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Relativistic quantum field theories become scale-free (conformal) at short distances. On the other hand, gravitational theories are expected to keep memory of the Planck scale at arbitrarily high energies. I will describe a class of exactly solvable relativistic models which exhibit a new type of asymptotic behavior and never loose memory of the fundamental scale. In spite of their simplicity, these models exhibit surprisingly many properties expected from a more mature gravitational theories. In particular, they do not allow for sharply defined local observables.

This is the Physics Department Colloquium. The topic is relevant for the Physics and Mathematics of the Universe Initiative.