In this paper we introduce a simple discrete stochastic model of eternal inflation that shares many of the most important features of the continuum theory as it is now understood. The model allows us to construct a multiverse and rigorously analyze its properties. Although simple and easy to solve, it has a rich mathematical structure underlying it. Despite the discreteness of the space-time the theory exhibits an unexpected nonperturbative analog of conformal symmetry that acts on the boundary of the geometry. The symmetry is rooted in the mathematical properties of trees, $p$-adic numbers, and ultrametric spaces; and in the physical property of detailed balance. We provide self-contained elementary explanations of the unfamiliar mathematical concepts, which have also appeared in the study of the $p$-adic string. The symmetry acts on a huge collection of very low-dimensional “multiverse fields” that are not associated with the usual perturbative degrees of freedom. They are connected with the late-time statistical distribution of bubble-universes in the multiverse. The conformal symmetry which acts on the multiverse fields is broken by the existence of terminal decays—to hats or crunches—but in a particularly simple way. We interpret this symmetry breaking as giving rise to an arrow of time. The model is used to calculate statistical correlations at late time and to discuss the measure problem. We show that the natural cutoff in the model is the analog of the so-called light-cone-time cutoff. Applying the model to the problem of the cosmological constant, we find agreement with earlier work.

• Received 1 December 2011

DOI:https://doi.org/10.1103/PhysRevD.85.063516