We study a model of one dimensional particles, performing geometrically weighted random walks that are conditioned not to intersect. The walkers start at equidistant points and end at consecutive integers. A naturally associated tiling model can be viewed as one of placing boxes on a staircase. For a particular value of the parameters we obtain a known model for the Schur measure, which has the sine kernel as a scaling limit. However, for other parameter values the process at the local scale, close to the starting points, does not fall in the universality class of the sine kernel. Instead, as the number of walkers tends to infinity we obtain a new family of kernels describing the local correlations. We shall describe these limits and some of their interesting features. This is joint work with Maurice Duits.