First passage percolation on the exponential of two-dimensional branching random walk

We consider the branching random walk $\{\mathcal R^N_z: z\in V_N\}$ with Gaussian increments indexed over a two-dimensional box $V_N$ of side length $N$, and we study the first passage percolation where each vertex is assigned weight $e^{\gamma \mathcal R^N_z}$ for $\gamma>0$. We show that for $\gamma>0$ sufficiently small but fixed, the expected FPP distance between the left and right boundaries is at most $O(N^{1 - \gamma^2/10})$.


Introduction
For n ∈ N and N = 2 n , let V N ⊆ Z 2 be a box of side length N whose lower left corner is located at the origin. Let BD j denote the collection of boxes of the form ([0, 2 j − 1] ∩ Z) 2 + (i 1 2 j , i 2 2 j ) where i 1 , i 2 and j are nonnegative integers. For v ∈ Z 2 , let BD j (v) be the (unique) box in BD j that contains v. Let {a B } k≥0,B∈BD k be a collection of i.i.d. standard Gaussian variables. We define a branching random walk (indexed over V N ) with Gaussian increments by (1.1) For any rectangle V ⊆ Z 2 , we denote by Π V LR the collection of all paths π in V (considered as a subgraph of Z 2 ) that connects the left boundary of V to its right boundary. We refer to such a path as a (left-right) crossing of V . For a fixed γ > 0 we define to be the first passage percolation distance between the two boundaries of V N where we assign each vertex v a weight of e γR N v .
Remark 1.2. Our proof in fact gives an upper bound on the expected weight for the geodesic crossing through a rectangle. As a result, one can show that the expected weight for the geodesic connecting two fixed vertices has exponent strictly less than 1, by constructing a sequence of (log N many) rectangles with geometrically growing size that connect these two vertices.

A brief discussion on motivation and background
In a broad context, our work is motivated by studying first passage percolation on random media with heavy correlation. The main contribution of our work, is to demonstrate an instance that in a hierarchical random field the exponent of the FPP distance can be strictly less than 1 (despite the fact that the a straight line has weight with exponent strictly larger than 1, and that the majority of the vertices are of values O( √ log N )). This is, of course, rather different from the classical FPP where the edge/vertex weights are independent and identically distributed. We refer to [2,11] and references therein for a review of classical first passage percolation.
In the specific instance of branching random walk, our work is closely related to the first passage percolation problem when the vertex weight is given by exponentiating a two-dimensional discrete Gaussian free field (GFF). Inspired by the current work, a similar upper bound on the FPP distance in this case has been recently obtained in [6]. The FPP on two-dimensional discrete GFF, as we expect, is of fundamental importance in understanding the random metric associated with the Liouville quantum gravity (LQG) [17,10,18]. We remark that the random metric of LQG is a major open problem, even "just" to make rigorous sense of it. In a recent series of works of Miller and Sheffield, much understanding has been obtained (more on the continuum set up) in the special case of γ = 8/3; see [15,14] and references therein. Another recent work [12] has provided some bounds on the scaling exponent for a type of LQG metric, though we emphasize that their definition is of very different flavor from the FPP perspective considered in our paper (in particular, no mathematical connection can be drawn between these works at the moment). We refrain ourselves from an extensive discussion on background of LQG in this article. As a final remark to the connection of LQG metric, we note that there are other candidate discrete metric (associated with LQG or GFF) that have been proposed, and it is possible that these metrics with suitable normalization would give a more desirable scaling limit than that of FPP. However, we feel that in the level of precision of the present article and that of [6], it is quite possible that the fundamental mathematical structures (and thus obstacles) are common for these related notions of metrics.
BRW is perhaps the simplest construction that approximates a log-correlated Gaussian field, of which the GFF is a special instance. A number of other properties, especially those regarding to the extreme values of the field, have been proved to exhibit universal behavior among log-correlated Gaussian fields (c.f., [13,7]). Indeed, it is typical that properties were first proved for BRW, and then later for GFF; and it is also typical that the proof in the case of GFF were substantially enlightened by the understanding on BRW. However, caution is required when drawing heuristic conclusion on GFF based on BRW for FPP: as demonstrated in [8], there exists a family of logcorrelated Gaussian fields whose exponent for FPP can be made arbitrarily close to 1 (as K grows to ∞) if one is allowed to perturb the covariance entry-wise by an additive amount that is at most K (and thus the scaling exponent for FPP is non-universal among log-correlated fields). That being said, we remark that while the mathematical details in [6] are substantially more involved and delicate, these two papers do share the same very basic multi-scale analysis proof framework.

Main ideas of our approach
Our approach is inspired by the rescaling argument employed in the proof of the connectivity of fractal percolation process [3,5,4], the idea of which went back to [1]. For the purpose of induction, we will in fact work with crossings through a rectangle rather than a square. In what follows, we should consider γ as a small positive number (but fixed as N → ∞).
We wish to carry out a inductive construction and proof for light path crossing the rectangle: we will take advantage of the hierarchical structure of BRW and will construct the BRW along the way as we construct light path in different scales. In order to properly describe our procedure, we need a number of definitions. Let Γ = Γ(γ) ∈ [α/γ 2 , α/γ 2 + 2] be an odd, positive integer for some α to be selected, and let V Γ The reason for choosing Γ odd is mere technical convenience which will become apparent later. We will also work under the assumption that γ ≪ 1/α ≪ 1 (small or large enough for our bounds or inequalities to hold although we keep this implicit in our discussions). We define a Gaussian process {R N z : z ∈ V Γ N } on V Γ N as follows: is basically a concatenation of Γ independent BRWs placed side by side. We can view V Γ N as a ΓN × N rectangle divided into Γ cells of dimension N × N . Similarly, V Γ 2N can be viewed as a 2Γ × 2 rectangle divided into 4 sub-rectangles of dimension ΓN × N each of which is a copy of V Γ N (see Figure 1 below for an illustration).
a B n;1,2,1 a B n;1,2,2 a B n;1,2,3 V Γ N ;1,2 a B n;1,1,3 a B n;1,1,2 a B n;1,1,1 V Γ N ;1,1 a B n;2,1,3 a B n;2,1,2 a B n;2,1,1 V Γ N ;2,1 a B n;2,2,1 a B n;2,2,2 a B n;2,2,3 We wish to construct a light path in V Γ 2N based on constructions in V Γ N . As the path crosses through the rectangle of V Γ 2N , we will make a choice of whether the path will stay in the upper or lower sub-rectangles based on the Gaussian values associated with the boxes of dimension N × N , and thus we will switch back and forth between top and bottom layers. Our goal is to show that the expected weight of the crossing we constructed expands by a factor less than 2 when the size of the rectangles doubles. Apart from doubling of dimension at each level,there are a number of reasons that the weight will expand: (1) For standard Gaussian variable, we have Ee γX = 1 + γ 2 2 + O(γ 3 ); (2) We will need to make a construction to connect the paths on the left half and the right half of the rectangle; (3) Every time we switch between top and bottom layer, we will have to add a top-down crossing in the switching location. What we would possibly gain is from the "variation" of these Gaussian variables associated with N × N boxes, and thus choosing the favorable layer would reduce the weight of our path. There are a couple of subtle points in carrying out this plan, as we explain now. In order to see how switching will reduce our cost, we first consider a toy problem where we simply compute E min(X, Y ) for independent standard Gaussian variables X and Y . It turns out for this purpose it is more convenient to write X = X+Y If we carry out inductive procedure in this fashion, what we gain from the switching will not be able to beat the aforementioned expansions. This is because, in the language of the toy model, min(X, Y ) is itself a random variable that can be written as where we gain − 1 √ π by averaging over (X − Y ) but we get nothing by averaging over X + Y (which is independent of (X − Y )). In fact, the better strategy would be to save the variation from (X + Y ) and use it in the optimizations in the next scale. But it is inconvenient to carry out this strategy, as it would require to do an induction based on quantities beyond the expected weight of the crossings in different scales. We resolve this issue by represent the BRW in a different way, which is tailored for our application here.
Another point is that, in the aforementioned optimization procedure, one has to be rather strategic in switchings as there is a cost for that. Eventually, this is reduced to a question of computing the following regularized total variation for the Brownian motion, defined by Here B [0,1] is a standard Brownian motion, and λ > 0 is the term that measures the switching cost. We will use a recent result of [9] on the asymptotic value of (1.2) as λ → 0.
Notation convention. In the rest of the paper we refer to the vertices of Z 2 as points. For any point z ∈ Z 2 , the horizontal and vertical coordinates of z are denoted by z x and z y respectively. If z y = 0, we represent z simply as z x , which should be clear from the context. A (finite) path π is a finite sequence (v 0 , v 1 , · · · , v m ) of points such that for each i ≥ 0, v i and v i+1 are neighbors in Z 2 .
In each of the pairs of non-negative integers (n, N ), (ℓ, L), (ℓ ′ , L ′ ) and (ℓ ′′ , L ′′ ), the second element is always assumed to be 2 raised to the power of first element. For functions F (.) and G(.), we Acknowledgement. We warmly thank Steve Lalley for numerous helpful discussions and constant encouragement on the project.

Coarsening of paths and L-segments
The L-coarsening of a (finite) path π = (v 0 , v 1 , · · · , v m ) in Z 2 is defined as follows. Cover the lattice Z 2 with L × L (non-overlapping) boxes such that one such box is V L . Denote the center of a box B as c(B). For v ∈ Z 2 , let B L (v) be the unique box containing v. Now define a sequence of integers m 0 , m 1 , · · · recursively as m 0 = 0 and, Here we adopt the standard convention that infimum of an empty set is ∞. Let p L,π be the last integer j such that m j is finite. The L-coarsening of π, denoted as π L,coarse , can now be defined as the sequence c(B L (v m 0 )), c(B L (v m 1 )), · · · , c(B L (v mp L,π )) . We can also treat π L,coarse as a path in R 2 that connects successive points in this list by a line segment. See Figure 2 for an illustration. Notice that the coarsening of a simple (i.e. self-avoiding) path is not necessarily simple. However if the L-coarsening of a (left-right) crossing π * of V Γ 2L is simple, then the 2L-coarsening of π * is also simple. We will use this fact when we carry out the induction step. An L-segment at level ℓ is a path π whose 2 ℓ -coarsening is one of two shapes shown in Figure 3.

Switched sign construction for branching random walk
We first introduce a few more notations. As described in the introduction, we view V Γ 2N as a 2Γ × 2 rectangle divided into 4 sub-rectangles of dimension ΓN × N each of which is a copy of V Γ N (see Figure 1 for an illustration). Denote these 4 subrectangles (or the copies of V Γ N ) as {V Γ N ;i,k } i∈ [2],k∈ [2] in the usual order. We can also shift the origin by u = (Γi2 n+1 , j2 n+1 ) for i, j ∈ N ∪ {0} and denote the corresponding subrectangles (respectively rectangles) by {V Γ,u N ;i,k } i∈ [2],k∈ [2] (respectively V Γ,u 2N ). Similarly we can define the fields We observe that the Gaussian field {R 2N,u Figure 1). We will omit the additional superscript u in all these notations when u = 0.
We will represent BRW by a construction in the fashion of switching-signs, which is tailored to our inductive construction for light crossings. To this end, we denote the collection of all rectangles of the form ([0, 2 ℓ − 1] × [0, 2 ℓ+1 − 1]) ∩ Z 2 + (i2 ℓ , j2 ℓ ) by BD k,2 where i, j, ℓ are nonnegative integers. Let A ℓ,Γ be the collection of all points of the form (Γi2 ℓ , j2 ℓ ). Denote by R u ℓ;k,j the rectangle formed by the boxes B u ℓ;1,k,j and B u ℓ;2,k,j when x ∈ A ℓ+1,Γ . Note that R u ℓ;k,j ∈ BD ℓ,2 . Let {a B } B∈BD k,2 ,k≥0 be a collection of i.i.d. standard normal random variables, that is independent of {a B } k≥0,B∈BD k . We will now construct a family of Gaussian fields {χ L,u z : z ∈ V Γ,u 2 ℓ } for u ∈ A ℓ,Γ and ℓ = 0, 1, 2, . . . , n recursively as follows: • χ 1,u z = 0 for all z ∈ V Γ,u 1 .
• Let u ∈ A ℓ+1,Γ and {χ L,u z;i,k : z ∈ V Γ,u L;i,k } be the field on V Γ,u L;i,k constructed in level ℓ. For z ∈ B u ℓ;2,k,j , define k,j is the rectangle defined as follows: For future reference we will denote Z ℓ;i,k,j = (−1) (k−1)Γ+j b ℓ aR ℓ;k,j + (−1) i c ℓ a B ℓ;2,k,j . Similarly as before, we will drop the superscript u whenever u = 0. See Figure 4 for an illustration of this construction at level ℓ + 1. Notice that since c 2 ℓ = 2b 2 ℓ , distribution of the field {χ N z : z ∈ V Γ N ;j } is symmetric with respect to symmetries of the box V Γ N ;j . This fact will be used later when we construct crossings through V Γ N ;j 's in vertical direction. Finally, we definẽ It is not difficult to check that the fields {χ N z : z ∈ V N ;j }'s are independent and identically distributed for j ∈ [Γ]. Hence it suffices to verify that we have the correct covariances between field values at all pairs of vertices inside V N . Toward this end take a pair (u, v) of vertices in V N which were separated until level ℓ, i.e., Cov(R N u , R N v ) = n − ℓ. The covariance between χ N u and χ N v is given by Hence the lemma follows.
Remark 2.2. It is clear that it suffices to work with {χ N z : z ∈ V Γ N } rather than the BRW in what follows. Using this construction, in every scale of our optimization, we crucially have effective variance c 2 ℓ ≈ 2 3 that is larger than 1/2 (which would be the variance we effectively used if we work with the more canonical construction of BRW).

The induction hypotheses
We will construct a light (left-right) crossing through V Γ L (or equivalently V Γ,u L ) inductively for L = 2 ℓ and ℓ = 0, . . . , n. Throughout this section whenever we refer to a crossing of V Γ,u L , it is implicitly assumed that the underlying field is {χ L,u z : z ∈ V Γ,u L } unless stated otherwise. For technical convenience, let us assume that for each ℓ ≥ 0 and u ∈ A ℓ+1,Γ we have an independent standard Brownian motion {W u,ℓ t } 0≤t≤2Γ that is also independent with {a B } B∈BD k,2 ,k≥0 . We sample a B u ℓ;2,k,j 's as increments of this process at appropriate time points. We are now ready to state our induction hypotheses. For ℓ = 0, we have only one option, i.e., to take the straight line as our crossing. This crossing has weight exactly Γ. Suppose that for each 1 ≤ ℓ ′ ≤ ℓ and u ∈ A ℓ ′ ,Γ , we have a crossing π * ,u,ℓ ′ of V Γ,u 2 ℓ ′ , identically distributed for different u, that satisfies the following properties: 1. π * ,u,ℓ ′ L ′ /2,coarse is simple and measurable relative to σ {W u,ℓ ′ −1 t } 0≤t≤2Γ .
One of the consequences of our induction hypotheses, which is crucial for our analysis, is that v u j,L,δL is uniformly distributed for all ℓ ′ ≤ ℓ as given by the following lemma.
Lemma 3.1. Let R u ℓ;i,k,j be the unique L 100 × L rectangle that shares its right boundary with with B u ℓ;i,k,j . Also let {B u ℓ;i,k,j,m } 1≤m≤L/L 100 be the collection of boxes in BD log 2 L 100 that comprise R u ℓ;i,k,j . Now if the induction hypotheses 1, 2 and 3 hold for all ℓ ′ ≤ ℓ, then v u j,L,L 100 is distributed uniformly on the set {c u ℓ;i,k,j,m } 1≤m≤L/L 100 , where c u ℓ;i,k,j,m is the center of the box B u ℓ;i,k,j,m (see Figure 5).
Proof. We get, from hypothesis 2, that for any 2 ≤ L ′ ≤ L, j ′ ∈ [Γ] and x ∈ A ℓ ′ ,Γ , v u j ′ ,L ′ ,L ′ /2 is equally likely to be above or below the line y = u y + 2 ℓ ′ −1 − 0.5. We can recursively apply this argument for all the levels down to log 2 L 100 in view of hypothesis 1 which gives that the L ′ /2coarsenings of crossings at level ℓ ′ are independent for different ℓ ′ . Combining these observations with hypothesis 3 completes the proof of the lemma.

The induction step
Now we will carry out the induction step. It suffices to produce a crossing for u = 0. As usual, we will drop u from all the superscripts when u is origin. Before going to more detailed discussions let us give a broad overview of the strategy that we follow.  First we decide the L-coarsening of π * ,ℓ+1 . It would consist of a sequence H 1 , H 2 , · · · , H R+1 of horizontal segments (possibly of length 0) from left to right such that every two successive segments are connected by a vertical segment. See Figure 6 for an illustration. Notice that this is the only possible option for π * ,ℓ+1 L,coarse , if it has to be simple as required by induction hypothesis 1. So we can "encode" π * ,ℓ+1 L,coarse as a sequence of {1, 2} valued random variables {i j,k } j∈[Γ],k∈ [2] , where i k,j = 1 or 2 accordingly as π * ,ℓ+1 L,coarse enters the rectangle R ℓ;k,j through B ℓ;1,k,j or B ℓ;2,k,j respectively. We refer to this as a switching strategy. Having defined L-coarsening of the crossing, we now move on to actual construction. From the previous level we have four crossings {π * ,ℓ i,k } i∈ [2],k∈ [2] respectively in {V Γ L;i,k } i∈ [2],k∈ [2] . We "join" the paths π * ,ℓ i,1 and π * ,ℓ i,2 into a crossing π * ,ℓ+1 i of V Γ 2L for i ∈ [2]. Since L-coarsening of crossings at level ℓ are also simple, it follows that the vertices of π * ,ℓ+1 i that lie within some B ℓ;i,k,j define a subpath of π * ,ℓ+1 i . Now we replace each H r with the subpath h * ,ℓ+1 r of π * ,ℓ+1 i that is contained within the L × L boxes that intersect H r . Here i is 1 or 2 accordingly as H r lies in the top or bottom row of V Γ 2L (See Figure 6 and the discussions preceding Figure 1). Finally we connect h * ,ℓ+1 r and h * ,ℓ+1 r+1 by an appropriate L-segment v * ,ℓ+1 r at level ℓ. The paths h * ,ℓ+1 R+1 define the crossing π * ,ℓ+1 . Note that L-coarsening of π * ,ℓ+1 is the very one that we started with.
We also need some notations in order to track the change in expected weight of crossings between L Figure 6 -L-coarsening of π * ,ℓ+1 . Here Γ = 3 and R = 2.
It is easy to see that E|Err γ,ℓ;k,2 | = d γ,ℓ O(γ 3 ). On the other hand, we get the following from expanding the terms in Err γ,ℓ;k,1 : Expectations of the first two terms in the expansion above are obviously 0. As to the third term, recall that our switching mechanism is independent of the random variables aR ℓ;k,j 's by induction hypothesis 2. Consequently EErr γ,ℓ;k,1 = 0. Thus we only need to reckon with the first term in the right hand side of (3.1) so far as the contributions from π * ,ℓ i,k 's are concerned. This will, of course, depend on the particular switching strategy we adopt. Finally we need to add the contributions from v * ,ℓ+1 r 's and the extra crossings that we make in order to link π * ,ℓ i,1 and π * ,ℓ i,2 for i ∈ [2]. Now we carry out the detailed computations. It is expected that d γ,ℓ,j is approximately uniform over j ∈ [Γ]. However, proving this requires some effort. It turns out easier to treat the "hypothetical" case that d γ,ℓ is dominated by a small number of d γ,ℓ,j 's. We also include a case to deal with small values of ℓ when c 2 ℓ is not big enough for our "main" strategy to work.
In this case our switching strategy is simple. For each k = 1, 2, we choose the rectangle that minimizes Γ j=1 d γ,ℓ,j Z ℓ;i,k,j . Thus, we have for k ∈ [2] where the last inequality follows from our assumption that Γ ≥ α/γ 2 . Let u k ∈ A ℓ,Γ be such that V Γ,u k L is the minimizer. In order to connect π * ,u 1 ,ℓ and π * ,u 2 ,ℓ into a crossing of V Γ 2L , we need some additional paths. To this end we add two vertical crossings through the rectangles We can now form π * ,ℓ+1 by concatenating π * ,u 1 ,ℓ , π * ,u 2 ,ℓ and these four crossings. See Figure 7 for an illustration. It is obvious from the construction that π * ,ℓ+1 L,coarse is simple, symmetric in law with respect to the line y = L − 0.5 and entirely determined by the random variables {a B ℓ;2,k,j } k∈ [2],j∈ [Γ] . The only possible case where induction hypothesis 3 could fail would be the line L ΓL . But we do not need to consider this line as Γ is odd. Finally notice that, since Γ ≫ 1 and minimum crossing weights through adjacent rectangles are super-additive, each of the four additional crossings will have expected weight bounded by, say, 2.1 Γ d γ,ℓ . Using the fact that α ≫ 1, we then deduce that in this case Case 2. If ℓ < 60, we use the strategy described in Case 1. It then follows from the previous discussions that d γ,ℓ+1 ≤ (2 + 2γ 2 )d γ,ℓ .
Since d γ,0 = Γ, we get by repeated application of the previous inequality

Case 3.
Γ j=1 d γ,ℓ,j 1 d γ,ℓ,j ≥Γ −2/3 d γ,ℓ ≤ d γ,ℓ,j Γ −1/10 and ℓ ≥ 60. This is the main and real case. We will also assume that for otherwise we can simply follow the strategy in Case 1. In this case our switching strategy will in fact be informed by the way we plan to contruct the L-segments v * ,ℓ+1 r 's. So we first give a detailed description of the latter. In what follows, we may assume without any loss of generality that δL ≥ 2 for δ = 2 −100 . We refer the reader to It suffices to consider the case when j r ≤ Γ. Let v u jr,L,δL and v d jr,L,δL respectively denote the last points along π * ,L,ℓ δL,coarse and π * ,ℓ δL,coarse that lie to the left of L jrL . Then v u jr,L,δL and v d jr,L,δL are centers of two δL × δL boxes, say, A u jr,L and A d jr,L respectively. LetÃ u jr,L (respectivelyÃ d jr,L ) be the box around A u jr,L (respectivelyÃ d jr,L ) of side length 2δL. Then we construct light contours in each of the annuliÃ u jr,L \ A u jr,L andÃ d jr,L \ A d jr,L . These can easily be achieved by concatenating four minimum weight crossings in four rectangles. In addition, we letÃ r be a rectangle whose left top corner is v u jr,L,δL + (0.5, δL − 0.5) and right bottom corner is v d jr,L,δL + (0.5, −δL + 0.5). We construct a minimum weight vertical crossing throughÃ r . Finally we concatenate these light crossings into v * ,ℓ+1 r . The total expected weight of these crossings can be bounded by our induction hypotheses. Notice that we can always concatenate these paths in a way so that our construction of v * ,ℓ+1 r obeys induction hypothesis 3. Let us now try to bound the expected weight of v * ,ℓ+1 r where we assign vertex weight based on variables associated with boxes of side length ≤ L. Notice that this does not require the knowledge of particular switching strategy we employ. Denote by D r the total weight of the light crossings in the two annuli (where we assign vertex weight based variables associated with boxes of side length ≤ δL/2). Since the δL coarsening of π * ,u,ℓ ′ is independent of all the Gaussians at level lower than δℓ, we see that ED r ≤ 33d γ,ℓ−log 2 (2/δ) Γ .

(3.4)
In both (3.3) and (3.4), we have silently used the fact that expected minimum weights of crossings through V m L ′ 's form a superadditive sequence in m. The following lemma says that we do not lose much in terms of the expected costs of these crossings after we take into account the contributions from Gaussians appearing in higher levels.  Proof. Let M = max i X i . It is clear that EM ≤ 200 log(1/γ), and also that P(M ≥ EM + u) ≤ e −u . The conclusion follows by a straightforward computation.
This completes the proof of Theorem 1.1.