We obtain sharp upper and lower bounds for the downward moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, which strengthen the approach developed in Asselah and Schapira (Sci Éc Norm Supér 50(4):755–786, 2017).

对于尺寸为5或更大的随机游走的范围的体积的向下适度偏差，我们获得了明显的上限和下限。我们的结果包括两个状态：一个用于小偏差的高斯方案，一个用于较大偏差的拉伸指数方案。在后一种情况下，我们表明，以中等偏差事件为条件，步行会将其范围的一小部分折叠成球状子集。此外，我们还在维度3中提供了新的路径属性。除了牛顿能力在这项研究中发挥的关键作用外，我们还介绍了两个具有普遍意义的原创思想，这些思想加强了在Asselah和Schapira中开发的方法（SciécNormSupér50（4）：755-786，2017）。