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In this note we give a(nother) combinatorial proof of an old result ofBaik--Rains: that for appropriately considered independent geometric weights,the generating series for last passage percolation polymers in a $2n \times n\times n$ quarter square (point-to-half-line-reflected geometry) splits as theproduct of two simpler generating series---that for last passage percolationpolymers in a point-to-line geometry and that for last passage percolation in apoint-to-point-reflected (half-space) geometry, the latter both in an $n \timesn \times n$ triangle. As a corollary, for iid geometric random variables---ofparameter $q$ off-diagonal and parameter $\sqrt{q}$ on the diagonal---we seethat the last passage percolation time in said quarter square obeysTracy--Widom $\mathrm{GOE}^2$ fluctuations in the large $n$ limit as both thepoint-to-line and the point-to-point-reflected geometries have known GOEfluctuations. This is a discrete analogue of a celebrated Baik--Rains theorem(the limit $q \to 0$) and more recently of results from Bisi's PhD thesis (thelimit $q \to 1$).

On the combinatorics of last passage percolation in a quarter square and $\mathrm{GOE}^2$ fluctuations