A universality property for large deviations of RWRE close to the axis

TL;DR

The authors establish a universality result for large deviations of random walks in a random environment on $\mathbb{Z}^2$ near the axis under i.i.d. elliptic environments with a finite $p$-th moment of $\log \omega$. By comparing RWRE to a corresponding edge-weighted LPP model and leveraging axis-near LPP results, they prove that for $0<a<\frac{3}{7}\left(1-\frac{2}{p}\right)$ the fluctuations of quenched probabilities converge to the directed landscape, with convergence strengthened to compact sense when $p>5$. The work also demonstrates KPZ-type behavior via the GUE Tracy–Widom distribution for appropriate scaling, and extends these universality results to special cases like the Beta RWRE and, via known LPP results, to directed polymers. The findings contribute a broad-Environment KPZ universality result for RWRE and establish convergence to the directed landscape in this regime, bridging RWRE with LPP and integrable and non-integrable universality classes. The methods highlight a robust link between random environment models and geometric growth models through LPP couplings.

Abstract

We establish a general version of the strong KPZ universality conjecture near the axis for random walks in a random environment (RWRE) on $\mathbb{Z}^2$. For an i.i.d. elliptic random environment, we consider the quenched large deviations probabilities for trajectories starting at the origin and arriving at time $n+[n^a]$ to the position $(n,[n^a])$ and show that, if the logarithm of the right-jump probability has a finite moment of order $p>2$, then for $a < \frac{3}{7}(1-\frac{2}{p})$ the fluctuations of these propabilities are asymptotically governed by the GUE Tracy-Widom distribution. Our results are based on a comparison between RWRE and a last passage percolation model, whose asymptotic fluctuations near the axis were previously established independently by Bodineau-Martin and Baik-Suidan. Furthermore, we obtain also the full convergence to the directed landscape in this regime based on the extension of the aforementioned results to this setting by McKeown and Zhang.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 2.2: MZ25
• Lemma 2.3
• proof : Proof of Lemma \ref{['lemma: L - L tilde']}
• proof : Proof of Theorem \ref{['theo: convergence to directed landscape']}
• Theorem 3.1
• proof : Proof of Theorem \ref{['theorem1']}