  Recently a dynamical selection mechanism for vacua based on search
optimization was proposed in the context of false-vacuum eternal inflation on
the landscape. The search algorithm, defined by local vacuum transitions, is
optimal in regions of the landscape where the dynamics are tuned at
criticality, with de Sitter vacua having an average lifetime of order their
Page time. The purpose of this paper is to shed light on the nature of the
dynamical phase transition at the Page lifetime. We focus on a finite region of
the landscape, which exchanges volume with the rest of the landscape and as
such acts as an open system. Through a change of variables the master equation
governing the comoving volume of de Sitter vacua is mapped to a stochastic
equation for coupled overdamped stochastic oscillators -- the well-known
Ornstein-Uhlenbeck process. The rest of the landscape, which acts as an
environment, is assumed to result in a non-vanishing driving term for all sites
in the region with uncorrelated, white noise fluctuations (though not
necessarily Gaussian). We first show that the static susceptibility of the
oscillators diverges as the average lifetime of de Sitter vacua approaches the
Page time. Thus, optimal regions of the landscape are most susceptible to
volume influx from their environing landscape. We then show that the
displacement fluctuations for the oscillators 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.

High Energy Physics - Theory