Models for RG running for Gravitational couplings and applications

The plan for this thesis is as follows. In the first part we discuss the relation between two different RG flows: functional and holographic one. The bigger emphasis is put on the novel holographic RG flow and we devote full third chapter for studying holographic RG flow geometries. We are not only interested in flows for gravitational couplings, we also consider standard RG flows from field theories with matter. The second part of this work is divided into two chapters. In the fourth chapter we study classicalization for nonlinear sigma model understood as a toy example before attacking more diffcult problems of full quantum gravity. We also point there possible relations between classicalization and asymptotic safety as between two similar in some conditions mechanisms for UV completion. In the fofth chapter we consider universal 1-loop effective action in system of gravitating scalar field. We use new methods to derive its IR limit and we compute few low-energetic observables in such effective field theory of gravitational interactions. Finally in the sixth chapter we shortly collect main obtained results and conclude. The material presented in this work is partially based on two scientific articles [42] and [63], I published during my PhD studies