For each of the four particle processes given by Dieker and Warren
[arXiv:0707.1843], we show the n-step transition kernels are given by the
(dual) (weak) refined symmetric Grothendieck functions up to a simple overall
factor. We do so by encoding the particle dynamics as the basis of free
fermions first introduced by the first author, which we translate into deformed
Schur operators acting on partitions. We provide a direct combinatorial proof
of this relationship in each case, where the defining tableaux naturally
describe the particle motions.Comment: 52 pages, 5 figures, 2 table