Interacting quantum scalar field theories in $dS_Dtimes M_d$ spacetime can be reduced to Euclidean field theories in $M_d$ space in the vicinity of $I_+$ infinity of $dS_D$ spacetime. Using this non-perturbative mapping, we analyze the critical behav
ior of Euclidean $lambdaphi_4^4$ theory in the symmetric phase and find the asymptotic behavior $beta(lambda)sim lambda$ of the beta function at strong coupling. Scaling violating contributions to the beta function are also estimated in this regime.
Dynamics of eternal inflation on the landscape admits description in terms of the Martin-Siggia-Rose (MSR) effective field theory that is in one-to-one correspondence with vacuum dynamics equations. On those sectors of the landscape, where transport
properties of the probability measure for eternal inflation are important, renormalization group fixed points of the MSR effective action determine late time behavior of the probability measure. I argue that these RG fixed points may be relevant for the solution of the gauge invariance problem for eternal inflation.
Disorder on the string theory landscape may significantly affect dynamics of eternal inflation leading to the possibility for some vacua on the landscape to become dynamically preferable over others. We systematically study effects of a generic disor
der on the landscape starting by identifying a sector with built-in disorder -- a set of de Sitter vacua corresponding to compactifications of the Type IIB string theory on Calabi-Yau manifolds with a number of warped Klebanov-Strassler throats attached randomly to the bulk part of the Calabi-Yau. Further, we derive continuum limit of the vacuum dynamics equations on the landscape. Using methods of dynamical renormalization group we determine the late time behavior of the probability distribution for an observer to measure a given value of the cosmological constant. We find the diffusion of the probability distribution to significantly slow down in sectors of the landscape where the number of nearest neighboring vacua for any given vacuum is small. We discuss relation of this slow-down with phenomenon of Anderson localization in disordered media.
