Can random walks be applied to String Theory in curved space

[jinawee]

If we study the high temperature limit (near Hagedorn) of a string gas, most of the energy is concentrated in a single long string. If we model the string by a fixed number of rigid links of length ls and calculate the number of possible configurations, we get the density of states:

$$\omega(E) \sim \frac{ e^{ \beta E} }{ E^{ 1+D/2 } }$$

Is it possible to generalize this method in curved space?

A possible way is to calculate the torus path integral of a string that wraps the euclidian periodic time in a curved background. At high temperatures this can be calculated from the path integral of a single non-relativistic particle, which gives the free energy and thus the density of states. This seems to be called the random walk model. References: http://arxiv.org/abs/1506.07798 and http://arxiv.org/abs/hep-th/0508148 .

But this seems totally different. A particle path can be related to a random walk, but one doesn't calculate the number of microstates from combinatoric reasoning. Is there a way to do something like that?

 

[AndyW]

Thank you for your question. The method you describe is commonly known as the random walk model in string theory, and it is indeed different from the method used to calculate the density of states in the high temperature limit. The random walk model is based on the idea that a string can be viewed as a collection of non-interacting particles moving in a curved background. This allows for the calculation of the string partition function and thus the density of states.

However, as you correctly pointed out, this method is not based on combinatoric reasoning and does not directly calculate the number of microstates. In order to generalize this method in curved space, one could potentially use the same approach of calculating the string partition function in a curved background, but with the inclusion of string interactions. This would require a more sophisticated mathematical framework and may not be as straightforward as the random walk model.

Another possible approach could be to use the formalism of statistical mechanics in curved space, which has been developed in the context of black hole thermodynamics. This formalism takes into account the effects of curvature on the statistical properties of a system and could potentially be applied to string theory in curved space.

In summary, while it is possible to generalize the random walk model in curved space, it may require the use of more advanced mathematical techniques and a deeper understanding of the statistical mechanics of curved systems. I hope this helps to answer your question.