  We analyze the ground state energy for N fermions in a two-dimensional box
interacting with an impurity particle via two-body point interactions. We allow
for mass ratios M > 1.225 between the impurity mass and the mass of a fermion
and consider arbitrarily large box sizes while keeping the Fermi energy fixed.
Our main result shows that the ground state energy in the limit of weak
coupling is given by the polaron energy. The polaron energy is an energy
estimate based on trial states up to first order in particle-hole expansion,
which was proposed by Chevy in the physics literature. For the proof we apply a
Birman-Schwinger principle that was recently obtained by Griesemer and Linden.
One main new ingredient is a suitable localization of the polaron energy.
