Interlacing and Scaling Exponents for the Geodesic Watermelon in Last Passage Percolation

Abstract

In a discrete planar last passage percolation (LPP), random values are assigned independently to each vertex in ${\mathbb {Z}}^2$, and each finite upright path in ${\mathbb {Z}}^2$ is ascribed the weight given by the sum of values attached to the vertices of the path. The weight of a collection of disjoint paths is the sum of its members’ weights. The notion of a geodesic, namely a path of maximum weight between two vertices, has a natural generalization concerning several disjoint paths. Indeed, a k-geodesic watermelon in $[1,n]^2\cap {\mathbb {Z}}^2$ is a collection of k disjoint upright paths contained in this square that has maximum weight among all such collections. While the weights of such collections are known to be important objects, the maximizing paths have remained largely unexplored beyond the $k=1$ case. For exactly solvable models, such as exponential and geometric LPP, it is well known that for $k=1$ the exponents that govern fluctuation in weight and transversal distance are 1/3 and 2/3; which is to say, the weight of the geodesic on the route $(1,1) \rightarrow (n,n)$ typically fluctuates around a dominant linear growth of the form $\mu n$ by the order of $n^{1/3}$; and the maximum Euclidean distance of the geodesic from the diagonal typically has order $n^{2/3}$. Assuming a strong but local form of convexity and one-point moderate deviation estimates for the geodesic weight profile—which are available in all known exactly solvable models—we establish that, typically, the k-geodesic watermelon’s weight falls below $\mu n k$ by order $k^{5/3}n^{1/3}$, and its transversal fluctuation is of order $k^{1/3}n^{2/3}$. Our arguments crucially rely on, and develop, a remarkable deterministic interlacing property that the watermelons admit. Our methods also yield sharp rigidity estimates for naturally associated point processes. These bounds improve on estimates obtained by applying tools from the theory of determinantal point processes available in the integrable setting.

Appendices

Appendix A. Exponential & Geometric LPP Satisfy the Assumptions

In this appendix, we cite the results which show that exponential and geometric LPP satisfy Assumptions 1, 2, and 3.

We start with the foundational result on the Tracy-Widom fluctuations of the maximal path weight, due to Johansson [Joh00a], and continue with moderate deviations estimates and expectation asymptotics for the maximal path weight.

Recall that $X_{n,\lfloor hn\rfloor }$ is the last passage value from (1, 1) to $(n,\lfloor hn\rfloor )$.

Theorem A.1

Suppose that the vertex weight law in LPP is exponential of mean one, and fix $\psi > 1$. As $n\rightarrow \infty$, uniformly over $h\in [\psi ^{-1}, \psi ]$,

$$\begin{aligned} \frac{X_{n, \lfloor hn\rfloor } - (1+\sqrt{h})^2n}{h^{-1/6}(1+\sqrt{h})^{4/3}n^{1/3}}{\mathop {\rightarrow }\limits ^{d}} F_{{\mathrm {TW}}} \, . \end{aligned}$$

Suppose instead that the weight law is geometric with parameter $p\in (0,1)$. Then, uniformly over $h\in [\psi ^{-1},\psi ]$, as $n\rightarrow \infty$,

$$\begin{aligned} \frac{X_{n, \lfloor hn\rfloor } - \omega (h,p)n}{\sigma (h,p)n^{1/3}}{\mathop {\rightarrow }\limits ^{d}} F_{{\mathrm {TW}}}, \end{aligned}$$

where

$$\begin{aligned} \omega (h,p) = \frac{(1+\sqrt{hp})^2}{1-p}\quad \text {and}\quad \sigma (h,p)=\frac{p^{1/6}h^{-1/6}}{1-p}(\sqrt{h} + \sqrt{p})^{2/3}(1+\sqrt{hp})^{2/3}. \end{aligned}$$

Here ${\mathop {\rightarrow }\limits ^{d}}$ denotes convergence in distribution, $F_{{\mathrm {TW}}}$ is the GUE Tracy-Widom distribution, and uniformly is meant in the sense of uniform (over h) convergence of distribution functions.

Proof

These results appear for fixed h in [Joh00a], the first as Theorem 1.6 there and the second as Theorem 1.2.

In fact, the proofs also establish the uniformity of the convergence for $h\in [\psi ^{-1},\psi ]$, though unfortunately this is not explicitly stated. For the aid of the reader, we give a few brief pointers to how this uniformity can be seen to hold in [Joh00a]. Let $F_{n,h}$ be the distribution function of $X_{n,\lfloor nh\rfloor }$. It is shown in [Joh00a] that $F_{n,h}$ converges pointwise to $F_{{\mathrm {TW}}}$ for fixed h (in [Joh00a], our h is denoted by $\gamma$).

In the geometric case, $F_{n,h}$ is explicitly written down in terms of a determinantal formula involving the Meixner polynomials; in the exponential case the same is done in terms of the Laguerre polynomials. The argument for the exponential case is the same as for the geometric case given ahead, with Laguerre polynomials replacing Meixner polynomials.

Lemma 3.1 of [Joh00a] asserts that if certain estimates are satisfied by determinantal formulas of this form, then they converge pointwise to $F_{{\mathrm {TW}}}$. That $F_{n,h}$ satisfies these estimates in the geometric case is then shown in [Joh00a, Sect. 5] by making use of asymptotics and estimates for Meixner polynomials. If these estimates hold uniformly over $h\in [\psi ^{-1}, \psi ]$, then the convergence claimed in [Joh00a, Lemma 3.1] holds uniformly.

That the estimates hold uniformly is a consequence of the fact that all the bounds in [Joh00a, Sect. 5] are continuous functions of h, and $[\psi ^{-1}, \psi ]$ is a compact set. The dependence on h in many of these bounds is not written explicitly, but it is straightforward to check that their dependence on h is continuous. $\square$

The next three statements establish that exponential and geometric LPP satisfy Assumptions 2 and 3a.

Theorem A.2

(Moderate deviation estimate). Consider exponential LPP. Fix $\psi >1$ and let $h\in [\psi ^{-1},\psi ]$. There exist $t_0=t_0(\psi ), n_0 = n_0(\psi )$ and $c = c(\psi )>0$ such that, for $n>n_0$ and $t > t_0$,

$$\begin{aligned} {\mathbb {P}}\left( X_{n, \lfloor hn\rfloor } - (1+\sqrt{h})^2 n > tn^{1/3}\right) \le \exp \left( -c\min (t^{3/2}, tn^{1/3})\right) \, . \end{aligned}$$

For the lower tail, for $t>t_0$,

$$\begin{aligned} {\mathbb {P}}\left( X_{n, \lfloor hn\rfloor } - (1+\sqrt{h})^2 < -tn^{1/3}\right) \le \exp \left( -ct^3\right) \, . \end{aligned}$$

Similarly, in geometric LPP of parameter $p\in (0,1)$, the above two displays hold with $\omega (h,p)$ (as in Theorem A.1) in place of $(1+\sqrt{h})^2$.

Proof

For exponential LPP, the lower tail for $t>t_0$ and the upper tail for $t_0< t < n^{2/3}$ are provided by Theorem 2 of [LR10]. For the remaining case of $t\ge n^{2/3}$ in the upper tail, see [Joh00a].

For geometric LPP, the upper tail bound is proved in [Joh00a] (combining Corollary 2.4 and equation (2.22) there), while the lower tail bound is implied by [BDM+01, Theorem 1.1]. $\square$

Theorem A.3

(Expected point-to-point weight). Fix $\psi >1$ and let $h \in [\psi ^{-1},\psi ]$. There exist $c_1 = c_1(\psi )>0$, $c_2=c_2(\psi )>0$, and $n_0=n_0(\psi )$ such that, for $n>n_0$, in exponential LPP,

$$\begin{aligned} \frac{{\mathbb {E}}[X_{n, \lfloor hn\rfloor }] - (1+\sqrt{h})^2n}{h^{-1/6}(1+\sqrt{h})^{4/3}n^{1/3}} \in (-c_1,-c_2), \end{aligned}$$

while for geometric LPP of parameter $p\in (0,1)$,

$$\begin{aligned} \frac{{\mathbb {E}}[X_{n, \lfloor hn\rfloor }] - \omega (h,p) n}{ \sigma (h,p)n^{1/3}} \in (-c_1,-c_2), \end{aligned}$$

with $\omega (h,p)$ and $\sigma (h,p)$ as in Theorem A.1.

Proof

This follows from Theorem A.1 and Theorem A.2, and that the GUE Tracy-Widom distribution has negative mean. Note that we need the uniform convergence of the distribution functions in Theorem A.1 in order to have $n_0$ depend on only $\psi$ and not h. $\square$

Assumption 2 can be verified in exponential and geometric LPP by replacing n by $n-xn^{2/3}$ and setting h to be $(n+xn^{2/3})/(n-xn^{2/3})$ in Theorem A.3.

We now note why the GUE Tracy-Widom distribution has negative mean, by pulling together known results to which Ivan Corwin has drawn our attention.

Lemma A.4

(Negative mean of GUE Tracy-Widom). Let $X_{{\mathrm {TW}}}$ be distributed according to the GUE Tracy-Widom distribution. Then we have ${\mathbb {E}}[X_{{\mathrm {TW}}}] < 0$.

Proof

Remark 1.15 and equation 1.29 of [QR19] show that $X_{{\mathrm {BR}}}$ strictly stochastically dominates $4^{1/3} X_{{\mathrm {GUE}}}$, where the law of $X_{{\mathrm {BR}}}$ is the Baik-Rains distribution (the cumulative distribution function of $X_{{\mathrm {BR}}}$ is labeled $F_{{\mathrm {stat}}}^0$ in [QR19, Equation 1.29]). Now [BR00, Proposition 2.1] asserts that ${\mathbb {E}}[X_{{\mathrm {BR}}}] = 0$, completing the proof. $\square$

Appendix B. Proofs of Basic Tools

This appendix contains some consequences of Assumptions 1, 2, and 3a that were used in the main text.

• Section B.1 proves Proposition 3.5, the upper tail bound on interval-to-interval weights;

• Section B.2 proves Proposition 3.6, concerning upper tails for point-to-line weights; and

• Section B.3 proves Proposition 3.7, concerning the lower tail and mean of the constrained point-to-point weights.

1.1 Interval-to-interval estimates

Proof of Proposition 3.5

We start with the bound (7) on ${{\widetilde{Z}}}$, and we treat only the case that $|z|\le \rho r^{1/3}$; when $|z|>\rho r^{1/3}$, our argument works by setting $z=\rho r^{1/3}$ and using the concavity of the limit shape posited in Assumption 2.

By considering the event that $\sup _{(u,v)\in {\mathcal {S}}(U)} X_{u \rightarrow v}$ is large and two events defined in terms of the environment outside of U, we find a point-to-point path which has large length. To define these events, first define points $\phi _{{\mathrm {low}}}$ and $\phi _{{\mathrm {up}}}$ on either side of A and B:

$$\begin{aligned} \phi _{{\mathrm {low}}}&:= \left( -\ell ^{3/2}r, -\ell ^{3/2}r\right) \\ \phi _{{\mathrm {up}}}&:= \left( (1+\ell ^{3/2})r-zr^{2/3}, (1+\ell ^{3/2})r +zr^{2/3}\right) . \end{aligned}$$

Let $u^*$ and $v^*$ be points on A and B where the suprema in the definition of ${{\widetilde{Z}}}$ are attained, and set

$$\begin{aligned} E_{{\mathrm {low}}}&= \left\{ X_{\phi _{{\mathrm {low}}}\rightarrow u^*-(0,1)}> \mu \ell ^{3/2}r - \frac{\theta \ell ^{1/2}}{3}r^{1/3}\right\} \quad \text {and}\\ E_{{\mathrm {up}}}&= \left\{ X_{v^*+(0,1) \rightarrow \phi _{{\mathrm {up}}}} > \mu \ell ^{3/2}r - \frac{\theta \ell ^{1/2}}{3} r^{1/3}\right\} . \end{aligned}$$

Using Assumption 2 to bound the expectation in going from the second to the third line of the following, we find that

$$\begin{aligned}&{\mathbb {P}}\bigg (\sup _{(u,v)\in {\mathcal {S}}(U)} X_{u\rightarrow v} > \mu r - G_2\frac{z^2r^{1/3}}{1+2\ell ^{3/2}}+\theta \ell ^{1/2}r^{1/3}, E_{{\mathrm {low}}}, E_{{\mathrm {up}}}\bigg )\nonumber \\&\quad \le {\mathbb {P}}\left( X_{\phi _{{\mathrm {low}}}\rightarrow \phi _{{\mathrm {up}}}} \ge \mu (1+2\ell ^{3/2})r - G_2\frac{z^2r^{1/3}}{1+2\ell ^{3/2}}+ \frac{\theta \ell ^{1/2}}{3} r^{1/3}\right) \nonumber \\&\quad \le {\mathbb {P}}\left( X_{\phi _{{\mathrm {low}}}\rightarrow \phi _{{\mathrm {up}}}} \ge {\mathbb {E}}\left[ X_{\phi _{{\mathrm {low}}}\rightarrow \phi _{{\mathrm {up}}}}\right] + \frac{\theta \ell ^{1/2}}{3} r^{1/3}\right) \le \exp \left( -c\min (\theta ^{3/2}, \theta r^{1/3})\right) , \end{aligned}$$

(51)

for $c>0$ independent of $\ell$. We used Assumption 2 for the former inequality of the final line, and Assumption 3a and the stationarity of the random environment for the latter.

Let us denote conditioning on the environment U by the notation ${\mathbb {P}}(\,\cdot \mid U)$. By this, we mean we condition on the weights of vertices interior to U as well as those on the lower and upper sides A and B. Then we see

$$\begin{aligned}&{\mathbb {P}}\bigg (\sup _{(u,v)\in {\mathcal {S}}(U)} X_{u \rightarrow v}> \mu r - G_2\frac{z^2r^{1/3}}{1+2\ell ^{3/2}}+\theta \ell ^{1/2}r^{1/3}, E_{{\mathrm {low}}}, E_{{\mathrm {up}}}\, \bigg \vert \, U\bigg )\\&\quad = {\mathbb {P}}\left( \sup _{(u,v)\in {\mathcal {S}}(U)} X_{u \rightarrow v} > \mu r - G_2\frac{z^2r^{1/3}}{1+2\ell ^{3/2}}+\theta \ell ^{1/2}r^{1/3} \, \bigg \vert \, U\right) \cdot {\mathbb {P}}\left( E_{{\mathrm {low}}}\mid U\right) \cdot {\mathbb {P}}\left( E_{{\mathrm {up}}}\mid U\right) . \end{aligned}$$

So with (51), all we need is a lower bound on ${\mathbb {P}}\left( E_{{\mathrm {low}}}\mid U\right)$ and ${\mathbb {P}}\left( E_{{\mathrm {up}}}\mid U\right)$. This is straightforward using independence of the environment between U and the regions above and below it:

$$\begin{aligned} {\mathbb {P}}\left( E^c_{{\mathrm {lower}}} \mid U\right)&\le \sup _{u\in A} \,{\mathbb {P}}\left( X_{\phi _{{\mathrm {low}}}\rightarrow u-(0,1)} \le \mu \ell ^{3/2}r - \frac{\theta \ell ^{1/2}}{3}r^{1/3}\right) \le \frac{1}{2} \end{aligned}$$

for large enough $\theta$ and r (depending on $\ell$), using Assumption 3a. A similar upper bound holds for ${\mathbb {P}}\left( E^c_{{\mathrm {upper}}}\mid U\right)$. Together this gives

$$\begin{aligned}&{\mathbb {P}}\bigg (\sup _{(u,v)\in {\mathcal {S}}(U)}X_{u \rightarrow v}> \mu r - G_2\frac{z^2r^{1/3}}{1+2\ell ^{3/2}}+\theta \ell ^{1/2}r^{1/3}, E_{{\mathrm {low}}}, E_{{\mathrm {up}}}\, \bigg \vert \, U\bigg )\\&\quad \ge \frac{1}{4}\cdot {\mathbb {P}}\bigg (\sup _{(u,v)\in {\mathcal {S}}(U)} X_{u \rightarrow v} >\mu r - G_2\frac{z^2r^{1/3}}{1+2\ell ^{3/2}}+ \theta \ell ^{1/2}r^{1/3}\bigg ) \, , \end{aligned}$$

and taking expectation on both sides, combined with (51), gives the bound (7) of Proposition 3.5.

We now treat the bound (8) on $Z^{{\mathrm {ext}}, \delta }$. Observe that

$$\begin{aligned} \left\{ Z^{{\mathrm {ext}},\delta }> \theta \right\} \subseteq \bigcup _{u\in {\mathbb {L}}_{{\mathrm {low}}}}\left( \left\{ X_{u \rightarrow (\delta r, 2r)}> \mu r + \theta r^{1/3}\right\} \cup \left\{ X_{u \rightarrow (2r, \delta r)} > \mu r + \theta r^{1/3}\right\} \right) \, . \end{aligned}$$

We bound the probability of $\bigcup _{u\in {\mathbb {L}}_{{\mathrm {low}}}}\left\{ X_{u \rightarrow (\delta r, 2r)} > \mu r + \theta r^{1/3}\right\}$; the full bound then follows by a symmetric argument and a union bound.

The point $(\delta r, 2r)$ can be written as $({{\tilde{r}}} -{{\tilde{z}}}, {{\tilde{r}}} +{{\tilde{z}}})$ with ${{\tilde{r}}} = (1+\delta /2)r$ and ${{\tilde{z}}} =(1-\delta /2)r$. By the definitions, we have that ${{\tilde{z}}}/{{\tilde{r}}} \ge \rho$ if $\delta$ is sufficiently small. So by the concavity guaranteed by Assumption 2 and by making $\delta$ small enough, we see that

$$\begin{aligned} {\mathbb {E}}\left[ X_{u \rightarrow (\delta r,2r)}\right] \le \mu {{\tilde{r}}} - G_2 \rho ^2{{\tilde{r}}}&\le \mu r +\frac{\mu \delta }{2}r- G_2\rho ^2\left( 1+\frac{\delta }{2}\right) r \le (\mu -\eta )r \end{aligned}$$

for some $\eta >0$. With such a value of $\delta$ fixed, note that $(\delta r, 2r)$ is such that we may apply Assumption 3a; we find by so doing that

$$\begin{aligned} {\mathbb {P}}\left( X_{u \rightarrow (\delta r,2r)} \ge \mu r+\theta r^{1/3}\right)&\le {\mathbb {P}}\left( X_{u \rightarrow (\delta r, 2r)} - {\mathbb {E}}[X_{u \rightarrow (\delta r, 2r)}] \ge \eta r+\theta r^{1/3}\right) \\&\le \exp \left( -cr-c\min (\theta ^{3/2}, \theta r^{1/3})\right) . \end{aligned}$$

Taking a union bound over the $O(r^{2/3})$ many u in ${\mathbb {L}}_{{\mathrm {low}}}$ and reducing the value of c to absorb this factor into the exponent complete the proof of Proposition 3.5. $\square$

1.2 Point-to-line estimates

Proof of Proposition 3.6

Note that we may assume $s\le r^{1/3}$. Let $\rho$ and $G_1$ be as in Assumption 2. Let $\Gamma$ be the leftmost path with endpoint not on the line segment connecting $(r-(s+t)r^{2/3}, r+(s+t)r^{2/3})$ and $(r+(s+t)r^{2/3}, r-(s+t)r^{2/3})$ whose weight is X; this is well-defined by the weight maximization property of $\Gamma$. Let $p_{{\mathrm {left}}}$ be the coordinates of the starting point of $\Gamma$ on ${\mathbb {L}}_{{\mathrm {left}}}$ (recall from (4)), and $p_{{\mathrm {right}}}$ be the coordinates of the endpoint of $\Gamma$ on the line $x+y=2r$. Let $y_j = (s+t+j)r^{2/3}$, ${\mathbb {L}}_j$ be the line segment joining the points

$$\begin{aligned} \left( r - y_j, r + y_j\right) \quad \text {and} \quad \left( r - y_{j+1}, r + y_{j+1}\right) , \end{aligned}$$

and let $A_j$ be the event that $p_{{\mathrm {right}}}\in {\mathbb {L}}_j$ and $(p_{{\mathrm {left}}}, p_{{\mathrm {right}}})$ are such that the slope of the line connecting them is not extreme enough to apply the second part of Proposition 3.5, i.e., it holds that $|(p_{{\mathrm {left}}}-p_{{\mathrm {right}}})_y|/(p_{{\mathrm {left}}}+p_{{\mathrm {right}}})_x \le 1-\delta /2$, for $j=0,\ldots , r^{1/3}$, with $\delta$ as in that proposition’s second part. Finally, let A be the event that $(p_{{\mathrm {left}}},p_{{\mathrm {right}}})$ satisfy $|(p_{{\mathrm {left}}}-p_{{\mathrm {right}}})_y|/(p_{{\mathrm {left}}}+p_{{\mathrm {right}}})_x>1-\delta /2$. Then clearly the whole probability space equals

$$\begin{aligned} \bigcup _{j=0}^{r^{1/3}}A_j \cup A \, . \end{aligned}$$

Thus we have

$$\begin{aligned} {\mathbb {P}}\Big (X -\mu r> \theta r^{1/3}-0.5c_1s^2r^{1/3}\Big ) \,\le & {} \, \sum _{j=0}^{r^{1/3}}{\mathbb {P}}\Big (X> \mu r + \theta r^{1/3}-0.5c_1s^2r^{1/3}, A_j\Big )\nonumber \\&+ {\mathbb {P}}\left( X > \mu r + \theta r^{1/3}-0.5c_1s^2r^{1/3}, A\right) .\nonumber \\ \end{aligned}$$

(52)

We will bound the two terms using the next two lemmas.

Lemma B.1

In the notation of Proposition 3.6 and under Assumptions 2 and 3a, there exist $c>0$, $c_1>0$, and $r_0$ such that, for $r>r_0$ and $j=0,1,\ldots , r^{1/3}$,

$$\begin{aligned} {\mathbb {P}}\left( X > \mu r + \theta r^{1/3} - 0.5c_1s^2r^{1/3}, A_j\right) \le \exp \left( -c(\min (\theta ^{3/2}, \theta r^{1/3})+s^3+j^3)\right) . \end{aligned}$$

Lemma B.2

In the notation of Proposition 3.6 and under Assumptions 2 and 3a, there exist constants $c>0$, $c_1>0$, $\theta _0$, $s_0$, and $r_0$ such that, for $r>r_0$, $s>s_0$, and $\theta _0<\theta < r^{2/3}$,

$$\begin{aligned} {\mathbb {P}}\left( X > \mu r +\theta r^{1/3} +\theta r^{1/3} - 0.5c_1s^2r^{1/3}, A\right) \le \exp \left( -c(\min (\theta ^{3/2}, \theta r^{1/3})+s^3)\right) . \end{aligned}$$

Before proving these lemmas, we show how the proof of Proposition 3.6 is completed using them. Lemma B.1 says that each individual summand in the first term of (52) is bounded by $\exp \left( -c(\min (\theta ^{3/2}, \theta r^{1/3})+s^3+j^3)\right)$, while Lemma B.2 says that the second term is bounded by $\exp \left( -c(\min (\theta ^{3/2}, \theta r^{1/3})+s^3)\right)$. So we have, by summing over j,

$$\begin{aligned} {\mathbb {P}}\left( X > \mu r + \theta r^{1/3}-0.5c_1s^2r^{1/3}\right) \le C\exp \left( -c(\min (\theta ^{3/2}, \theta r^{1/3})+s^3)\right) \end{aligned}$$

for some $C<\infty$. Reducing the value of c completes the proof. $\square$

In seeking to prove Lemmas B.1 and B.2, we wish to show that when the endpoint of a particular geodesic is sufficiently extreme, it suffers a weight loss with high probability. Lemma B.1 addresses the case that the slope between the points is bounded away from 0 and $\infty$, while Lemma B.2 addresses when the slope between the points is extreme.

Proof of Lemma B.1

We fix j. We divide ${\mathbb {L}}_{{\mathrm {left}}}$ into segments of size at most $r^{2/3}$ each, indexed by i as ${\mathbb {L}}_{{\mathrm {left}}}^i$. Thus there are at most t segments.

In the notation of Proposition 3.5, we have that z is bounded uniformly away from r, and so we may bound X on $A_j$ by using Assumption 2 and Proposition 3.5. We set $c_1 = G_2\rho ^2/3$ with $G_2$ as in Assumption 2. Then,

$$\begin{aligned} {\mathbb {P}}\Big (X> \mu r +\theta r^{1/3}-0.5c_1s^2r^{1/3}, A_j\Big )&\le {\mathbb {P}}\bigg (\sup _{\begin{array}{c} y\in {\mathbb {L}}_{{\mathrm {left}}},\\ w\in {\mathbb {L}}_j \end{array}} X_{y \rightarrow w}> \mu r +\theta r^{1/3}-0.5c_1s^2r^{1/3}\bigg )\\&\le \sum _{i=1}^t{\mathbb {P}}\bigg (\sup _{\begin{array}{c} y\in {\mathbb {L}}_{{\mathrm {left}}}^i,\\ w\in {\mathbb {L}}_j \end{array}} X_{y \rightarrow w} > \mu r +\theta r^{1/3}-0.5c_1s^2r^{1/3}\bigg ). \end{aligned}$$

Now note that $0.5c_1s^2 < \frac{1}{6}G_2\left( (s+j)\wedge \rho r^{1/3}\right) ^2$ by our choice of $c_1$ and since $s \le r^{1/3}$. Thus each summand in the previous display is in turn bounded by

$$\begin{aligned}&{\mathbb {P}}\Bigg (\smash {\sup _{\begin{array}{c} y\in {\mathbb {L}}_{{\mathrm {left}}}^i,\\ w\in {\mathbb {L}}_j \end{array}}} \bigg (X_{y \rightarrow w}-\mu r + \frac{1}{3}G_2\left( (s+j)\wedge \rho r^{1/3}\right) ^2r^{1/3}\bigg )\\&\quad > \theta r^{1/3}+ \frac{1}{6}G_2\left( (s+j)\wedge \rho r^{1/3}\right) ^2r^{1/3}\Bigg )\\&\quad \le \exp \left( -c(\min (\theta ^{3/2}, \theta r^{1/3})+s^3+j^3)\right) . \end{aligned}$$

The last inequality is via the first part of Proposition 3.5, and holds for both (i) $s=0$ and $\theta$ sufficiently large, as well as for (ii) $\theta =0$ and $s>s_0$ (with the earlier mentioned assumption that $s\le r^{1/3}$, which also allows us to put $s^3+j^3$ in the exponent without a $\rho$ term). Using that $t\le s^2$ and reducing the value of c complete the proof. $\square$

Remark B.3

Note that we only use the upper bound on t at the very last step, and a weaker bound such as $t\le s^{\alpha }$ for some $\alpha \ge 2$ in Proposition 6.1 would also work (we require $\alpha \ge 2$ in applications of the proposition, for example to prove Theorem 3.3).

Proof of Lemma B.2

Here the endpoints are not bounded uniformly away from the coordinate axes, and so we will make use of the second part of Proposition 3.5, which yields

$$\begin{aligned} {\mathbb {P}}\left( X > \mu r +\theta r^{1/3} - 0.5c_1s^2r^{1/3}, A\right)&\le \exp \left( -c (r+ \min (\theta ^{3/2}, \theta r^{1/3}))\right) \\&\le \exp \left( -0.5c(r + \min (\theta ^{3/2}, \theta r^{1/3})+s^3)\right) . \end{aligned}$$

The last inequality is from the fact that $s\le r^{1/3}$ (as otherwise the statement is trivial) and $\theta < r^{2/3}$, and again holds for both choices of parameters (i) and (ii) in Proposition 3.6. $\square$

1.3 Lower tail and mean of constrained point-to-point

Proof of Proposition 3.7

We first derive the lower tail bound (9). Fix $J = \theta ^{1/2}/\ell$ and define $u_j = \left( J^{-1}\cdot j(r-z-hr^{2/3}), J^{-1}\cdot j(r+z+hr^{2/3})\right)$ for $j=0,\ldots , J$. By the stationarity of the random field and the union bound, we have

$$\begin{aligned} {\mathbb {P}}\left( X_{u \rightarrow u^{\prime }}^{U} \le \mu r-\theta r^{1 / 3}\right)&\le J\cdot {\mathbb {P}}\left( X_{u \rightarrow u_1}^{U} \le \frac{1}{J}\mu r -\frac{\theta }{J} r^{1 / 3}\right) . \end{aligned}$$

We also have

$$\begin{aligned} {\mathbb {P}}\left( X_{u \rightarrow u_1}^{U} \le \frac{1}{J}\mu r -\frac{\theta }{J} r^{1 / 3}\right)&\le {\mathbb {P}}\left( X_{u \rightarrow u_1} \le \mu \frac{r}{J}-\frac{\theta }{J^{2/3}}\cdot \left( \frac{r}{J}\right) ^{1 / 3}\right) \\&\quad + {\mathbb {P}}\left( X_{u \rightarrow u_1}> \mu \frac{r}{J}-\frac{\theta }{J^{2/3}}\cdot \left( \frac{r}{J}\right) ^{1 / 3}, {{\,\mathrm{TF}\,}}(\Gamma _{u,u_1})\right. \\&\left. > \ell J^{2/3}\left( \frac{r}{J}\right) ^{2/3}\right) \\&\le \exp \left( -c\theta ^{3/2}/J\right) + \exp \left( -c\ell ^3 J^{2}\right) \\&\le 2\exp \left( -c\ell \theta \right) , \end{aligned}$$

for sufficiently large $\theta$ depending on K. For the second-to-last inequality we have used Assumption 3a for the first term and Theorem 3.3 for the second. The parameters (marked here with tildes to avoid confusion) for the application of Theorem 3.3 are as follows: ${{\tilde{r}}} = r/J$, ${{\tilde{t}}} = KJ^{2/3}/J = K/J^{1/3}$, and ${{\tilde{s}}} = \theta ^{1/2}/J^{1/3}$. It is easy to check that the conditions of Theorem 3.3 are met with these parameter choices for sufficiently large $\theta$. For Theorem 3.3 to apply, we also need $\ell J^{2/3} \ge {{\tilde{s}}} = \theta ^{1/2}/ J^{1/3},$ which is satisfied with equality by our choice of J. This completes the proof of the lower tail estimate.

For the lower bound (10) on ${\mathbb {E}}[X_{u \rightarrow u'}^U]$, we have

$$\begin{aligned} {\mathbb {E}}[X_{u \rightarrow u'}^{U}]&= {\mathbb {E}}[X_{u \rightarrow u'}] - {\mathbb {E}}[X_{u \rightarrow u'} - X_{u \rightarrow u'}^U]\\&\ge \mu r - G_1z^2r^{1/3} - g_1r^{1/3} - {\mathbb {E}}[X_{u \rightarrow u'} - X_{u \rightarrow u'}^U], \end{aligned}$$

using the lower bound for the first term from Assumption 2. Note that the second term is the expectation of a positive random variable. We claim that the expectation is bounded above by $Cr^{1/3}$ for some $C<\infty$. This follows from the tail probability formula for expectation:

$$\begin{aligned} r^{-1/3}{\mathbb {E}}[X_{u \rightarrow u'} - X_{u \rightarrow u'}^U] = \int _0^{\infty }{\mathbb {P}}\left( X_{u \rightarrow u'} - X_{u \rightarrow u'}^U > tr^{1/3}\right) \, dt. \end{aligned}$$

Bounding the integrand by ${\mathbb {P}}(X_{u \rightarrow u'} > \mu r + 0.5 tr^{1/3}) + {\mathbb {P}}(X_{u \rightarrow u'}^U < \mu r - 0.5 tr^{1/3})$ and using the bounds from Assumption 3a and the tail bound from above to show that this expression is integrable, with the bound on the integral being independent of r, complete the proof. $\square$

Appendix C. Proof of a Crude Upper Bound Under a Convexity Assumption

In this appendix, we complete the proof of Lemma 3.4 under the upper tail convexity hypothesis, Assumption 3b.

Proof of Lemma 3.4 under Assumption 3b

As in the proof of Lemma 3.4, for any fixed k-geodesic watermelon ${\mathcal {W}}$, let $\smash {X_{n,{\mathcal {W}}}^{k,j}}$ be the weight of its $j^{\mathrm{th}}$ heaviest curve. For $J\subseteq \llbracket 1,k \rrbracket$ and ${{\overline{\delta }}} = (\delta _1,\ldots ,\delta _k)$ such that $\delta _j \ge 0$ for each $j\in J$ and $\delta _j = 0$ for $j\in \llbracket 1,k \rrbracket \setminus J$, define

$$\begin{aligned} A_{{{\overline{\delta }}}, J} = \left\{ \exists {\mathcal {W}}: X_{n, {\mathcal {W}}}^{k,j} > \mu n + \delta _j n^{1/3} \text { for all }j\in J\right\} . \end{aligned}$$

We claim that if $\sum _{j\in J}\delta _j = \lceil tk^{5/3}\rceil$, then, for an absolute constant $c>0$ and for $k\le t^{-3/2}n$,

$$\begin{aligned} {\mathbb {P}}(A_{{{\overline{\delta }}}, J}) \le \exp \left( -ct^{3/2}k^2\right) . \end{aligned}$$

(53)

This follows from the BK inequality and Assumption 3b, which gives that

$$\begin{aligned} {\mathbb {P}}\left( A_{{{\overline{\delta }}}, J}\right)&\le {\mathbb {P}}\Big (\exists \text { disjoint paths } \{\gamma _j: j\in J\} \text { with } \ell (\gamma _j) \ge \mu n + \delta _j n^{1/3} \ \ \text {for all } j\in J\Big )\\&\le \exp \bigg (-\sum _{j\in J} I_n(\delta _j)\bigg ). \end{aligned}$$

Since $I_n$ is convex and $\sum _{j\in J}\delta _j = \lceil tk^{5/3}\rceil$, we get

$$\begin{aligned} \sum _{j\in J} I_n(\delta _j) \ge k I_n\bigg (\frac{1}{k}\sum _{j\in J} \delta _j\bigg ) = k I_n\left( k^{-1}\lceil tk^{5/3}\rceil \right)&\ge c'\min (t^{3/2} k^2, tk^{5/3}n^{1/3})\\&\ge c't^{3/2}k^2 \end{aligned}$$

as $I_n(x) \ge c'\min (xn^{1/3},x^{3/2})$ for all $x\ge 0$ from Assumption 3b, and $k\in \llbracket 1,\lfloor \min (1,t^{-3/2})n\rfloor \rrbracket$. Let D be the set of ${{\overline{\delta }}}$ defined by

$$\begin{aligned} D = \left\{ {{\overline{\delta }}}: \sum _{j=1}^k\delta _j = \lceil tk^{5/3}\rceil , \delta _j \in {\mathbb {N}}\cup \{0\} \ \ \forall j\in \llbracket 1,k \rrbracket \right\} . \end{aligned}$$

Now we observe ${\mathbb {P}}\left( X_{n}^k> \mu nk + tk^{5/3}n^{1/3}\right) = {\mathbb {P}}\big (\exists {\mathcal {W}}: \sum _{j=1}^k (X_{n,{\mathcal {W}}}^{k,j} -\mu n)n^{-1/3} > tk^{5/3}\big ).$ By considering the ceiling of each summand in the event of this last probability, we obtain that the last expression is bounded by

$$\begin{aligned}&{\mathbb {P}}\left( \exists {\mathcal {W}}: \sum _{j=1}^k \lceil (X_{n, {\mathcal {W}}}^{k,j} -\mu n)n^{-1/3}\rceil> tk^{5/3}\right) \\&\quad \le {\mathbb {P}}\left( \bigcup _{{{\overline{\delta }}}\in D}\left\{ \exists {\mathcal {W}}\text { and } J\subseteq \llbracket 1,k \rrbracket : \lceil (X_{n, {\mathcal {W}}}^{k,j} -\mu n)n^{-1/3}\rceil > \delta _j \ \ \forall j\in J\right\} \right) \\&\quad \le 2^k\cdot |D|\cdot \exp \left( -ct^{3/2}k^2\right) , \end{aligned}$$

where the first inequality is seen by considering J to be the set of indices j such that $\lceil (X_{n, {\mathcal {W}}}^{k,j} -\mu n)n^{-1/3}\rceil > 0$ and taking $\delta _j$ to be an integer at most the latter quantity such that $\sum _{j\in J} \delta _j = \lceil tk^{5/3}\rceil$ for $j\in J$, and $\delta _j = 0$ for $j\in \llbracket 1,k \rrbracket \setminus J$. The second inequality follows by the union bound; the bound (53) on ${\mathbb {P}}\left( A_{{\bar{\delta }},J}\right)$; and the cardinality of the number of subsets of $\llbracket 1,k \rrbracket$. The cardinality of D is the number of non-negative solutions to $x_1+\ldots +x_k = tk^{5/3}$, which is $\left( {\begin{array}{c}tk^{5/3}+k-1\\ k-1\end{array}}\right) \le \left( {\begin{array}{c}2(tk^{5/3}\vee k)\\ k\end{array}}\right)$. Using the trivial bound $\left( {\begin{array}{c}n\\ k\end{array}}\right) \le n^k$ and simplifying, we obtain that $|D| \le \exp (\frac{1}{2} ct^{3/2}k^2)$ for k larger than some absolute $k_0$ depending on t, so the proof is completed by reducing the value of c. $\square$