  We study scalar fields propagating on Euclidean dynamical triangulations
(EDT). In this work we study the interaction of two scalar particles, and we
show that in the appropriate limit we recover an interaction compatible with
Newton's gravitational potential in four dimensions. Working in the quenched
approximation, we calculate the binding energy of a two-particle bound state,
and we study its dependence on the constituent particle mass in the
non-relativistic limit. We find a binding energy compatible with what one
expects for the ground state energy by solving the Schr\"{o}dinger equation for
Newton's potential. Agreement with this expectation is obtained in the
infinite-volume, continuum limit of the lattice calculation, providing
non-trivial evidence that EDT is in fact a theory of gravity in four
dimensions. Furthermore, this result allows us to determine the lattice spacing
within an EDT calculation for the first time, and we find that the various
lattice spacings are smaller than the Planck length, suggesting that we can
achieve a separation of scales and that there is no obstacle to taking a
continuum limit. This lends further support to the asymptotic safety scenario
for gravity.
