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.

最近，在景观上虚假的永恒膨胀的背景下，提出了一种基于搜索优化的动态选择机制。由局部真空转变定义的搜索算法在动态性处于临界状态的地形区域中是最佳的，de Sitter vacua的平均寿命约为其Page time。本文的目的是阐明Page寿命期间动态相变的性质。我们专注于景观的有限区域，该区域与景观的其余部分交换体积，因此充当开放系统。通过变量的变化，将控制de Sitter vacua协同运动量的主方程映射到耦合过阻尼随机振荡器的随机方程，即著名的Ornstein-Uhlenbeck过程。假定作为环境的景观的其余部分将导致该区域中所有站点具有不相关的白噪声波动（尽管不一定是高斯噪声）而不会消失。我们首先显示，随着de Sitter vacua的平均寿命接近Page time，振荡器的静态磁化率发生了变化。因此，景观的最佳区域最容易受到周围景观的大量涌入的影响。然后，我们证明了振荡器的位移波动表现出一个 我们首先显示，随着de Sitter vacua的平均寿命接近Page time，振荡器的静态磁化率发生了变化。因此，景观的最佳区域最容易受到周围景观的大量涌入的影响。然后，我们证明了振荡器的位移波动表现出一个 我们首先显示，随着de Sitter vacua的平均寿命接近Page time，振荡器的静态磁化率发生了变化。因此，景观的最佳区域最容易受到周围景观的大量涌入的影响。然后，我们证明了振荡器的位移波动表现出一个1 /楼功率谱在很宽的频率范围内，精确地处于临界的Page寿命分布。一个1 /楼功率谱是非平衡系统处于临界状态的标志。与阿贝尔沙堆中的雪崩或大脑中的神经元雪崩类似，处于临界状态的de Sitter vacua可以认为是经历了尺度不变的体积波动雪崩。