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Moderate deviations in first-passage percolation for bounded weights

Wai-Kit Lam, Shuta Nakajima

TL;DR

This work analyzes moderate deviations in first-passage percolation on Z^d with bounded edge-weights, focusing on the regime between typical fluctuations and large deviations. By combining a multi-scale slab/decomposition approach with refined concentration assumptions and curvature conditions on the limit shape, the authors derive explicit upper and lower bounds for both upper- and lower-tail moderate deviations around the time constant, with exponents governed by the fluctuation exponent χ. They further connect near-zero rate-function behavior to these MD exponents and discuss the implications for the limiting distribution and universality, including a heuristic relation to KPZ-type scaling. Importantly, several results are proven rigorously without unverified assumptions, while the full MD picture remains contingent on widely believed but unproven curvature and concentration hypotheses in higher dimensions.

Abstract

We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on $\mathbb{Z}^d$ for $d \geq 2$. Write $T(\mathbf{x}, \mathbf{y})$ for the first-passage time and denote by $μ(\mathbf{u})$ the time constant in direction $\mathbf{u}$. In this paper, we establish that, if one assumes that the sublinear error term $T(\mathbf{0}, N\mathbf{u}) - Nμ(\mathbf{u})$ is of order $N^χ$, then under some unverified (but widely believed) assumptions, for $χ< a < 1$, \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) > Nμ(\mathbf{u}) + N^a\bigr) = \exp{\Big(-\,N^{\frac{d(1+o(1))}{1-χ}(a-χ)}\Big)},\end{align*} \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) < Nμ(\mathbf{u}) - N^a\bigr) = \exp{\Big(-\,N^{\frac{1+o(1)}{1-χ}(a-χ)}\Big)}, \end{align*} with accompanying estimates in the borderline case $a=1$. Moreover, the exponents $\frac{d}{1-χ}$ and $\frac{1}{1-χ}$ also appear in the asymptotic behavior near $0$ of the rate functions for upper and lower tail large deviations. Notably, some of our estimates are established rigorously without relying on any unverified assumptions. Our main results highlight the interplay between fluctuations and the decay rates of large deviations, and bridge the gap between these two regimes. A key ingredient of our proof is an improved concentration via multi-scale analysis for several moderate deviation estimates, a phenomenon that has previously appeared in the contexts of two-dimensional last-passage percolation and two-dimensional rotationally invariant FPP.

Moderate deviations in first-passage percolation for bounded weights

TL;DR

This work analyzes moderate deviations in first-passage percolation on Z^d with bounded edge-weights, focusing on the regime between typical fluctuations and large deviations. By combining a multi-scale slab/decomposition approach with refined concentration assumptions and curvature conditions on the limit shape, the authors derive explicit upper and lower bounds for both upper- and lower-tail moderate deviations around the time constant, with exponents governed by the fluctuation exponent χ. They further connect near-zero rate-function behavior to these MD exponents and discuss the implications for the limiting distribution and universality, including a heuristic relation to KPZ-type scaling. Importantly, several results are proven rigorously without unverified assumptions, while the full MD picture remains contingent on widely believed but unproven curvature and concentration hypotheses in higher dimensions.

Abstract

We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on for . Write for the first-passage time and denote by the time constant in direction . In this paper, we establish that, if one assumes that the sublinear error term is of order , then under some unverified (but widely believed) assumptions, for , \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) > Nμ(\mathbf{u}) + N^a\bigr) = \exp{\Big(-\,N^{\frac{d(1+o(1))}{1-χ}(a-χ)}\Big)},\end{align*} \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) < Nμ(\mathbf{u}) - N^a\bigr) = \exp{\Big(-\,N^{\frac{1+o(1)}{1-χ}(a-χ)}\Big)}, \end{align*} with accompanying estimates in the borderline case . Moreover, the exponents and also appear in the asymptotic behavior near of the rate functions for upper and lower tail large deviations. Notably, some of our estimates are established rigorously without relying on any unverified assumptions. Our main results highlight the interplay between fluctuations and the decay rates of large deviations, and bridge the gap between these two regimes. A key ingredient of our proof is an improved concentration via multi-scale analysis for several moderate deviation estimates, a phenomenon that has previously appeared in the contexts of two-dimensional last-passage percolation and two-dimensional rotationally invariant FPP.

Paper Structure

This paper contains 33 sections, 23 theorems, 268 equations, 6 figures.

Table of Contents

  1. Model and main results
  2. Introduction
  3. Notations
  4. Main results
  5. Upper bound results for upper tail moderate deviations
  6. Refinement of concentration condition for upper tail
  7. Lower bound results for upper tail moderate deviations
  8. Results for lower tail moderate deviations
  9. Discussion on the assumptions
  10. A heuristic argument for the main results
  11. A conjecture on the limiting distribution
  12. Related work
  13. Organization of the paper
  14. Upper bound for upper tail moderate deviations
  15. Idea of the proof
  16. ...and 18 more sections

Key Result

Theorem 1.1

Assume Assumptions assum: initial concentration and assum: finite curvature with $\overline{\chi}\in (0,1)$.

Figures (6)

  • Figure 1: A depiction of the limit shape (left) and a tilted cylinder (right). Here, the boundary of the limit shape is smooth, and at any direction $\mathbf{u}$ we can find a unique tangent plane $H_\mathbf{u}$ that touches $\mathbf{u}$ at the boundary. In this case, $\widetilde{\mathbf{u}}_2$ is a unit vector in $H_\mathbf{u}$. As for the cylinder, note that the directions other than $\mathbf{v}$ are spanned by the $\widetilde{\mathbf{u}}_j$, which might not be orthogonal to $\mathbf{v}$, and, in particular, a tilted cylinder may not be a rotated rectangle.
  • Figure 2: Depiction of the sets $L_m$ and $R_m$. $L_m$ goes from left to right as $m$ increases, while $R_m$ goes from right to left, but both $L_m$ and $R_m$ are "moving along" the direction $\mathbf{v}$ as $m$ changes, and their lengths increase as $m$ increases. The distance between $L_{M-1}$ and $R_{M-1}$ is $\Delta_{M-1} = N - 2N^{\frac{M-1}{M}}$.
  • Figure 3: Depiction of the set $\mathcal{Z}_{N,m}(b)$ for $m\leq M-2$. Each intersection is a point in $\mathcal{Z}_{N,m}(b)$.
  • Figure 4: A depiction of the points $(\overline{\mathbf{x}}_m)$ with $M = 4$. $\overline{\mathbf{x}}_1, \ldots, \overline{\mathbf{x}}_{M-1}$ go from $\mathbf{0}$ towards $N\mathbf{v}$ (and "drifted" by the direction $\mathbf{x}$), while $\overline{\mathbf{x}}_{2M-1}, \ldots, \overline{\mathbf{x}}_{M}$ go from $N\mathbf{v}$ back to $\mathbf{0}$ (also "drifted" by the direction $\mathbf{x}$).
  • Figure 5: The path from $\mathbf{0}$ to $N\mathbf{e}_1$ is a geodesic. It intersects the faces $j\widehat{K}H_{\mathbf{e}_1}$, $j=0, 1, \ldots, \widehat{J} - 1$. $\mathbf{x}_{t_j}$ is the first time the path touches $j\widehat{K}H_{\mathbf{e}_1}$, considering the path starting from $\mathbf{0}$.
  • ...and 1 more figures

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 2.1
  • Lemma 2.2: Lower tail large deviation estimates
  • ...and 32 more