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Using the standard formulation of algebraic random walks (ARWs) viacoalgebras, we consider ARWs for co-and Hopf-algebraic structures in the ringof symmetric functions. These derive from different types of products bydualisation, giving the dual pairs of outer multiplication and outer coproduct,inner multiplication and inner coproduct, and symmetric function plethysm andplethystic coproduct. Adopting standard coordinates for a class of measures(and corresponding distribution functions) to guarantee positivity and correctnormalisation, we show the effect of appropriate walker steps of the outer,inner and plethystic ARWs. If the coordinates are interpreted as heights oroccupancies of walker(s) at different locations, these walks introducetranslations, dilations (scalings) and inflations of the height coordinates,respectively.

Algebraic random walks in the setting of symmetric functions