The string landscape, together with the mechanism of eternal inflation for populating vacua, leads to the seemingly inescapable conclusion that we are part of a vast multiverse. As an inhabitant of the multiverse, how should we reason probabilistically about the expected physical properties of our observable universe? Probabilities in eternal inflation are usually defined in terms of frequencies, but this approach has failed to yield a unique answer. In this talk, I will present a different approach to the problem, based on Bayesian reasoning. By adopting the least informative priors on the model parameters, we will be led to well-defined, time-reparametrization invariant (and non-anthropic) probabilities for occupying different vacua. Remarkably, these probabilities favor vacua whose surrounding landscape topography is that of a deep valley or funnel, akin to folding funnels of naturally-occurring proteins. Furthermore, by modeling the landscape of vacua as a random network, I will show that our probabilities favor regions that are close to the directed percolation phase transition. As usual, the predictive power of criticality lies in scale invariant observables characterized by critical exponents. As an example, I will show that the probability distribution for the cosmological constant is power-law, with a particular critical exponent, and favors a naturally small and positive vacuum energy. Tantalizingly, this hints at a deep connection between non-equilibrium critical phenomena on the landscape and the near-criticality of our universe.
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