We study random circuit models of constrained fracton dynamics. In this model, particles on a one-dimensional lattice undergo random local motion subject to both charge and dipole moment conservation. The constitutive space of this system provides a continuous phase transition between weakly fragmented (“thermalized”) and strongly fragmented (“unthermalized”) phases as a function of particle number density. indicate. We now identify the exact solution for the critical density $n_c$ by mapping it to two different problems of combinatorics. Specifically, if evolution is driven by operators acting on $\ell$ adjacent sites, the critical density is given by $n_c = 1/(\ell -2)$. We identify a significant scaling near the transition and show that there is a universal value of the correlation length exponent $\nu = 2$. Confirm theoretical results with numerical simulations. At the thermalization stage, the kinetic exponent is quasi-diffusive: $z=4$, but increases to $z_c \gtrsim 6$ at the critical point.

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