Directed Percolation Criticality in Eternal Inflation

False-vacuum eternal inflation can be described as a random walk on the network of vacua of the string landscape. In this paper we show that the problem can be mapped naturally to a problem of directed percolation. The mapping relies on two general and well-justified approximations for transition rates: 1.~the downward approximation, which neglects ``upward" transitions, as these are generally exponentially suppressed; 2. the dominant decay channel approximation, which capitalizes on the fact that tunneling rates are exponentially staggered. Lacking detailed knowledge of the string landscape, we model the network of vacua as random graphs with arbitrary degree distribution, including Erd\"os-R\'enyi and scale-free graphs. As a complementary approach, we also model regions of the landscape as regular lattices, specifically Bethe lattices. We find that the uniform-in-time probabilities proposed in our previous work favor regions of the landscape poised at the directed percolation phase transition. This raises the tantalizing prospect of deriving universal statistical distributions for physical observables, characterized by critical exponents that are insensitive to the details of the underlying landscape. We illustrate this with the cosmological constant, and show that the resulting distribution peaks as a power-law for small positive vacuum energy, with a critical exponent uniquely determined by the random graph universality class.