## 1 Introduction  
Semiconductor micro-cavities operating in the strong coupling regime are optical resonators in which the _eigenmodes_ are no longer purely excitonic or photonic, but a mixed light matter state is occurring. This evolution has finally led to polariton lasing [1; 2; 3; 4; 5; 6; 7; 8; 9]. Random lasing [10] has been claimed to be fundamentally incompatible with polaritons. From recent research developments however it turns out that it is not yet clear whether random lasers are either pure photon lasers or pure polariton lasers, or whether they even have both characteristics at the same time [11]. Thresholds like in conventional lasers may be observed [12; 13]. We show in this article that for excitations of the quantum many-body system of ZnO nano-resonators by experimentally feasible threshold intensities of the optical pump found in random laser experiments we derive theoretically well confined peaks in terms of gain narrowing in the spectrum of the lifetime for the cavity polariton in the non-equilibrium. These peaks give evidence that such a sample may undergo a transition towards exciton-polariton lasing rather than a transition to exciton-photon lasing, at least a coexistence of both regimes may occur. ZnO is known for excellent light-matter coupling characteristics [7; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]; it is predestined for this study. It has been experimentally derived that ZnO can be considered as a Mott insulator under certain conditions [16; 18]. A feature of ZnO nano-structures is a transition from the non-centrosymmetric wurtzite structure to the centrosymmetric rocksalt configuration with a variation of temperature and pressure, see Figure 1. When the system is externally pumped, Figure 1c, lasing may occur eventually. The propagating light intensity in the random laser may experience Mie resonances as a whispering gallery resonance of light at the inner surface of the individual nano-structure. In a complex medium, which pumped ZnO in the non-equilibrium certainly is, the resonant Mie mode forms a light matter bound state in the form of a Fano resonance. It leads to quantum many body physics and the renormalization of the bands under excitations. We consider (**i**) the semiconductor material as it is subjected to the strong external AC field of the optical pump in the sense of a topological excitation [14; 15]. We model the coupling of the classical laser, \(\hbar\Omega_{L}\,=\,1.75\,\)eV, to the quantum many body system as well as higher-order photon absorption processes, \(\hbar\Omega_{L}\,=\,3.5\,\)eV etc., by means of the Floquet matrix [28; 29; 30; 31]. The band structure and the lifetimes of light-matter coupled states are derived by the Dynamical Mean Field Theory (DMFT) [32; 33; 34; 35] in the sense of a generalized Hubbard-Hamiltonian out-of thermodynamical equilibrium. The Hubbard Hamiltonion includes strictly local interactions between electrons with opposite spins. It is predestined to describe the exciton dynamics of excited matter in a physical time range where additional impurity scattering is not expected. Bulk ZnO develops under excitation in the near band gap region an exciton dynamics. In the second step of this work (**ii**) we additionally consider photons which populate the Mie resonance of the individual nano-resonator and we study the formation of polaritons. The spectral features are compared to experimental emission spectra of ZnO random lasers and polariton lasers and we find a very good agreement in either case.  
Figure 1: \(Cont.\)