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

Dan Betea|arXiv (Cornell University)|Sep 18, 2018

Random Matrices and Applications41 references3 citations

TL;DR

This paper provides a combinatorial proof that the last passage percolation (LPP) time in a point-to-half-line-reflected geometry on a $2n\times n\times n$ quarter square decomposes into a product of two independent LPP distributions: one for point-to-line geometry and one for point-to-point-reflected (half-space) geometry, both on $n\times n\times n$ triangles. As a result, the limiting fluctuation distribution is Tracy–Widom $\mathrm{GOE}^2$, derived purely from the product structure without requiring asymptotic analysis.

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 $2n imes n imes n$ 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 $n imes n imes n$ 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 $q o 0$) and more recently of results from Bisi's PhD thesis (the limit $q o 1$).

Motivation & Objective

• To establish a combinatorial decomposition of the generating function for last passage percolation in a point-to-half-line-reflected geometry on a $2n\times n\times n$ quarter square.
• To show that this generating function factors into the product of generating functions for point-to-line and point-to-point-reflected geometries in $n\times n\times n$ triangles.
• To recover Tracy–Widom $\mathrm{GOE}^2$ fluctuations in the $n\to\infty$ limit as a direct consequence of the product structure and known asymptotics of the components.
• To demonstrate that the $\mathrm{GOE}^2$ fluctuation result follows without requiring complex asymptotic analysis, relying instead on bijective and representation-theoretic techniques.

Proposed method

• The proof uses the Robinson–Schensted–Knuth (RSK) correspondence and its column-insertion variant ($\mathrm{colRSK}$) to relate path weights to Schur function identities.
• It applies bounded Littlewood identities for Schur functions to establish a skew Cauchy identity that underlies the factorization of generating functions.
• The local growth rules of $\mathrm{rowRSK}$ and $\mathrm{colRSK}$ are used to track interlacing partitions and path weights in discrete percolation models.
• The method relies on Greene’s theorem, which connects the sum of the first $k$ parts of the output partition to the maximal weight of $k$ non-intersecting paths.
• The geometry is defined via three discrete domains: $D^{\mathrm{p2hlr}}_n$ (quarter square), $D^{\mathrm{p2pr}}_n$ (half-space), and $D^{\mathrm{p2l}}_n$ (point-to-line), each with independent geometric weights.
• The weight assignment assigns $x_i x_j$ to off-diagonal squares and $\sqrt{q}$ on the diagonal, with $q$ as the off-diagonal parameter.

Experimental results

Research questions

• RQ1 Does the generating function for last passage percolation in a point-to-half-line-reflected $2n\times n\times n$ quarter square factor into simpler components corresponding to point-to-line and point-to-point-reflected geometries?
• RQ2 Can the Tracy–Widom $\mathrm{GOE}^2$ fluctuation scaling in the large $n$ limit be derived combinatorially from such a factorization?
• RQ3 What role do the RSK and $\mathrm{colRSK}$ bijections play in establishing the product structure of the LPP time distribution?
• RQ4 How do bounded Littlewood identities and skew Schur function identities support the decomposition of the generating function?
• RQ5 Can the $\mathrm{GOE}^2$ limit be recovered without asymptotic analysis, relying solely on the factorization and known asymptotics of the components?

Key findings

• The generating function for last passage percolation in the $2n\times n\times n$ point-to-half-line-reflected geometry factors as the product of the generating functions for point-to-line and point-to-point-reflected geometries on $n\times n\times n$ triangles.
• The probability distribution of the last passage time in the quarter square is the product of the distributions in the point-to-line and half-space geometries.
• The $\mathrm{GOE}^2$ fluctuation scaling in the $n\to\infty$ limit follows directly from the product structure and the known $\mathrm{GOE}$ fluctuations of the component geometries.
• The result is established combinatorially using RSK and $\mathrm{colRSK}$ bijections, along with skew Cauchy identities and bounded Littlewood identities.
• The derivation of $\mathrm{GOE}^2$ fluctuations does not require Toeplitz+Hankel determinants or Riemann–Hilbert analysis, unlike previous approaches.
• The method suggests a pathway to generalizing results to non-free fermionic models via representation-theoretic and bijective techniques.