Random matrices

Abstract

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.

Defects at the Intersection: the Supergroup Side

“…It turns out that the two are essentially related by analytic continuation in one of the equivariant parameters. This seems to parallel what is known from other situations [5]: the S 3 U(N 1 ) × U(N 2 ) ABJ(M) and the L(2, 1) U(N 1 + N 2 ) Chern-Simons matrix models are related by analytic continuation N 2 → −N 2 , while from the combinatorial perspective the role of matrix sizes is played by 1/ ln q, −1/ ln t [39]. Furthermore, we considered yet another Higgsing of the 5d U(1|1) theory, and the partition function of the resulting single component defect theory resembles that of refined Chern-Simons on L(2, 1).…”

Sphere and disk partition functions in Liouville and in matrix integrals

“…The explicit values (3.25) determine a potential exactly, which in turn determines ρ 0 exactly, and together these quantities determine the free energy F 0 via for example (3.9) or a somewhat more efficient version of this formula in [25]. Using Mathematica, we found the first few cases: Here we have boxed the first nonanalytic term in each case.…”

The Yang-Mills duals of small AdS black holes

“…Let us consider the holomorphic matrix model where the W (z) possibly contains complex-valued coupling constants, and Γ is not necessarily unit circle but the N-product of a contour γ in the complexified z-plane C, properly chosen to make the above partition function convergent (We directly adopt the eigenvalue representation without further explanation. For more detail on holomorphic matrix models and their complex saddles, see [44,45,46,47] for instance). This partition function can be written as…”