  An end-grafted flexible polymer chain in 3d space between two pistons
undergoes an abrupt transition from a confined coil to a flower-like
conformation when the number of monomers in the chain, N, reaches a critical
value. In 2d geometry, excluded volume interactions between monomers of a chain
confined inside a strip of finite length 2L transform the coil conformation
into a linear string of blobs. However, the blob picture raises questions on
the nature of this escape transition. To check the theoretical predictions
based on the blob picture we study 2d single polymer chains with excluded
volume interactions and with one end grafted in the middle of a strip of length
  2L and width H by simulating self-avoiding walks on a square lattice with the
pruned-enriched-Rosenbluth method (PERM). We estimate the free energy, the
end-to-end distance, the number of imprisoned monomers, the order parameter,
and its distribution. It is shown that in the thermodynamic limit of large N
and L but finite L/N, there is a small but finite jump in several average
characteristics, including the order parameter. We also present a theoretical
description based on the Landau free energy approach, which is in good
agreement with the simulation results. Both simulation results and the
analytical theory indicate that the 2d escape transition is a weak first-order
phase transition.
