External Source

Abstract

An external, i.e. deterministic, source matrix A is now coupled to the random matrices with weight

$$\begin{aligned} P_A(M) = \frac{1}{Z_A} {\mathrm{exp}} \left[ -\frac{\lambda }{2} {\mathrm{tr}} M^2 + {\mathrm{tr}} M A\right] \end{aligned}$$

The matrix A is an $N\times N$ Hermitian matrix. The normalization $Z_A$, up to a trivial constant, is proportional to $\mathrm{exp}[{\frac{1}{2\lambda } {\mathrm{tr}} A^2}]$.