A note on last passage percolation and Schur processes

TL;DR

The authors provide a concise, self-contained derivation of the distributional equivalence between geometric last passage percolation along general down-right paths and Schur processes in full-space, and Pfaffian Schur processes in half-space. Their approach hinges on generalized RSK correspondences via growth diagrams and Greene's theorem, enabling a Markovian dynamics of RSK-type that preserves Schur-process structure. The main results (Theorems T1 and T2) express the joint distributions along specified down-right paths in terms of Schur process measures with explicit normalizations and one-variable specializations. This work strengthens the bridge between LPP, symmetric function theory, and integrable probability, providing accessible, rigorous derivations under broad parameter regimes and path geometries. It also clarifies how RSK-type dynamics can reproduce determinantal and Pfaffian structures observed in full-space and half-space LPP models, respectively.

Abstract

In this note we provide a short proof of the distributional equality between last passage percolation with geometric weights along a general down-right path and Schur processes. We do this in both the full-space and half-space settings, and for general parameters. The main inputs for our arguments are generalizations of the Robinson-Schensted-Knuth correspondence and Greene's theorem due to Krattenthaler, which are based on Fomin's growth diagrams.

Figures (6)

• Figure 1: The left side depicts the array $W = (w_{i,j}: i,j \geq 1)$, an up-right path $\pi$ and a NE-chain $\chi$. The vertices in $\mathbb{Z}^2$ correspond to the midpoints of the squares and are not drawn. The right side depicts $\min(m,n)$ pairwise disjoint up-right paths, with $\pi_i$ connecting $(1,i)$ to $(m, n-k+i)$ that cover the whole $n \times m$ rectangle, for $(m,n) = (7,3)$ and $(m,n) = (3,7)$.
• Figure 2: The left side depicts a down-right path $\gamma$ that connects $(0,n)$ to $(m,0)$ for $n = 4$, $m = 6$. The set $Y(\gamma)$ in (\ref{['Eq.FerresShapeDef']}) is enclosed by bold black lines. The word corresponding to $\gamma$ is $RRDRDDRRRD$. The right side depicts the down-right path $\gamma'$ obtained from $\gamma$ by replacing the second $RD$ in the word corresponding to $\gamma$ with $DR$. The star indicates the unique vertex in $Y(\gamma) \setminus Y(\gamma')$.
• Figure 3: Young diagrams for $\lambda = (5,3,1)$, $\mu = (3,3,2,1)$, and their maximum $\max(\lambda, \mu)$ and minimum $\min(\lambda, \mu)$. The interiors $\mathring{\lambda}$ and $\mathring{\mu}$ are in gray; the boundaries $\partial \lambda$ and $\partial \mu$ are outlined with dashed boxes.
• Figure 4: The top-left depicts $\lambda = (5,4,4,3,2)$ with three pairwise disjoint NE-chains $\chi_1,\chi_2, \chi_3$ drawn inside. The top three diagrams show the triplet of NE-chains $(\chi^m_1, \chi^m_2, \chi^m_3)$ in the proof of Lemma \ref{['Lem.Layers']} for $m = 0,1,2$. The dashed lines indicate $\lambda(\chi^m_3)$ for $m = 0,1,2$. The bottom-left depicts the result of applying the induction hypothesis to $(\chi^3_1, \chi^3_2)$. The bottom-right depicts the $\lambda^1, \lambda^2, \lambda^3$ from Lemma \ref{['Lem.Layers']} in bold lines, and we have $\lambda^1 = (2)$, $\lambda^2 = (3,3,1)$, $\lambda^3 = (5,4,4,2)$.
• Figure 5: The left side depicts the partition $R$ in Lemma \ref{['Lem.OffDiag']} and the vertices in (\ref{['Eq.OffDiag']}) that are contained in $\sqcup_{i = 1}^k \chi_i$ for $m = 7$, $n = 5$ and $k = 3$. The right side depicts the up-right paths $s_1, \dots, s_k$ and $\tilde{s}_1, \dots, \tilde{s}_k$ in the proof of Lemma \ref{['Lem.OffDiag']}. The circles indicate the locations of the points $(i,u_i)$ for $i = 1, \dots, k$.

Theorems & Definitions (25)

• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• Theorem 2.4
• Remark 2.5
• Remark 2.6
• Theorem 2.7
• Remark 2.8
• Remark 3.1
• Proposition 3.2