Last passage percolation in hierarchical environments

TL;DR

This work analyzes last passage percolation in hierarchical, log-correlated environments, highlighting that KPZ scaling can fail at critical points and logarithmic corrections emerge. Using a robust multi-scale skeleton framework, the authors derive polylogarithmic lower bounds and concentration for LPP in two canonical hierarchical settings: (i) heavy-tailed i.i.d. weights with tail exponent $\alpha=2$, and (ii) the branching random walk (BRW) as a hierarchical proxy for log-correlated Gaussian fields. They further demonstrate that finite second moment alone does not guarantee finite linear growth of the LPP energy, by constructing a tail correction that yields superlinear growth, and extend the analysis to higher dimensions with dimension-dependent log corrections. The results illuminate the delicate role of scale interactions in hierarchical environments and connect to broader themes in log-correlated models and Liouville quantum gravity.

Abstract

Last passage percolation (LPP) is a model of a directed metric and a zero-temperature polymer where the main observable is a directed path evolving in a random environment accruing as energy the sum of the random weights along itself. When the environment has light tails and a fast decay of correlation, the fluctuations of LPP are predicted to be explained by the Kardar-Parisi-Zhang (KPZ) universality theory. However, the KPZ theory is not expected to apply for many natural environments, particularly "critical" ones exhibiting a hierarchical structure often leading to logarithmic correlations. In this article, we initiate a novel study of LPP in such hierarchical environments by investigating two particularly interesting examples. The first is an i.i.d. environment but with a power-law distribution with an inverse quadratic tail decay which is conjectured to be the critical point for the validity of the KPZ scaling relation. The second is the Branching Random Walk which is a hierarchical approximation of the two-dimensional Gaussian Free Field. The second example may be viewed as a high-temperature (weak coupling) directed version of Liouville Quantum Gravity, which is a model of random geometry driven by the exponential of a logarithmically-correlated field. Due to the underlying fractal structure, LPP in such environments is expected to exhibit logarithmic correction terms with novel critical exponents. While discussions about such critical models appear in the physics literature, precise predictions about exponents seem to be missing. Developing a framework based on multi-scale analysis, we obtain bounds on such exponents and prove almost optimal concentration results in all dimensions for both models. As a byproduct of our analysis we answer a long-standing question of Martin on necessary and sufficient conditions for the linear growth of the LPP energy in i.i.d. environments.

Figures (10)

• Figure 1: A simulation of the directed geodesic from $(0,0)$ to $(2^{15},2^{15})$ in an environment of i.i.d. weights with power-law tails of exponent $2$. The left figure depicts the geodesic (blue) with two of its "skeletons" superimposed on top. The $k$th skeleton is obtained from the geodesic by linearly interpolating between the weights that exceed $2^{15-k}$. The right figure depicts the same situation as the left, but with more skeletons and with each path plotted on its own copy of the plane. (The skeletons corresponding to $k=6,9,12$ are visually quite similar to the geodesic, and their inclusion in the left figure would obscure the geodesic.)
• Figure 2: Simulations of the directed geodesic in an environment of i.i.d. weights with power-law tails of exponent $\alpha=2$.
• Figure 3: A simulation of the directed geodesic from $(0,0)$ to $(2^{13}, 2^{13})$ on the branching random walk.
• Figure 4: Depicted are $[0,n]^2$ (black square) and the cylinder (shaded gray) around the diagonal of width $\sqrt{\delta}$, divided into smaller cylinders of length $\frac{1}{\sqrt{\delta}}$.
• Figure 5: The black rectangle is $\mathsf{Rect}(a,b)$. The shaded gray region is the cylinder $\mathsf{Cyl}_r(\mathsf{Rect}(a,b))$, for some $r\in(0,1)$.

Theorems & Definitions (43)

• Definition 1.1: Last passage percolation
• Theorem 1: Logarithmic correction lower bound for $\alpha=2$
• Theorem 2: Concentration for $\alpha=2$
• Remark 1.2
• Remark 1.3
• Theorem 3: Logarithmic correction lower bound for BRW
• Theorem 4: Concentration and fluctuations for BRW
• Theorem 5
• Remark 1.4
• Definition 3.1: Rectangles, slopes, and cylinders