  Understanding the nature of entanglement growth in many-body systems is one
of the fundamental questions in quantum physics. Here, we study this problem by
characterizing the entanglement fluctuations and distribution of $(d+1)$ qubit
lattice evolved under a random unitary circuit. Focusing on Clifford gates, we
perform extensive numerical simulations of random circuits in $1\le d\le 4$
dimensions. Our findings demonstrate that properties of growth of bipartite
entanglement entropy are characterized by the roughening exponents of a
$d$-dimensional membrane in a $(d+1)$ elastic medium.
