  We introduce a family of discrete determinantal point processes related to
orthogonal polynomials on the real line, with correlation kernels defined via
spectral projections for the associated Jacobi matrices. For classical weights,
we show how such ensembles arise as limits of various hypergeometric orthogonal
polynomials ensembles.
  We then prove that the q-Laplace transform of the height function of the ASEP
with step initial condition is equal to the expectation of a simple
multiplicative functional on a discrete Laguerre ensemble --- a member of the
new family. This allows us to obtain the large time asymptotics of the ASEP in
three limit regimes: (a) for finitely many rightmost particles; (b) GUE
Tracy-Widom asymptotics of the height function; (c) KPZ asymptotics of the
height function for the ASEP with weak asymmetry. We also give similar results
for two instances of the stochastic six vertex model in a quadrant. The proofs
are based on limit transitions for the corresponding determinantal point
processes.
