  We extend results of Zeitouni-Zelditch on large deviations principles for
zeros of Gaussian random polynomials $s$ in one complex variable to certain
non-Gaussian ensembles that we call $P(\phi)_2$ random polynomials. The
probability measures are of the form $e^{- S(f)} df$ where the actions $S(f)$
are finite dimensional analgoues of those of $P(\phi)_2$ quantum field theory.
The speed and rate function are the same as in the associated Gaussian case. As
a corollary, we prove that the expected distribution of zeros in the
$P(\phi)_2$ ensembles tends to the same equilibrium measure as in the Gaussian
case.
