We study energy transport between two quantum systems coupled through a random interaction. The central feature of our approach is to model the coupling as a Gaussian random matrix, which enables a simple and systematic perturbative expansion. In the large-$N$ limit, we derive explicit expressions for the energy transfer rate and heat conductance up to second order in the coupling strength. Using spectral methods and diagrammatic expansions, we obtain the leading-order and next-to-leading-order contributions to the energy transfer rate. We illustrate our results through explicit calculations for Gaussian, constant, semicircular, and gamma densities of states.

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• Heat transfer
• Disordered systems
• Perturbative methods