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The complexity of quantum many-body systems is manifested in the vastdiversity of their correlations, making it challenging to distinguish thegeneric from the atypical features. This can be addressed by analyzingcorrelations through ensembles of random states, chosen to faithfully embodythe relevant physical properties. Here, we focus on spins with localinteractions, whose correlations are extremely well captured by tensor networkstates. Adopting an operational perspective, we define ensembles of randomtensor network states in one and two spatial dimensions that admit a sequentialgeneration. As such, they directly correspond to outputs of quantum circuitswith a sequential architecture and random gates. In one spatial dimension, theensemble explores the entire family of matrix product states, while in twospatial dimensions, it corresponds to random isometric tensor network states.We extract the scaling behavior of the average correlations between twosubsystems as a function of their distance. Using elementary concentrationresults, we then deduce the typical case for measures of correlation such asthe von Neumann mutual information and a measure arising from theHilbert-Schmidt norm. We find for all considered cases that the typicalbehavior is an exponential decay (for both one and two spatial dimensions). Weobserve the consistent emergence of a correlation length that depends only onthe underlying spatial dimension and not the considered measure. Remarkably,increasing the bond dimension leads to a higher correlation length in onespatial dimension but has the opposite effect in two spatial dimensions.

Typical Correlation Length of Sequentially Generated Tensor Network States