Matrix Models 2025

Special course for 4th year students at LMTP MIPT

Table of Contents

Course plan

Lecture 1. Introduction, motivation and course outline

Historical motivation (non-perturbative string theory description) along the lines of [FGZ95, pp.4–12]. Modern development according to [Mor25].

Lecture 2. Feynman graphs in QFT

Feynman graphs, Wick theorem, asymptotic series, connected and disconnected diagrams [Ski18, pp.13–26].

Lecture 3. Ribbon Feynman graphs. QCD. Large limit

Ribbon Feynman graphs — [FGZ95, sec.1.2]. To understand the connection with fancy, though accurate eq. (2.39) and (2.40) from [Ski18] (Lecture 2) see also [EKR15, sec.2.1], especially eqns. (2.1.20) and (2.1.21). QCD, large limit and planar diagrams — [Ton18, pp.305–313].

Lecture 4. Eigenvalue matrix models. Leading behaviour

You'll learn quick and dirty hack of obtaining leading behaviour of one-point correlators (as well as directly related one-point eigenvalue distributions). Main reference — [Zee10, pp.396–402]. For more convincing Vandermonde determinant derivations see footnote 8 in [FGZ95], as well as Appendix C.

Just in case of questions about the so-called Haar measure, see Example 5 here.

For more convincing Coloumb gas method explanations see also [FGZ95, sec.2.1].

Lecture 5. Loop equations. Subleading behaviour

Subleading behaviour can be obtained by straightforward expansion around saddle point from previous lecture. The recipe is as follows: expand the exponent around equillibrium . You'll have something like

(1)

so will be the propagator and all the rest — vertices. Next, you can calculate all the correlators using this Feyman diagrams technique. This procedure is quite tedious and even if you are patient enough to calculate first few orders, you may notice the beautiful pattern behind, which can be discovered independently and which is the subject of the current lecture.

Main goal is to understand [Mor25, sec.2.2]. It can be done with the help of [Mor94, sec.2.2], elementary matrix calculus [PP12] could also be of use. As a glimpse into the realm of AMM-CEO topological recursion describe the recursive procedure for obtaining all the correlators of the matrix model with the generating function given as sum over Young diagrams

(2)

by substituting it in the Virasoro constraints. For a result obtained from somewhat different persperctive see eq. (4.1.10) in [EKR15].

Lecture 6. Origin of loop equations. QCD

Makeenko-Migdal loop equations in QCD are well described in section 1 and appendix A of [Mig83]. Loop equations in 2 quantum gravity [Kaz89].

Lecture 7. -representation

The subject of the lecture is [Mor25, sec.2.3]. For the origin of this approach, as well as the notion of and matrix calculus derivation, see [MS09]. Grading and dilatation operator are also explained in my notes.

Lecture 8. Integrability

We need to understand [Mor25, sec.2.4]. For details consult [Mir94, secs.4.2, 4.5].

Lectures 9–11. Superintegrability

[MO22, sec.2] is the way to go. Also consult my note.

Lecture 12. Multi-matrix models. Ising model on a random 2 surface

See [Kaz86] and [EKR15, sec.2.2.1].

Lecture 13. Matrix model in external field. Kontsevich model

[Kon92, sec.4]: Feynman rules for cubic model in external field, its determinant formula and proof that it's a -function. [Mor94, sec.5]: Double scaling limit of one matrix model equals to Kontsevich one. [Wit90]: motivation.

Lecture 14. Unitary models. Itzykson-Zuber integral. Large 2 lattice QCD

See [IZ80]. Character expansion of the Itzykson-Zuber integral is also explained in my note. For determinant formula consult either paper above, or Terry Tao's blog post. Uniform model is described in works [BG80; GW80] and used for description of large 2 lattice QCD. Itzykson-Zuber integral is the "building block" of Kazakov-Migdal lattice -dimensional QCD [KM93].

Further reading

See [ABF15] and references therein. Especially [Mar18].

Problem set

Problems 1-7. (Matrix) models

For the Gaussian Hermitian matrix model calculate the generating function up to the grading 4 via

Problem 8. Connected vs. disconnected

Find the expression for the GHMM generating function of cumulants up to 8-th order (the least order, where disconnected diagrams appear in ). Check that correlators obtained from this generating function actually count connected only ribbon graphs.

Problem 9. Two cuts

For the quartic model

(3)

find the large eigenvalue distribution .

Grading

Both talks and HW are graded on 5-point scale at first. Talks grade is just an average of individual talks grades. Homework grade is calculated as follows

(4)

Next, raw grade is calculated

(5)

and converted in final grade on 10-point scale

(6)

This function almost perfectly maps unsatisfactory, satisfactory, good, and excellent grades from one scale to another.

Table 1: Individual results
  Аня Радомир Саша Юля Вася Пупкин
Talk 1 5 5 5 5 5
Talk 2 5 5 5 5 5
Talk 3 5   5 5 5
Problem 1 1       1
Problem 2 1       1
Problem 3 1       1
Problem 4         1
Problem 5         1
Problem 6 1       1
Problem 7         1
Problem 8         1
Problem 9         1
Grade 5 2 3 3 10

References

[FGZ95]
P. Francesco, P. Ginsparg, and J. Zinn-Justin, “2D gravity and random matrices,” Phys. Rep., vol. 254, no. 1-2, pp. 1–133, 1995, doi: 10.1016/0370-1573(94)00084-g. arXiv: hep-th/9306153.
[Mor25]
A. Morozov, “Integrability and matrix models,” in Encyclopedia of mathematical physics, in Encyclopedia of mathematical physics. , Elsevier, 2025, pp. 168–174. doi: 10.1016/b978-0-323-95703-8.00040-9. arXiv: 2212.02632.
[Ski18]
D. Skinner, “Quantum field theory ii,” 2018. Available: https://www.damtp.cam.ac.uk/user/dbs26/AQFT.html
[EKR15]
B. Eynard, T. Kimura, and S. Ribault, “Random matrices,” 2015. arXiv: 1510.04430.
[Ton18]
D. Tong, “Gauge theory,” 2018. Available: https://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf
[Zee10]
A. Zee, Quantum field theory in a nutshell, 2nd ed. in In a nutshell Princeton. Princeton University Press, 2010.
[Mor94]
A. Y. Morozov, “Integrability and matrix models,” Phys.-Uspekhi, vol. 37, no. 1, pp. 1–55, 1994, doi: 10.1070/pu1994v037n01abeh000001. arXiv: hep-th/9303139.
[PP12]
K. B. Petersen and M. S. Pedersen, “Matrix cookbook,” 2012. Available: https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf
[Mig83]
A. Migdal, “Loop equations and 1n expansion,” Phys. Rep., vol. 102, no. 4, pp. 199–290, 1983, doi: 10.1016/0370-1573(83)90076-5.
[Kaz89]
V. Kazakov, “The appearance of matter fields from quantum fluctuations of 2d-gravity,” Mod. Phys. Lett. A, vol. 04, no. 22, pp. 2125–2139, 1989, doi: 10.1142/s0217732389002392.
[MS09]
A. Morozov and S. Shakirov, “Generation of matrix models by $ w$-operators,” J. High Energy Phys., vol. 2009, no. 04, p. 064, 2009, doi: 10.1088/1126-6708/2009/04/064. arXiv: 0902.2627.
[Mir94]
A. Mironov, “2D gravity and matrix models i: 2d gravity,” Int. J. Modern Phys. A, vol. 09, no. 25, pp. 4355–4405, 1994, doi: 10.1142/s0217751x94001746. arXiv: hep-th/9312212.
[MO22]
V. Mishnyakov and A. Oreshina, “Superintegrability in \(\)-deformed Gaussian Hermitian matrix model from \(W\)-operators,” Eur. Phys. J. C, vol. 82, no. 6, p. 548, 2022, doi: 10.1140/epjc/s10052-022-10466-y. arXiv: 2203.15675.
[Kaz86]
V. Kazakov, “Ising model on a dynamical planar random lattice: Exact solution,” Phys. Lett. A, vol. 119, no. 3, pp. 140–144, 1986, doi: 10.1016/0375-9601(86)90433-0.
[Kon92]
M. Kontsevich, “Intersection theory on the moduli space of curves and the matrix airy function,” Commun. Math. Phys., vol. 147, no. 1, pp. 1–23, 1992, doi: 10.1007/bf02099526.
[Wit90]
E. Witten, “Two-dimensional gravity and intersection theory on moduli space,” Surv. Differ. Geom., vol. 1, no. 1, pp. 243–310, 1990, doi: 10.4310/sdg.1990.v1.n1.a5.
[IZ80]
C. Itzykson and J.-B. Zuber, “The planar approximation. ii,” J. Math. Phys., vol. 21, no. 3, pp. 411–421, 1980, doi: 10.1063/1.524438.
[GW80]
D. J. Gross and E. Witten, “Possible third-order phase transition in the large-\(N\) lattice gauge theory,” Phys. Rev. D, vol. 21, no. 2, pp. 446–453, 1980, doi: 10.1103/physrevd.21.446.
[BG80]
E. Brezin and D. J. Gross, “The external field problem in the large \(N\) limit of QCD,” Phys. Lett. B, vol. 97, no. 1, pp. 120–124, 1980, doi: 10.1016/0370-2693(80)90562-6.
[KM93]
V. Kazakov and A. Migdal, “Induced gauge theory at large n,” Nucl. Phys. B, vol. 397, no. 1-2, pp. 214–238, 1993, doi: 10.1016/0550-3213(93)90342-m. arXiv: hep-th/9206015.
[ABF15]
G. Akemann, J. Baik, and P. D. Francesco, The oxford handbook of random matrix theory, Online version. Oxford University Press, USA, 2015.
[Mar18]
M. Marino Beiras, “String theory,” in The oxford handbook of random matrix theory, 2018, p. 31.