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

Abstract

In this note we give a(nother) combinatorial proof of an old result of Baik--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 the product of two simpler generating series---that for last passage percolation polymers in a point-to-line geometry and that for last passage percolation in a point-to-point-reflected (half-space) geometry, the latter both in an $n\times n\times n$ triangle. As a corollary, for iid geometric random variables---of parameter $q$ off-diagonal and parameter $\sqrt{q}$ on the diagonal---we see that the last passage percolation time in said quarter square obeys Tracy--Widom ${\mathrm{GOE}}^2$ fluctuations in the large $n$ limit as both the point-to-line and the point-to-point-reflected geometries have known GOE fluctuations. 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 (the limit $q\to 1$).

16 pages, 4 figures; v2: improved references, corrected off-by-two error

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Cited by in corpus (1)

• Discrete and continuous Muttalib--Borodin processes I: the hard edge