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We provide an exact solution of large random arrowhead Hamiltonians withdiagonal disorder, a minimal model for inhomogeneously broadened quantumemitters coupled to a cavity mode. We find that the distribution of energyspacing can be continuously tuned between Poisson statistics and a distributionthat is very close to semi-Poisson statistics - the latter being usuallyassociated to the critical point of "Anderson" localization-delocalizationtransitions. We demonstrate that all the eigenstates - including two polaritonsand a continuum of dark states - are multifractal, which indicates theexistence of a critical "semi-localized" phase for all values of thelight-matter coupling strength, where dark states are localized over multiple,arbitrarily-distant sites. By computing the escape probability from an initialsite, we find that the system has a peculiar diffusive-like behavior with anescape probability growing linearly with time for any finite coupling strength,and that the escape rate can be controlled by selecting the energy of theinitial site. The escape rate averaged over the disorder configurations isfound to exhibit a maximum for intermediate coupling strengths, beforesaturating at a lower value in the collective strong coupling limit - a "cavityprotection" effect. Surprisingly, we show that the saturation value increaseswith the disorder, indicating that the cavity does not only protect transportagainst disorder but can also turn the latter into an ally improving transport.We finally investigate the system in a two-terminal configuration, and showthat the steady-state excitation current exhibits similar features as theescape probability, thereby extending our cavity-protected transport scenarioto out-of-equilibrium situations. We finally demonstrate that dark states canprovide the major contribution to long-distance transport in disorderedsystems.

Large Random Arrowhead Matrices: Multifractality, Semi-Localization, and Protected Transport in Disordered Quantum Spins Coupled to a Cavity