Lattice Paths, Sampling without Replacement, and the Kernel Method

In this work we consider weighted lattice paths in the quarter plane N0 × N0. The steps are given by (m, n) → (m − 1, n), (m, n) → (m, n − 1) and are weighted as follows: (m, n)→ (m− 1, n) by m/(m + n) and step (m, n)→ (m, n− 1) by n/(m + n). The considered lattice paths are absorbed at lines y = x/t− s/t with t ∈ N and s ∈ N0. We provide explicit formulæ for the sum of the weights of paths, starting at (m, n), which are absorbed at a certain height k at lines y = x/t−s/t with t ∈ N and s ∈ N0, using a generating functions approach and the kernel method. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, surprisingly obtaining a total of five phase changes.