Landscape cosmology posits the existence of a convoluted, multidimensional, scalar potential -- the "landscape" -- with vast numbers of metastable minima. Random matrices and random functions in many dimensions provide toy models of the landscape, allowing the exploration of conceptual issues associated with these scenarios. We compute the relative number and slopes of minima as a function of the vacuum energy $\Lambda$ in an $N$-dimensional Gaussian random potential, quantifying the associated probability density, $p(\Lambda)$. After normalisations $p(\Lambda)$ depends only on the dimensionality $N$ and a single free parameter $\gamma$, which is related to the power spectrum of the random function. For a Gaussian landscape with a Gaussian power spectrum, the fraction of positive minima shrinks super-exponentially with $N$; at $N=100$, $p(\Lambda>0) \approx 10^{-780}$. Likewise, typical eigenvalues of the Hessian matrices reveal that the flattest approaches to typical minima grow flatter with $N$, while the ratio of the slopes of the two flattest directions grows with $N$. We discuss the implications of these results for both swampland and conventional anthropic constraints on landscape cosmologies. In particular, for parameter values when positive minima are extremely rare, the flattest approaches to minima where $\Lambda \approx 0$ are much flatter than for typical minima, increasingly the viability of quintessence solutions.

中文翻译：

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真空在随机景观势中的分布

景观宇宙学假设存在一个复杂的、多维的、标量势——“景观”——具有大量亚稳态最小值。许多维度的随机矩阵和随机函数提供了景观的玩具模型，允许探索与这些场景相关的概念问题。我们将最小值的相对数量和斜率计算为 $N$ 维高斯随机势中真空能量 $\Lambda$ 的函数，量化相关的概率密度 $p(\Lambda)$。归一化后 $p(\Lambda)$ 仅取决于维度 $N$ 和单个自由参数 $\gamma$，后者与随机函数的功率谱有关。对于具有高斯功率谱的高斯景观，正极小值的比例以 $N$ 超指数地缩小；在 $N=100$ 时，$p(\Lambda>0) \大约 10^{-780}$。同样，Hessian 矩阵的典型特征值表明，典型最小值的最平坦方法随着 $N$ 变得更加平坦，而两个最平坦方向的斜率之比随着 $N$ 增长。我们讨论了这些结果对沼泽地和传统人为约束对景观宇宙学的影响。特别是，对于正极小值极少的参数值，最平坦的极小值方法比典型的极小值更平坦，提高了典型解的可行性。我们讨论了这些结果对沼泽地和传统人为约束对景观宇宙学的影响。特别是，对于正极小值极少的参数值，最平坦的极小值方法比典型的极小值更平坦，提高了典型解的可行性。我们讨论了这些结果对沼泽地和传统人为约束对景观宇宙学的影响。特别是，对于正极小值极少的参数值，最平坦的极小值方法比典型的极小值更平坦，提高了典型解的可行性。