  The coupling between gravity and matter provides an intriguing length scale
in the infrared for theories of gravity within Einstein-Hilbert action and
beyond. In particular, we will show that such an infrared length scale is
determined by the number of gravitons $N_{g}\gg1$ associated to a given mass in
the non-relativistic limit. After tracing out the matter degrees of freedom,
the graviton vacuum is found to be in a displaced vacuum with an occupation
number of gravitons $N_{g}\gg1$. In the infrared, the length scale appears to
be $L=\sqrt{N_{g}}\ell_{p}$, where $L$ is the new infrared length scale, and
$\ell_{p}$ is the Planck length. In a specific example, we have found that the
infrared length scale is greater than the Schwarzschild radius for a slowly
moving in-falling thin shell of matter. We will argue that the appearance of
such an infrared length scale in higher curvature theories of gravity, such as
in quadratic and cubic curvature theories of gravity, is also expected.
Furthermore, we will show that gravity is fundamentally different from the
electromagnetic interaction where the number of photons, $N_{p}$, is the fine
structure constant after tracing out an electron wave function.
