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Curve separation in supercritical half-space last passage percolation

Evgeni Dimitrov, Zhengye Zhou

TL;DR

This work analyzes line ensembles arising from half-space geometric LPP with symmetric weights in the supercritical regime $c>1$. By leveraging a distributional identity with the Pfaffian Schur process, the authors obtain a Pfaffian point-process representation and a Gibbs-interlacing structure, enabling precise kernel asymptotics via steepest-descent analyses. They show a full-curve phase separation: the top curve converges to a Brownian motion under $N^{1/2}$-horizontal and $N$-scale adjustments, while the remaining curves converge to the Airy line ensemble under $N^{1/3}$ vertical and $N^{2/3}$ horizontal scaling. This establishes a KPZ-class curve-separation phenomenon and provides a robust framework for handling both determinantal and Pfaffian line ensembles through kernel convergence and Gibbsian tightness arguments.

Abstract

We study line ensembles arising naturally in symmetrized/half-space geometric last passage percolation (LPP) on the $N \times N$ square. The weights of the model are geometrically distributed with parameter $q^2$ off the diagonal and $cq$ on the diagonal, where $q \in (0,1)$ and $c \in (0, q^{-1})$. In the supercritical regime $c > 1$, we show that the ensembles undergo a phase transition: the top curve separates from the rest and converges to a Brownian motion under $N^{1/2}$ fluctuations and $N$ spatial scaling, while the remaining curves converge to the Airy line ensemble under $N^{1/3}$ fluctuations and $N^{2/3}$ spatial scaling. Our analysis relies on a distributional identity between half-space LPP and the Pfaffian Schur process. The latter exhibits two key structures: (1) a Pfaffian point process, which we use to establish finite-dimensional convergence of the ensembles, and (2) a Gibbsian line ensemble, which we use to extend convergence uniformly over compact sets.

Curve separation in supercritical half-space last passage percolation

TL;DR

This work analyzes line ensembles arising from half-space geometric LPP with symmetric weights in the supercritical regime . By leveraging a distributional identity with the Pfaffian Schur process, the authors obtain a Pfaffian point-process representation and a Gibbs-interlacing structure, enabling precise kernel asymptotics via steepest-descent analyses. They show a full-curve phase separation: the top curve converges to a Brownian motion under -horizontal and -scale adjustments, while the remaining curves converge to the Airy line ensemble under vertical and horizontal scaling. This establishes a KPZ-class curve-separation phenomenon and provides a robust framework for handling both determinantal and Pfaffian line ensembles through kernel convergence and Gibbsian tightness arguments.

Abstract

We study line ensembles arising naturally in symmetrized/half-space geometric last passage percolation (LPP) on the square. The weights of the model are geometrically distributed with parameter off the diagonal and on the diagonal, where and . In the supercritical regime , we show that the ensembles undergo a phase transition: the top curve separates from the rest and converges to a Brownian motion under fluctuations and spatial scaling, while the remaining curves converge to the Airy line ensemble under fluctuations and spatial scaling. Our analysis relies on a distributional identity between half-space LPP and the Pfaffian Schur process. The latter exhibits two key structures: (1) a Pfaffian point process, which we use to establish finite-dimensional convergence of the ensembles, and (2) a Gibbsian line ensemble, which we use to extend convergence uniformly over compact sets.

Paper Structure

This paper contains 33 sections, 28 theorems, 355 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Half-space geometric last passage percolation
  3. Main results
  4. Main ideas and paper outline
  5. Pfaffian Schur processes
  6. Definitions
  7. Pfaffian point process structure
  8. Parameter scaling for the bottom curves
  9. Parameter scaling for the top curve
  10. Preliminary estimates
  11. Power series expansions
  12. Estimates along circles
  13. Estimates along large contours
  14. Kernel convergence for the bottom curves
  15. Function bounds
  16. ...and 18 more sections

Key Result

Theorem 1.2

Assume the same notation as in Definition Def.TopCurveScaledLPP, and let $(B_t: t \geq 0)$ be a standard Brownian motion. Then, $\mathcal{U}_1^{\mathrm{top},N} \Rightarrow W$ in $C\left((\kappa_0, 1)\right)$ (with the topology of uniform convergence over compact sets), where $W_t = B_{t - \kappa_0}$

Figures (5)

  • Figure 1: The left side depicts the array $W = (w_{i,j}: i,j \geq 1)$ and an up-right path $\pi$ (in gray) that connects $(1,1)$ to $(7,5)$. 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.
  • Figure 2: The figure depicts the top three curves in $\{\lambda_i(m,n): i \geq 1, 0 \leq m \leq n\}$.
  • Figure 3: The figure depicts the top several curves of $\{\lambda_i(m,n): i \geq 1, 0 \leq m \leq n\}$ for $q = 0.5$, $n = 500$, and $c = 0.8$ (left) and $c = 1.4$ (right). In the subcritical regime (left side), the curves concentrate around the function $n h^{\mathrm{bot}}(m/n)$ on $[0,n]$, where $h^{\mathrm{bot}}(\kappa) = \frac{ q (q + 2 \sqrt{\kappa} + q \kappa)}{1-q^2}$. In the supercritical regime (right side), they concentrate around the same curve on $[0, \kappa_0 n]$. On the interval $[\kappa_0 n, n]$ the top curve separates from the rest and follows the straight line $n h^{\mathrm{top}}(m/n)$, where $h^{\mathrm{top}}(\kappa) = \frac{q}{c-q} + \frac{qc \kappa}{1-qc}$, while the remaining curves continue to follow $n h^{\mathrm{bot}}(m/n)$.
  • Figure 4: The left side depicts the contours $C(x,\theta, R,r)$ from Definition \ref{['Def.ContoursBulk']}. The right side depicts the contours $\mathcal{C}_a^{\varphi}[r]$, defined above (\ref{['Eq.I11Vanish']}).
  • Figure 5: The figure depicts the points $\zeta_1, \zeta_2$, and the angles $\phi_1, \phi_2$ when $\theta_1 = \pi/3$ and $\theta_2 = \pi/2$ (left), and when $\theta_1 = \pi/4$ and $\theta_2 = \pi/3$ (right).

Theorems & Definitions (79)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 2.1
  • Remark 2.2
  • ...and 69 more