  Studies of the effective regime of loop quantum gravity (LQG) revealed that,
in the limit of Planckian curvature scales, spacetime may undergo a transition
from the Lorentzian to Euclidean signature. This effect is a consequence of
quantum modifications of the hypersurface deformation algebra, which in the
linearized case is equivalent to a deformed version of the Poincar\'e algebra.
In this paper the latter relation is applied to the LQG-inspired hypersurface
deformation algebra that is characterized by the above mentioned signature
change.
  While the exact form of the deformed Poincar\'e algebra is not uniquely
determined, the algebra under consideration is representative enough to capture
a number of qualitative features. In particular, the analysis reveals that the
signature change can be associated with two symmetric invariant energy scales,
which separate three physically disconnected momentum subspaces.
  Furthermore, the invariant measure on momentum space is derived, which allows
to properly define the average return probability, characterizing a fictitious
diffusion process on spacetime. The diffusion is subsequently studied in the
momentum representation for all possible variants of the model. Finally, the
spectral dimension of spacetime is calculated in each case as a function of the
scale parameter. In the most interesting case the deformation is of the
asymptotically ultralocal type and the spectral dimension undergoes a reduction
to $d_S = 1$ in the UV limit.
