The Physical Meaning of Variance of Random Matrix Entries I am trying to make some physical sense of the Hamiltonian described on pages 1, 2 here. The part I don't get is in the image attached below. I understand what the variance of each entry term tells me physically, but I have no clue as to what the stated variances dictate. Any hints will help.

 A: Random matrix theory was extensively developed in connection to transport in mesoscopic systems, such as quantum dots, quantum wires, quantum rings, etc. Like everywhere, there are inherent sources of randomness in these systems, which, however, due to the long coherence scales need to be taken into account in fully quantum way (rather than simply via perturbation theory, as is often done in bulk approaches.) In particular, randomness can originate from:

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*Averaging over ensemble of system with slightly varying parameters (values of energy levels and tunneling matrix elements, like $W$ in the OP)

*Averaging over uncontrollable distribution of impurities in the system

*Voltage fluctuations in the leads or on the gate electrodes forming the system (which translates into variation of level energies and matrix elements mentioned above, but possibly in real time or when the experiment is repeatedly performed on the same device

*A system may be extended with physical parameters varying through the system, so that the global properties involve averaging over them  - self-averaging.

So variance is literally the variance of these quantities in experimental device or from one device to another.
Classical reference is Random-Matrix Theory of Quantum Transport by Carlo Beenakker (published in Reviews of Modern Physics, but available also from arxive.) A more general text is Joe Imry's Introduction to mesoscopic physics.
Remark
Regarding the Emergence of Fermi's Golden Rule, which is the subject of the article linked in the OP - while I do agree that FGR involves transitions to a continuum, modeling the continuum via random matrix reflects the author's background, and is by no means general. E.g., in atomic physics broadening into continuum would be due to coupling to photon vacuum, just as in many other situations it could be due to coupling to some kind of bath.
