  We investigate the highly polarized limit of a two-dimensional (2D) Fermi
gas, where we effectively have a single spin-down impurity atom immersed in a
spin-up Fermi sea. By constructing variational wave functions for the impurity,
we map out the ground state phase diagram as a function of mass ratio M/m and
interaction strength. In particular, we determine when it is favorable for the
dressed impurity (polaron) to bind particles from the Fermi sea to form a
dimer, trimer or even larger clusters. Similarly to 3D, we find that the Fermi
sea favors the trimer state so that it exists for M/m less than the critical
mass ratio for trimer formation in the vacuum. We also find a region where
dimers have finite momentum in the ground state, a scenario which corresponds
to the Fulde-Ferrell-Larkin-Ovchinnikov superfluid state in the limit of large
spin imbalance. For equal masses (M=m), we compute rigorous bounds on the
polaron-dimer transition, and we show that the polaron energy and residue is
well captured by the variational approach, with the former quantity being in
good agreement with experiment. When there is a finite density of impurities,
we find that this polaron-dimer transition is preempted by a first-order
superfluid-normal transition at zero temperature, but it remains an open
question what happens at finite temperature.
