Anticoncentration and state design of random tensor networks

We investigate quantum random tensor network states where the bond dimensions scale polynomially with the system size, $N$. Specifically, we examine the delocalization properties of random Matrix Product States (RMPS) in the computational basis by deriving an exact analytical expression for the Inverse Participation Ratio (IPR) of any degree, applicable to both open and closed boundary conditions. For bond dimensions $\chi \sim \gamma N$, we determine the leading order of the associated overlaps probability distribution and demonstrate its convergence to the Porter-Thomas distribution, characteristic of Haar-random states, as $\gamma$ increases. Additionally, we provide numerical evidence for the frame potential, measuring the $2$-distance from the Haar ensemble, which confirms the convergence of random MPS to Haar-like behavior for $\chi \gg \sqrt{N}$. We extend this analysis to two-dimensional systems using random Projected Entangled Pair States (PEPS), where we similarly observe the convergence of IPRs to their Haar values for $\chi \gg \sqrt{N}$. These findings demonstrate that random tensor networks with bond dimensions scaling polynomially in the system size are fully Haar-anticoncentrated and approximate unitary designs, regardless of the spatial dimension.