This thesis focuses primarily on false-vacuum eternal inflation on the string landscape. Its main goal is to shed light on a dynamical phase transitiondiscovered recently, which occurs when de Sitter vacua in a region of the landscape have an average lifetime of order the de Sitter Page time. We first aim to understand the mechanism driving the inflationary multiverse to the critical point. For this purpose, we focus on a finite region of the landscape,which exchanges volume with surrounding vacua, and as such acts as an open system. The surrounding vacua, which play the role of an environment, result in a driving term for vacua in the region, with uncorrelated, white noise fluctuations. We show that the static susceptibility of the region diverges as the average lifetime of de Sitter vacua approaches the Page lifetime. Thus optimal regions of the landscape are most susceptible to volume influx from their environing landscape. We then show that the rescaled comoving volume fluctuations exhibit a~$1/f$ power spectrum over a broad range of frequencies, precisely at the critical Page lifetime distribution. A~$1/f$ power spectrum is a hallmark of non-equilibrium systems at criticality. In analogy with sand avalanches in the abelian sandpile or neuronal avalanches in the brain, de Sitter vacua at criticality can be thought of as undergoing scale invariant volume fluctuation avalanches. We then study the critical properties of the inflationary multiverse from the information-theoretic perspective. We show that the string landscape has emergent computational capabilities at the critical Page distribution. Specifically, we find that integrated information storage is enhanced when the average lifetime of the vacua is of the order of the Page time. Other indications of criticality, such as optimization between the integrated storage and information storage rates or competition between information storage and transfer, are also discussed. At last, we study the separate topic of the Bose-Einstein condensation and superfluidity at non-zero temperature. We derive the equation of state of a Bose gas accounting for thermal and quantum corrections up to 1-loop order. We work in the Hartree-Fock-Bogoliubov approximation and follow Yukalov's prescription of introducing two chemical potentials, one for the condensed phase and another one for the excited phase, to circumvent the well-known Hohenberg-Martin dilemma. Finally, we connect our results to the hydrodynamical, two-fluid model for finite-temperature superfluids. Our results are relevant for the phenomenology of Bose-Einstein condensate and superfluid dark matter candidates, as well as the color-flavor locking phase of quark matter in neutron stars.