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New stochastic approaches for the computation of electronic excitations aredeveloped within the many-body perturbation theory. Three approximations to theelectronic self-energy are considered: $G_0W_0$, $G_0W_0^tc$, and$G_0W_0^{tc}\Gamma_x$. All three methods are formulated in the time domain andthe latter two incorporate non-local vertex corrections. In case of$G_0W_0^{tc}\Gamma_x$, the vertex corrections are included both in the screenedCoulomb interaction and in the expression for the self-energy. Theimplementation of the three approximations is verified by comparison todeterministic results for a set of small molecules. The performance fullystochastic implementation is tested on acene molecules, C$_{60}$ andPC$_{60}$BM. The vertex correction appears crucial for the description ofunoccupied states. Unlike conventional (deterministic) approaches, all threestochastic methods scale linearly with the number of electrons.

Stochastic vertex corrections: linear scaling methods for accurate quasiparticle energies