Rotationally invariant first passage percolation: Breaking the $n/\log n$ variance barrier

Abstract

For first passage percolation (FPP) on Euclidean lattices $\mathbb{Z}^d$ with $d\ge 2$, it is expected that the variance of the first passage time between two points grows sublinearly in the distance with a universal exponent strictly smaller than $1$. Following Kesten's $O(n)$ upper bound (Ann. Appl. Probab., 1993) on the variance, Benjamini, Kalai and Schramm (Ann. Probab., 2003) used hypercontractivity to obtain an improvement of a factor of $\log n$ when passage times take two values with equal probability. This was later extended to more general classes of passage time distributions. However, unlike in exactly solvable planar models in last passage percolation where the variance is known to be $Θ(n^{2/3})$, the best known upper bound for the variance of passage times has remained $O(n/\log n)$ in all non-trivial variants of FPP. For a class of rotationally invariant Riemannian FPP on the plane, we show that the variance is $O(n^{1-\varepsilon})$ for some $\varepsilon>0$. Our argument uses fluctuation estimates for passage times and geodesics derived in Basu, Sidoravicius and Sly (2023) together with a multi-scale argument to establish that the geodesic exhibits disorder chaos, i.e., upon resampling a small fraction of the underlying randomness, the updated geodesic has on average a small overlap with the original one; this, established at a large number of scales, leads to a polynomial improvement of the variance bound.

Figures (23)

• Figure 1: Construction of alternative paths at different scales. Proposition \ref{['p:chaos']} states that once a small fraction of the $n\times W_n$ blocks have their randomness resamples, the expected number of blocks that intersect the geodesic from $\bf 0$ to $(Mn,0)$ both before and after resampling is $o(M)$; this essentially boils down to showing that on most vertical columns of width $n$, the geodesics before and after resampling are separated. To show the latter fact, we prove Theorem \ref{['t:Pplusintro']} which shows that at every given length scale with large probability there are a constant fraction ($\delta_0$) of locations where the geodesics before and after resampling are separated. In the figure, the black path shows the original geodesic whereas the red, green, and blue paths show alternative paths at different scales which are better then the original geodesic (or any other path passing close to the original geodesic at that location) after the resampling, thus guaranteeing the geodesics before and after are separated at that location. Starting from the largest scale $\ell_{\max}$, (which corresponds to columns of width $n\Phi^{\ell_{\max}}\approx M^{1/100}n$) we find a constant fraction of columns where geodesics (before and after resampling) are separated; then we find another constant fraction of columns at then next scale, and so on, eventually showing that geodesics are separated at most columns since $\ell_{\max}$ was chosen appropriately large.
• Figure 2: Building blocks of the multiscale estimate: the plane is divided into boxes $\Lambda_{i,j}=[(i-1)n,in]\times[(j-1)W_n,jW_n]$. The event that the geodesic comes within distance $1$ of $\Lambda_{ij}$ is denoted $\mathcal{U}_{ij}$. Proposition \ref{['p:uij.bounds']} has various estimates showing certain weighted sums of $I(\mathcal{U}_{ij})$ cannot be too large. To this end we denote the $j$-index of the block where the geodesic $\gamma$ depicted in the picture enters the column $\Lambda_i$ by $J_{i-1}$. Therefore all the blocks $\Lambda_{ij}$ with $j$ between $J_{i-1}$ and $J_{i}$ will witness $\mathcal{U}_{ij}$ (these blocks are marked in pink in the figure). However, there may also other other blocks in $\Lambda_i$ where $\mathcal{U}_{ij}$ holds. These blocks, called overhang blocks, are marked in green in the figure. The contributions from these two types of blocks are controlled separately by observing that the segment of the geodesics within a column remains sandwiched between the two blue paths marked in the figure. The number of overhang blocks can then be controlled just by looking at the transversal fluctuations of the blue paths.
• Figure 3: The regions where different events are defined at coordinate $i,j$ at length scale $r$. In the vertical direction, the centres of the different columns shown in the figure is located at $jW_r$. The column at the middle (central column) corresponds to $[(i-1)r,ir]$, the columns marked in light blue are the intermediate columns ($[(i-2)r,(i-1)r]$ and $[ir,(i+1)r]$) and and the outer columns correspond to $[(i-3)r,(i-2)r]$ and $[(i+1)r, (i+2)r]$. The different colours in the central and outer columns represents regions where we ask for different type of events. The yellow columns flanking these five columns are referred to as wings on a series of different scales and we ask them to satisfy certain typical events called wing conditions.
• Figure 4: The event $\mathcal{B}_{i,j}$ for the central column $[(i-1)r,ir]$. The different panels show different subevents referring to different regions of the column and the first panel illustrates them combined. The first event $\mathcal{B}_{i,j}^{(1)}$ asks that the passage time across blue region in the middle for both ${\mathcal{X}}$ and ${\mathcal{X}}'$ is not too larger than typical. The second event $\mathcal{B}_{i,j}^{(2)}$ asks that in the corresponding light red region, none of the paths are too short and any path across with large vertical change will have large excess length both before and after resampling. The third event $\mathcal{B}_{i,j}^{(3)}$ asks that paths across the corresponding light red region are not too short both before and after the resampling. The fourth event $\mathcal{B}_{i,j}^{(4)}$ asks that across the green region marked (which we shall refer to as the gadget), none of the paths are too short before resampling, while after resampling there is a very good path. The fifth event $\mathcal{B}_{i,j}^{(5)}$ asks that the $\mathcal{M}$ event holds in the yellow region before and after resampling, i.e., the passage time across the yellow region and the passage time across the green region are not too different. The sixth event $\mathcal{B}_{i,j}^{(6)}$ asks that the $\mathcal{K}$ event holds for the horizontal lines marked in red both before and after resampling. The seventh event $\mathcal{B}_{i,j}^{(7)}$ asks that the regions marked in dark red are barriers, i.e., all paths across these regions are atypically large (given by $\mathcal{I}^+$ and $\mathcal{J}$ events). The regions $V_{ij}$ in the final panel will be used for conditioning and separating out the likely part of the $\mathcal{B}$ events, see Section \ref{['s:eventperc']}.
• Figure 5: The event $\mathcal{D}_{i,j}$ for the rectangle $[(i-1)r,ir]\times [(j-L_2)W_r,(j+L_2)W_r]$. As in Figure \ref{['f:EventB']}, the different panels show different subevents referring to different regions of the rectangle and the first panel illustrates them combined. The first event $\mathcal{D}_{i,j}^{(1)}$ asks that the passage time across blue region in the middle for both ${\mathcal{X}}$ and ${\mathcal{X}}'$ is not too larger than typical. The second event $\mathcal{D}_{i,j}^{(2)}$ asks that in the corresponding light red region, none of the paths are too short both before and after resampling. The third event $\mathcal{D}_{i,j}^{(3)}$ (not shown) asks for not having any very good paths across the whole column (with some tolerance as we go away from the centre of the rectangle as in event $\mathcal{A}$) both before and after resampling. The fourth event $\mathcal{D}_{i,j}^{(4)}$ asks that the $\mathcal{K}$ event holds for the horizontal lines marked in red both before and after resampling. The seventh event $\mathcal{D}_{i,j}^{(3)}$ asks that the regions marked in dark red are barriers, i.e., all paths across these regions are atypically large (given by $\mathcal{I}^+$ and $\mathcal{J}$ events). Finally, the regions $\widehat{V}'_{ij}$ in the final panel will be used for conditioning and separating out the likely part of the $\mathcal{D}$ events, see Section \ref{['s:eventperc']}.

Theorems & Definitions (175)

• Theorem 1
• Theorem 2.1: BSS23
• Proposition 2.2: BSS23
• Lemma 2.3: BSS23
• Theorem 2.4: BSS23
• Lemma 2.5: BSS23
• Proposition 2.6
• Lemma 2.7
• Lemma 2.8
• Proposition 2.9