Quantum entanglement is the most surprising feature of quantum mechanics. Ground states of quantum many-body systems typically exhibit the area law behavior in the entanglement entropy, which measures the amount of entanglement between a subsystem and the rest of the system. Recently, a class of solvable one-dimensional spin models with local interactions has been constructed, in which the ground state is expressed as a superposition of random walks, and has much larger entanglement entropy proportional to a square root of the volume. It seems to share some features of random geometry with matrix models. In this talk, I explain properties of these model and their implication to quantum gravity. In computation of the Renyi entropy for them, a new phase transition takes place in varying a parameter in the definition of the Renyi entropy.