  We develop a "self-avoiding walks" approximation and use it to calculate the
spectral function of a single hole introduced into the 2D square lattice Ising
antiferromagnet. The obtained local spectral function qualitatively agrees with
the exact diagonalisation result and is largely incoherent. Such a result stays
in contrast with the spectrum obtained on a Bethe lattice, which consists of
the well-separated quasiparticle-like peaks and stems from the motion of a hole
in an effective linear potential. We determine that this onset of the
incoherent spectrum on a square lattice (i) is not triggered by the so-called
Trugman loops but (ii) originates in the warping of the linear potential by the
interactions between magnons created along the tangential paths of the moving
hole.
