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## 2. Random matrix models with external source

1. ### Random matrices with external source. More general cases

#### Problem 2.1.

[Pavel Bleher] Study the random matrix model with probability measure $$\frac{1}{Z_N}e^{-N\textrm{tr}(V(M)-AM)},$$ where $A$ is an $N\times N$ diagonal matrix with eigenvalues $\pm a$ (each with equal multiplicity $N/2$, assuming $N$ is even), and $V(M)$ is a general polynomial.
A more general situation involves an external source term $A$ with several different eigenvalues $\{a_i\}_{i=1}^{j}$, and multiplicities $\{n_i\}_{i=1}^j$, where $n=n_1+n_2+\ldots+n_j$. It is assumed that all limits $c_j=\lim_{n\to\infty} \frac{n_j}{n}$ exist.
• ### $2+1/2$ random matrix models

#### Problem 2.2.

[A. Martínez-Finkelshtein] A general question about Riemann-Hilbert techniques to study the two-matrix plus external source model.

Cite this as: AimPL: Vector equilibrium problems and random matrix models, available at http://aimpl.org/vectorequilib.