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Introduction to Monte Carlo for Matrix Models
by Raghav G. Jha
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Submission summary
| Authors (as registered SciPost users): | Raghav Govind Jha |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202111_00064v1 (pdf) |
| Code repository: | https://github.com/rgjha/MMMC |
| Date submitted: | 2021-11-30 20:32 |
| Submitted by: | Jha, Raghav Govind |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We consider a wide range of matrix models and study them using the Monte Carlo technique in the large N limit. The results we obtain agree with exact analytic expres- sions and recent numerical bootstrap methods for models with one and two matrices. We then present new results for several unsolved multi-matrix models where no other tool is yet available. In order to encourage an exchange of ideas between different numerical approaches to matrix models, we provide programs in Python that can be easily modified to study potentials other than the ones discussed. These programs were tested on a laptop and took between a few minutes to several hours to finish depending on the model, N, and the required precision.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-2-19 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202111_00064v1, delivered 2022-02-19, doi: 10.21468/SciPost.Report.4475
Report
I think this review and sample programs are useful materials for students and researchers interested in numerical approaches to matrix models. However, I would like to suggest a few points which can be improved.
Sec.3.1 seems to be a little bit out of place. Matrix bootstrap is a numerical method, but not the main topic of this article (Monte Carlo methods). Perhaps it can be moved to Sec.2. (Then the title of Sec.2 has to be modified as well.)
The HMC algorithm is explained in Sec.3.2. Although the algorithm is clearly explained, laypeople may wonder why the right ensemble is obtained by using this algorithm. Probably it is better to introduce the Metropolis algorithm as the simplest example of the Markov Chain Monte Carlo methods and explain the basic logic behind such algorithms. Then, as a byproduct, the advantage of the HMC algorithm can be illuminated.
(A question which is somewhat related to this part:
The HMC algorithm is particularly useful when dynamical fermions are involved. For the bosonic matrix models, is it possible to use the heat-bath algorithm?)
In Sec.3.2.1, it is better to introduce the word "Box-Muller algorithm". (This name is used in sample code.)
Reference [39] did not study the D0-brane quantum mechanics. Better references are hep-th/0803.4273 and hep-th/0707.4454. Another reference that achieved good numerical precision is 1503.08499[hep-lat].
In this context, it would be good to refer to hep-th/9910001 and hep-th/hep-th/0007051. They used the Gaussian approximation method to study D0-brane quantum mechanics numerically and demonstrated the power of the numerical approach to holography.
In Appendix G, would it be possible to show the solutions to all exercises?
Requested changes
Please find the suggestions in the report.
Report #1 by Anonymous (Referee 1) on 2022-1-25 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202111_00064v1, delivered 2022-01-25, doi: 10.21468/SciPost.Report.4237
Strengths
The lecture notes provide a clear and gentle introduction into numerical methods for large N matrix models. It incorporates also nicely some very recent developments concerning a bootstrap approach to matrix models. It can sometimes be hard for someone interested in this sort of material to get into it as it requires a lot of programming. The lecture notes remove this drawback as they come with the codes and a short accompanying explanation. This allows the researcher to immediately understand the physics she/he is ultimately interested in.
Weaknesses
This is not really a weakness, but personally I always like to have a short section in a lecture note about the 'way forward' or 'current issues'. This would explain a little bit what the methods that are explained in the lecture notes are currently used for and what the obstacles are. This might spark the reader into going beyond the things discussed in the lecture notes.
Report
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Anonymous on 2022-02-11 [id 2185]
We thank the referee for the report and comments. We will be happy to add a short section on future directions (in the final version of submitted article), with a clear description of where these methods are used at the moment and what the problems are going ahead.