  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.
