  We construct a vertex model whose partition function is a refined dual
Grothendieck polynomial, where the states are interpreted as nonintersecting
lattice paths. Using this, we show refined dual Grothendieck polynomials are
multi-Schur functions and give a number of identities, including a Littlewood
and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection
between dual Grothendieck polynomials and the last passage percolation (LPP)
stochastic process discussed by Johansson. By refining algebraic techniques of
Johansson, we show Jacobi-Trudi formulas for skew refined dual Grothendieck
polynomials conjectured by Grinberg and recover a relation between LPP and the
Schur process due to Baik and Rains. Lastly, we extend our vertex model
techniques to show some identities for refined Grothendieck polynomials,
including a Jacobi-Trudi formula.
