In the present article we extend the Random Matrix Theory to the theory of matrix-valued random fields. We derive the "equilibrium" probability distribution and "equilibrium" ensembles of matrix-valued random fields applying Maximum Entropy Principle. The inferred "equilibrium" probability density functionals are applied to Euclidean Quantum Field Theory and to the theory of two-dimensional Euclidean quantum gravity.
Random matrix ensembles (RME) of quantum statistical Hamiltonians, Gaussian random matrix ensembles (GRME) and Ginibre random matrix ensembles (Ginibre RME), found applications in study of following quantum statistical systems: molecular systems, nuclear systems, disordered materials, random Ising spin systems, quantum chaotic systems, two-dimensional quantum gravity and two-dimensional electron systems (Wigner-Dyson electrostatic analogy). Quantum statistical information functional is defined as negentropy (minus entropy). Entropy is neginformation (minus information). The distribution functions for the various random matrix ensembles are derived from the maximum entropy principle.
The random matrix ensembles are applied to the quantum statistical two-dimensional systems of electrons. The quantum systems are studied using the finite dimensional real, complex and quaternion Hilbert spaces of the eigenfunctions. The linear operators describing the systems act on these Hilbert spaces and they are treated as random matrices in generic bases of the eigenfunctions. The random eigenproblems are presented and solved. Examples of random operators
are presented with connection to physical problems.