We describe the formation and properties of Holstein polarons in the entire parameter regime. Our presentation focuses on the polaron mass and radius, which we obtain with an improved numerical technique. It is based on the combination of variational
exact diagonalization with an improved construction of phonon states, providing results even for the strong coupling adiabatic regime. In particular we can describe the formation of large and heavy adiabatic polarons. A comparison of the polaron mass for the one and three dimensional situation explains how the different properties in the static oscillator limit determine the behavior in the adiabatic regime. The transport properties of large and small polarons are characterized by the f-sum rule and the optical conductivity. Our calculations are approximation-free and have negligible numerical error. This allows us to give a conclusive and impartial description of polaron formation. We finally discuss the implications of our results for situations beyond the Holstein model.
