  Observations of mixed modes in evolved low-mass stars enable us to probe the
properties of not only the outer envelope of these stars, but also their deep
layers. Among the seismic parameters associated with mixed modes, the gravity
offset, denoted with $\varepsilon_{\rm g}$, is expected to reveal information
on the boundaries of the inner buoyancy resonant cavity. This parameter was
recently measured for hundreds of stars observed by the Kepler satellite and
its value was shown to change during the evolution on the red giant branch. In
this article, we theoretically investigate the reasons for such a variation in
terms of structure properties. Using available asymptotic analyses and a simple
model of the Brunt-V\"ais\"al\"a and Lamb frequencies, we derived an analytical
expression of $\varepsilon_{\rm g}$ for dipolar modes and compared its
predictions to observations. First, we show that the asymptotic value of
$\varepsilon_{\rm g}$ well agrees with the mean value observed at the beginning
of the ascent of the red giant branch, which results from the high density
contrast between the helium core and the base of the convective envelope.
Second, we demonstrate that the predicted value also explains the sharp
decrease in $\varepsilon_{\rm g}$ observed for the more luminous red giant
stars of the sample. This rapid drop turns out to occur just before the
luminosity bump and result from the kink of the Brunt-V\"ais\"al\"a frequency
near the upper turning point associated with the buoyancy cavity as stars
evolve and this latter becomes close to the base of the convective envelope.The
observed variation in $\varepsilon_{\rm g}$ and its link with the internal
properties on the red giant branch are now globally understood. This motivates
further analyses of the potential of this parameter as a seismic diagnosis of
the region located between the helium core and the convective envelope.
