Entropy of random entangling surfaces

We consider the situation when a globally defined four-dimensional field system is separated on two entangled sub-systems by a dynamical (random) two-dimensional surface. The reduced density matrix averaged over ensemble of random surfaces of fixed area and the corresponding average entropy are introduced. The average entanglement entropy is analyzed for a generic conformal field theory in four dimensions. Two important particular cases are considered. In the first, both the intrinsic metric on the entangling surface and the spacetime metric are fluctuating. An important example of this type is when the entangling surface is a black hole horizon, the fluctuations of which cause necessarily the fluctuations in the spacetime geometry. In the second case, the spacetime is considered to be fixed. The detail analysis is carried out for the random entangling surfaces embedded in flat Minkowski spacetime. In all cases the problem reduces to an effectively two-dimensional problem of random surfaces which can be treated by means of the well-known conformal methods. Focusing on the logarithmic terms in the entropy we predict the appearance of a new $\ln\ln(A)$ term.