole horizons

“…A final characterization of this value is found by looking at the continuum. Assuming that the continuous quantum number s does not give a contributing τ 2 -dependent exponential correction (as we have seen happens in fact for the linear dilaton background discussed elsewhere [29]), we find that the continuum state becomes marginal when 40 1 4(k − 2) + k 4 = 1 ⇔ k = 3 (C. 15) and analogously for the type II superstring. We remark that for both k larger and smaller than this critical value, this state is non-tachyonic, it can only become marginal when this critical value of k is reached.…”

“…At the same time, the asymptotically linear dilaton thermal scalar (coming from the continuous part of Z) becomes massless. The random walk of the latter looks asymptotically like the one studied 29 In fact, we believe this continuous process sheds some light on the nature of the thermal scalar on black hole horizons. We discussed earlier that the interpretation of the genus one partition function in terms of the Hamiltonian trace is obscured since possibly interactions are included with an open string gas on the horizon.…”

“…This raises some questions regarding the thermodynamic interpretation of this theory, as usually these are attributed to corrections to MaxwellBoltzmann statistics [29].…”

“…29 We can now give a full picture of thermodynamics on the SL(2, R)/U(1) cigar (or the near-extremal NS5 brane solution) as one varies k.…”

“…30 This continuum part comes from the phase shift contribution in the density of states (or equivalently the reflection amplitude of the gravitational potential). in [29] for the linear dilaton background. Not just the thermal scalar, but in fact all bound states disappear from the spectrum at this point.…”