Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension

First, let us highlight the main differences between the method in this paper and the methods used in the earlier works [ding-dunlap-lqg-fpp, ding-dunlap-lgd, df-lqg-metric, dddf-lfpp, dg-supercritical-lfpp] to establish the tightness of approximations of exponential metrics for log-correlated fields in two-dimensions. All the results in two dimensions rely crucially on RSW estimates, which give up-to-constants comparisons between quantiles of $D_{n}$-crossing lengths of rectangles in the “easy direction” and the “hard direction”, see e.g. Section 3 of [dddf-lfpp]. The arguments to prove these estimates are based on either approximate conformal invariance or on forcing paths to cross each other, neither of which works in higher dimensions. For this reason, we will use a fundamentally different approach to prove tightness which bypasses any direct proof of RSW estimates.

The first difference in our approach as compared to the two-dimensional case is that we initially use the median of the point-to-point distance, namely $\lambda_{n}$ from (1.7), as the normalizing constant. In contrast, previous works use the median of the left-right crossing distance within a box as their normalizing constant (although these two medians are eventually proved to be equivalent up to a constant). The point-to-point distance is typically larger than the left-right crossing distance, which makes it easier to upper-bound other types of distances in terms of $\lambda_{n}$. We choose to work with the internal point-to-point distance inside a box to ensure that we have long-range independence, which allows us to employ percolation arguments. To use the percolation argument, we will actually work with the $q$-quantile of $D_{n}(0,e_{1};B_{2}(0))$ for $q$ close to one, but not depending on $n$, in most parts of the proof.

The second difference arises from the lack of concentration results for dimension $\geq 3$. In previous works, the authors have derived upper tail estimates for the left-to-right crossing distance, see Sections 4 and 5 of [dddf-lfpp] and Section 3 of [dg-supercritical-lfpp] for these types of results. These estimates are based on the RSW estimates, percolation arguments, and the Efron-Stein inequality. Despite the fact that the RSW argument is not applicable in our case, we can still hope to use the percolation argument to achieve an upper tail estimate in our setting. Simplistically, if we can compare $\lambda_{n}$ and $\lambda_{n-k}$ (which should differ by at most a constant if $k$ is fixed), then we can divide a box into $2^{k}$ pieces. By using the scaling property of $h_{n}$ (Lemma 2.2), the definition (1.7) of $\lambda_{n-k}$ (actually we will use a large quantile instead of the median) and percolation arguments, we can deduce an upper tail estimate for the $D_{n}$-distance across a hypercubic shell⁶⁶6A hypercubic shell is the domain between two concentric boxes, which is the $d$-dimensional analog of a square annulus., e.g. $D_{n}(\partial B_{1}(0),\partial B_{2}(0))$, in terms of $\lambda_{n}$. It turns out that a specific comparison bound between $\lambda_{n}$ and $\lambda_{n-k}$ for all integers $1\leq k\leq n$, as detailed in Proposition LABEL:prop:compare, is sufficient to achieve an upper tail estimate for the diameter of a box. This in turn ensures the tightness of the metric. Deriving this comparison is the most technical part of this paper and is detailed in Section LABEL:sec:compare. We will actually derive a comparison between the metrics $D_{n}$ and $D_{n-k}$ in that section, which may also be of independent interest.

The third difference also arises from the lack of concentration results for dimension $\geq 3$ and our choice of the normalizing constant. In previous works, a lower tail estimate for the left-right crossing distance follows from the RSW argument, see Section 4 of [dddf-lfpp], and this implies that each subsequential limit is a metric. Here we will use a different approach to demonstrate this. Note that, a prior, the point-to-point distance can be much larger than the left-to-right crossing distance of a box. Our strategy begins by showing that the distance across a hypercubic shell is positive with non-zero probability. Combining with a zero-one law argument (Lemma LABEL:lem:zero-one), we can increase this probability to one, thereby establishing that each subsequential limit is a metric. This will be detailed in Subsection LABEL:subsec:non-degenerate.

Next, we describe our strategy in more detail and outline the content of each section. More comprehensive overviews can be found at the beginning of each respective section and subsection.

In Section 2, we provide preliminaries and fix some notation. Let $W(dx,dt)$ be the space-time white noise. Throughout this paper, we will work with the approximation of the log-correlated Gaussian field

for integers $n>m\geq 0$ (note that $h_{0,n}=h_{n}$, as defined in (1.4)). Some basic properties and estimates of $h_{m,n}$ are provided in Subsection 2.2. In Subsection 2.3, we define $D_{m,n}$ as the exponential metrics associated with $h_{m,n}$ and establish some basic properties of these metrics, including a Gaussian concentration bound (Lemma 2.7). Subsection 2.4 collects basic arguments about percolation with finite range of dependence, which will play a crucial role in Sections 3 and LABEL:sec:bound-distance.

In Section 3, we will prove Proposition 3.1, which establishes the existence of an exponent $Q$ satisfying (1.8). This follows from the subadditivity inequality: $\lambda_{n}\leq e^{Cn^{2/3}}\lambda_{m}\lambda_{n-m}$ for integers $n>m\geq 1$, and the proof is similar to Proposition 2.5 of [dg-supercritical-lfpp]. This inequality requires constructing a path connecting $0$ and $e_{1}$ of typical $D_{n}$-length. The construction essentially follows two steps. The first step is to construct a path on $2^{-m}\mathbb{Z}^{d}$ such that its $D_{m}$-length can be upper-bounded. The second step is to modify the path locally so that its $D_{n}$-length can be controlled using a percolation argument on a refined lattice. To use the percolation argument, we actually consider a large quantile of $D_{n}(0,e_{1};B_{2}(0))$. In Lemma LABEL:lem:Q-lower, we will establish basic properties of $Q(\xi)$, based on estimates for $h_{n}$. This will conclude the proof of Proposition 1.1.

In Section LABEL:sec:bound-distance, we will establish a chaining argument similar to the ones in Section 6.3 of [ding-dunlap-lqg-fpp] and Section 6.1 of [df-lqg-metric], and derive bounds for different types of distances. First, we present the chaining argument in Subsection LABEL:subsec:chaining. We use paths of typical $D_{n}$-length at different scales to connect any two points in a box, and thus establish an upper-bound for the $D_{n}$-diameter of a box in terms of the large quantiles of $D_{n-m}(0,e_{1};B_{2}(0))$ for $0\leq m\leq n$. This result will be used subsequently in two places. First, we will use it in Subsections LABEL:subsec:bound-diameter and LABEL:subsec:cross to show that the medians of the diameter of a box or distance across a hypercubic shell all satisfy the relation in (1.8). Secondly, this result will be used to prove the tightness of the metric in Subsection LABEL:subsec:tightness after a comparison between quantiles of $D_{n}(0,e_{1};B_{2}(0))$ for different values of $n$ is achieved. In Subsection LABEL:subsec:super-exponential, we will establish super-exponential concentration bounds for distances across and around hypercubic shells, which will be used in Section LABEL:sec:compare.

In Section LABEL:sec:compare, we will compare the metrics $D_{n}$ and $D_{n+k}$ for integers $n\geq k\geq 1$. We will briefly describe the strategy here, and refer to Subsection LABEL:subsec:sec5-strategy for a more detailed outline of this section. The comparison is achieved by controlling the behavior of the field $h_{n,n+k}$ (note that $D_{n+k}$ is obtained from $D_{n}$ by adding $h_{n,n+k}$ to the field). In most parts of the space, $h_{n,n+k}$ behaves well, and $D_{n+k}$ and $D_{n}$ satisfy the desired bound given in Proposition LABEL:prop:compare. However, there are places where $h_{n,n+k}$ does not behave well, and a priori, it is possible that a $D_{n}$- or $D_{n+k}$-geodesic spends most of its time in these problematic regions. Our main effort is to address these regions. The proof essentially involves two steps. In the first step, we use a coarse-graining argument to show that, with high probability, we can find boxes at different scales to cover the problematic regions. Importantly, all these boxes satisfy the condition that the $D_{n}$-distance around the hypercubic shell enclosing the box can be upper-bounded by the $D_{n}$-distance across a larger hypercubic shell. In the second step, we use this condition to show that the ill-behaved field within these boxes has a minor impact on the metric $D_{n}$. Specifically, paths can be modified to avoid these boxes, and their $D_{n}$-length will increase by no more than a constant factor. Moreover, for paths entirely contained within the domain where $h_{n,n+k}$ behaves well, by adjusting the paths, their $D_{n}$-length and $D_{n+k}$-length satisfy the desired bound. This leads to a comparison between $D_{n}$ and $D_{n+k}$.

In Section LABEL:sec:final-proof, we prove Theorem 1.2. The proof consists of two parts. In Subsection LABEL:subsec:tightness, we combine results from the chaining argument in Subsection LABEL:subsec:chaining and the comparison of quantiles from Section LABEL:sec:compare to demonstrate the tightness of $D_{n}$ when normalized by the $q$-quantile of $D_{n}(0,e_{1};B_{2}(0))$, where $q$ is close to one but independent from $n$. In Subsection LABEL:subsec:non-degenerate, we will establish that each possible subsequential limit is a metric. From the definition of quantiles and the positive association (FKG) property for positively correlated Gaussian processes, we first show that the distance across a hypercubic shell is bounded away from zero with positive probability. Another crucial input is a zero-one law (Lemma LABEL:lem:zero-one), which is derived from the locality property of the metric. By using this argument, we can increase the probability to one. Applying this to countably many hypercubic shells shows that the subsequential limit is a metric, which in turn implies an up-to-constants comparison between the median $\lambda_{n}$ and the $q$-quantile of $D_{n}(0,e_{1};B_{2}(0))$. This gives tightness when we normalize by $\lambda_{n}$ instead of by the $q$-quantile.

In Section LABEL:sec:open-problem, we list some open problems related to the metric we constructed. Appendix LABEL:appendix:index includes a list of notation that we will use in this paper.

We write $\mathbb{N}=\{1,2,\ldots\}$. Without specific mention, the logarithm in this paper will be taken with respect to the base $e$. For $a\in\mathbb{R}$, we use $\lfloor a\rfloor$ to represent the largest integer not greater than $a$. For a random variable $X$, we will use ${\rm Med}(X)$ to represent its median.

Let $(X,D)$ be a metric space. For a curve $P:[a,b]\rightarrow X$, the $D$-length of $P$ is defined as

where the supremum is taken over all partitions $T:a=t_{0}<t_{1}<\ldots<t_{n}=b$ of $[a,b]$. The $D$-length of a curve may be infinite.

For a curve $P:[a,b]\rightarrow X$ and a set $Y\subset X$, consider the pre-image $P^{-1}(Y)\subset[a,b]$. Write the interior of $P^{-1}(Y)$ as the disjoint union of countably many open intervals $\{(a_{i},b_{i})\}_{i\geq 1}$. We define the restriction of $P$ to $Y$ as $P|_{Y}:=\cup_{i\geq 1}P[a_{i},b_{i}]$, which is the union of a family of curves, and its length is defined as

Note that $P|_{Y}$ and $P(P^{-1}(Y))$ are the same up to the inclusion of end points (of intervals in $P^{-1}(Y)$) or single points (i.e., each interval containing them is not a subset of $P^{-1}(Y)$). For the sets $Y$ that we will consider in this paper, the lengths of $P|_{Y}$ and $P(P^{-1}(Y))$ will be the same.

For $Y\subset X$, the internal metric of $D$ on $Y$ is defined as

where the infimum is taken over all paths $P$ in $Y$ from $x$ to $y$. Then $D(\cdot,\cdot;Y)$ is a metric on $Y$, allowing the distance between two points to be infinite.

We say $D$ is a length metric if for all $x,y\in X$ and $\delta>0$, there exists a curve with $D$-length at most $D(x,y)+\delta$ connecting $x$ and $y$. We say $D$ is a geodesic metric if for each $x,y\in X$, there exists a curve with $D$-length precisely $D(x,y)$ connecting $x$ and $y$.

In this paper, we consider the space $\mathbb{R}^{d}$ where $d\geq 2$ is a fixed dimension. For $z\in\mathbb{R}^{d}$, we write $z=(z_{1},\ldots,z_{d})$ for its coordinates. We use the notation $|\cdot|_{1}$, $|\cdot|_{2}$, and $|\cdot|_{\infty}$ to represent the $l^{1}$-, $l^{2}$-, and $l^{\infty}$-norms, respectively. We use $\mathfrak{d}_{1}$, $\mathfrak{d}_{2}$, and $\mathfrak{d}_{\infty}$ to denote the distances associated with these norms. Without specific mention, the distance that we use is the $l^{\infty}$-distance. For a set $A\subset\mathbb{R}^{d}$ and $r>0$, we define the $l^{\infty}$-neighborhood

As in (1.3), for $z\in\mathbb{R}^{d}$, we write $B_{r}(z)=B_{r}(\{z\})=z+(-r,r)^{d}$ for the open box centered at $z$ with side-length $2r$. We call a domain $A\subset\mathbb{R}^{d}$ a hypercubic shell if $A=B_{r_{1}}(x)\backslash B_{r_{2}}(x)$ for some $x\in\mathbb{R}^{d}$ and $r_{1}>r_{2}>0$.

We extend the notation of $|\cdot|_{\infty}$, $|\cdot|_{1}$, $\mathfrak{d}_{\infty}$, and $\mathfrak{d}_{1}$ to the integer lattice $\mathbb{Z}^{d}$. For $x\in\mathbb{Z}^{d}$ and an integer $n\geq 0$, we define $B_{n}(x)$ as the box centered at $x$ with side-length $2n$. Namely,

We will clarify in the context whether we are considering $x$ as a point in $\mathbb{R}^{d}$ or as a vertex in $\mathbb{Z}^{d}$. For an integer $n\geq 0$, define the set

Typically, we consider ${\hyperref@@ii[eq:def-rescaled-lattice]{\mathscr{L}}}_{n}$ as a subset of $\mathbb{R}^{d}$. However, when analyzing ($*$-)paths or ($*$-)clusters on the rescaled lattice $2^{-n}\mathbb{Z}^{d}$, as defined in Subsection 2.4, we view ${\hyperref@@ii[eq:def-rescaled-lattice]{\mathscr{L}}}_{n}$ as a subset of $2^{-n}\mathbb{Z}^{d}$. In this paper, we will also consider the graph distance on the rescaled lattice $\varepsilon\mathbb{Z}^{d}$ which is defined as $1/\varepsilon$ times the $l^{\infty}$-distance when considering $\varepsilon\mathbb{Z}^{d}$ as a subset of $\mathbb{R}^{d}$.

Constants like $c,c^{\prime},C,C^{\prime}$ may change from place to place, while constants with subscripts like $c_{1},C_{1}$ remain fixed throughout the article. All constants may implicitly rely on the dimension $d$, the kernel K, r0, and $\xi$. The dependence on additional variables will be indicated at the first occurrence of each constant.