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Introduction to Monte Carlo for Matrix Models

by Raghav G. Jha

This Submission thread is now published as SciPost Phys. Lect. Notes 46 (2022)

Submission summary

As Contributors: Raghav Govind Jha
Preprint link: scipost_202111_00064v2
Code repository: https://github.com/rgjha/MMMC
Date accepted: 2022-03-25
Date submitted: 2022-03-18 19:32
Submitted by: Jha, Raghav Govind
Submitted to: SciPost Physics Lecture Notes
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
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.

Published as SciPost Phys. Lect. Notes 46 (2022)



Author comments upon resubmission

version2

List of changes

1. We have merged Sec.3.1 with Sec.2 and renamed the section.
2. We have added 1-2 paragraphs regarding the Metropolis-Hastings algorithm as the simplest example of MCMC (Markov chain Monte Carlo).
3. For the bosonic fields, it is certainly possible to use heat-bath, but the article introduces HMC since it is more natural when advancing to field theories with fermions where usually RHMC is used (rational HMC). We have made a comment about this in the article.
4. We have introduced “Box-Muller algorithm” in Sec.3.2.1.
5. We have corrected the fact that Ref. [39] did not study D0 model and added the four references.
6. We have added future directions as suggested by the first referee and added references.

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Comments

Anonymous on 2022-03-18  [id 2301]

We thank both the referees for their comments and address them below, starting with the second referee report.

  1. We have merged Sec.3.1 with Sec.2 and renamed the section.
  2. We have added 1-2 paragraphs regarding the Metropolis-Hastings algorithm as the simplest example of MCMC (Markov chain Monte Carlo).
  3. For the bosonic fields, it is certainly possible to use heat-bath, but the article introduces HMC since it is more natural when advancing to field theories with fermions where usually RHMC is used (rational HMC). We have made a comment about that in the article.
  4. We have introduced “Box-Muller algorithm” in Sec.3.2.1.
  5. We have corrected the fact that Ref. [39] did not study D0 model and added the four references.
  6. The solutions to most of the exercises is given. Those which are not given is 1,6,7,10. Out of these, 1 is a cumbersome 1-2 pages of algebra which is best done by people reading this review. The codes for 6 and 10 are given in the paper. It is a numerical exercise. And for 7, we just need to complete the square, like we typically do when dealing with quadratic equations. So, I think we would like to keep the solution section as is.
  7. We have added future directions as suggested by the first referee and added references.

I hope these changes make the version2 of the article (attached) ready to be published. Thank you.