  A linear polymer grafted to a hard wall and underneath an AFM tip can be
modelled in a lattice as a grafted lattice polymer (or self-avoiding walk)
compressed underneath a piston approaching the wall. As the piston approaches
the wall the increasingly confined polymer escapes from the confined region to
explore conformations beside the piston. This conformational change is believed
to be a phase transition in the thermodynamic limit, and has been argued to be
first order, based on numerical results in reference [12]. In this paper a
lattice self-avoiding walk model of the escape transition is constructed. It is
proven that this model has a critical point in the thermodynamic limit
corresponding to the escape transition of compressed grafted linear polymers.
This result relies on the analysis of ballistic self-avoiding walks in slits
and slabs in the square and cubic lattices. Additionally, numerical estimates
of the location of the escape transition critical point is reported based on
Monte Carlo simulations of self-avoiding walks in slits and in slabs.
