Shocks and instability in Brownian last-passage percolation

Abstract

For stochastic Hamilton-Jacobi (SHJ) equations, instability points are the space-time locations where two eternal solutions with the same asymptotic velocity differ. Another crucial structure in such equations is shocks, which are the space-time locations where the velocity field is discontinuous. In this work, we provide a detailed analysis of the structure and relationships between shocks, instability, and competition interfaces in the Brownian last-passage percolation model, which serves as a prototype of a semi-discrete inviscid stochastic HJ equation in one space dimension. Among our findings, we show that the shock trees of the two unstable eternal solutions differ within the instability region and align outside of it. Furthermore, we demonstrate that one can reconstruct a skeleton of the instability region from these two shock trees.

Figures (9)

• Figure 2.1: Left: An illustration of an up-right path. According to Proposition \ref{['nodouble']}, up-right paths do not jump twice in a row like the path in the right panel.
• Figure 3.1: Left: An illustration of a maximal $\theta$-instability interval. In reality, the set of endpoints of the vertical edges has a Hausdorff dimension $\frac{1}{2}$. Thus, each vertical edge has uncountably many vertical edges near it, similar to the way the zeros of standard Brownian motion behave. There are no $\theta$-instability points from which emanate both a vertical edge going upward and another going downward. Center: An illustration of the web-like structure. Points ${\mathbf x}$ and ${\mathbf y}$ have a common NE ancestor ${\mathbf z}$ and a common SW descendant ${\mathbf z}'$. Right: An illustration of the shocks tree structure. When $\theta\not\in\Theta^\omega$, the coloring is immaterial. When $\theta\in\Theta^\omega$ and ${\scaleobj{0.9}{\square}}=+$, red depicts the subtree of points in ${\mathrm{NU}}_1^{\theta+}\setminus{\mathrm{NU}}_1^{\theta-}$ and blue depicts the "bushes" of points in ${\mathrm{NU}}_1^{\theta+}\cap{\mathrm{NU}}_1^{\theta-}$. Similarly, when $\theta\in\Theta^\omega$ and ${\scaleobj{0.9}{\square}}=-$, red depicts the subtree of points in ${\mathrm{NU}}_1^{\theta-}\setminus{\mathrm{NU}}_1^{\theta+}$ and blue depicts the "bushes" of points in ${\mathrm{NU}}_1^{\theta+}\cap{\mathrm{NU}}_1^{\theta-}$.
• Figure 4.1: Left: An illustration of a point $(m,s)\in{\mathrm{NU}}_1^{\theta{\scaleobj{0.8}{\square}}}$. Right: An illustration of a point $(m,s)\in{\mathrm{NU}}_0^{\theta{\scaleobj{0.8}{\square}}}\setminus{\mathrm{NU}}_1^{\theta{\scaleobj{0.8}{\square}}}$. Proposition \ref{['nodouble']} prevents the existence of points of the latter type. In both figures, the top up-right path is $\Gamma_{(m,s)}^{L, \theta{\scaleobj{0.8}{\square}}}$ and the bottom up-right path is $\Gamma_{(m,s)}^{R, \theta{\scaleobj{0.8}{\square}}}$.
• Figure 7.1: An illustration of the proof of Lemma \ref{['geodefinstabedge1']}. Left: The proof of the first direction. The two geodesics in the middle are $\Gamma_{(m,t)}^{L, \gamma-}$ and $\Gamma_{(m,t)}^{R, \theta+}$. They separate immediately and do not meet again, thus separating the two outer geodesics $\Gamma_{(m,r')}^{R, \theta+}$ and $\Gamma_{(m,r")}^{L, \gamma-}$. Middle: The first case in the second direction. Right: The second and third cases in the second direction.
• Figure 8.1: An illustration of Theorem \ref{['inst:summary.int']}.

Theorems & Definitions (147)

• Remark 1.1
• Theorem 2.1: Theorem 4.2 in Alb-Ras-Sim-20
• Definition 2.2
• Remark 2.3
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 3.4
• Remark 3.5
• Theorem 3.6