  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.
