Tensor network states are used extensively as a mathematically convenient description of physically relevant states of many-body quantum systems. Those built on regular lattices, i.e. matrix product states (MPS) in dimension 1 and projected entangled pair states (PEPS) in dimension 2 or higher, are of particular interest in condensed matter physics. The general goal of this work is to characterize which features of MPS and PEPS are generic and which are, on the contrary, exceptional. This problem can be rephrased as follows: given an MPS or PEPS sampled at random, what are the features that it displays with either high or low probability? One property which we are particularly interested in is that of having either rapidly decaying or long-range correlations. In a nutshell, our main result is that translation-invariant MPS and PEPS typically exhibit exponential decay of correlations at a high rate. We have two distinct ways of getting to this conclusion, depending on the dimensional regime under consideration. Both yield intermediate results which are of independent interest, namely: the parent Hamiltonian and the transfer operator of such MPS and PEPS typically have a large spectral gap. In all these statements, our aim is to get a quantitative estimate of the considered quantity (generic correlation length or spectral gap), which has the best possible dependency on the physical and bond dimensions of the random MPS or PEPS.

张量网络状态被广泛用作对多体量子系统的物理相关状态的数学上方便的描述。那些建立在规则晶格上的，即维度 1 中的矩阵乘积状态 (MPS) 和维度 2 或更高维度中的投影纠缠对状态 (PEPS)，在凝聚态物理中特别有趣。这项工作的总体目标是表征 MPS 和 PEPS 的哪些特征是通用的，而相反，哪些是例外的。这个问题可以重新表述如下：给定一个随机采样的 MPS 或 PEPS，它以高概率或低概率显示哪些特征？我们特别感兴趣的一个特性是具有快速衰减或长程相关性的特性。简而言之，我们的主要结果是平移不变的 MPS 和 PEPS 通常以高速率呈现相关性的指数衰减。我们有两种不同的方法可以得出这个结论，这取决于所考虑的维度制度。两者都产生具有独立兴趣的中间结果，即：这种 MPS 和 PEPS 的父哈密顿量和转移算子通常具有很大的光谱间隙。在所有这些陈述中，我们的目标是对所考虑的数量（通用相关长度或光谱间隙）进行定量估计，它对随机 MPS 或 PEPS 的物理和键合尺寸具有最佳的依赖性。两者都产生具有独立兴趣的中间结果，即：这种 MPS 和 PEPS 的父哈密顿量和转移算子通常具有很大的光谱间隙。在所有这些陈述中，我们的目标是对所考虑的数量（通用相关长度或光谱间隙）进行定量估计，它对随机 MPS 或 PEPS 的物理和键合尺寸具有最佳的依赖性。两者都产生具有独立兴趣的中间结果，即：这种 MPS 和 PEPS 的父哈密顿量和转移算子通常具有很大的光谱间隙。在所有这些陈述中，我们的目标是对所考虑的数量（通用相关长度或光谱间隙）进行定量估计，它对随机 MPS 或 PEPS 的物理和键合尺寸具有最佳的依赖性。