Optimal tail exponents in general last passage percolation via bootstrapping & geodesic geometry

Abstract

We consider last passage percolation on $\mathbb {Z}^2$ with general weight distributions, which is expected to be a member of the Kardar-Parisi-Zhang (KPZ) universality class. In this model, an oriented path between given endpoints which maximizes the sum of the i.i.d. weight variables associated to its vertices is called a geodesic. Under natural conditions of curvature of the limiting geodesic weight profile and stretched exponential decay of both tails of the point-to-point weight, we use geometric arguments to upgrade the tail assumptions to prove optimal upper and lower tail behavior with the exponents of 3/2 and 3 for the weight of the geodesic from (1, 1) to (r, r) for all large finite r, and thus unearth a connection between the tail exponents and the characteristic KPZ weight fluctuation exponent of 1/3. The proofs merge several ideas which are not reliant on the exact form of the vertex weight distribution, including the well known super-additivity property of last passage values, concentration of measure behavior for sums of stretched exponential random variables, and geometric insights coming from the study of geodesics and more general objects called geodesic watermelons. Previous proofs of such optimal estimates have relied on hard analysis of precise formulas available only in integrable models. Our results illustrate a facet of universality in a class of KPZ stochastic growth models and provide a geometric explanation of the upper and lower tail exponents of the GUE Tracy-Widom distribution, the conjectured one point scaling limit of such models. The key arguments are based on an observation of general interest that super-additivity allows a natural iterative bootstrapping procedure to obtain improved tail estimates.

Appendix A. Transversal fluctuation, interval-to-interval, and crude lower tail bounds

In this appendix we explain how to obtain the first, second, and fourth tools of Sect. 1.8, i.e., Theorem 1.9 and Propositions 1.10 and 1.12, and provide the outstanding proofs of Lemmas 3.5, 4.2, and 4.5 from the main text. The third tool was already explained in Sect. 4.

The proofs of the first and fourth tools follow verbatim from corresponding results in [7] by replacing the upper and lower tails used there with Assumption 3; the parabolic curvature assumption there is provided by our Assumption 2. In particular, Theorem 1.9 follows from [7, Theorem 3.3] and Proposition 1.12 from [7, Proposition 3.7].

The proof of the second tool, Proposition 1.10, will be addressed in Section A.1, after we next provide the outstanding proofs of Lemmas 3.5, 4.2, and 4.5 from the main text.

We start with the proofs of Lemmas 3.5 and 4.5 on an upper tail bound of the interval-to-interval weight; this largely follows the proof of [7, Proposition 3.5]. The strategy is to back up from the intervals appropriately and consider a point-to-point weight for which we have tail bounds by hypothesis; this strategy was illustrated in Fig. 5 and in the proof of Lemma 4.4.

Proofs of Lemmas 3.5 and 4.5

We prove Lemma 3.5 and indicate at the end the modifications for Lemma 4.5. We set $\lambda = \lambda _j$ and $\lambda ' = \lambda _{j+1}$ to avoid confusion later when we describe the modifications for Lemma 4.5. Note that $\lambda ' < \lambda$.

Recall that Z is defined as the largest last passage value between a pair of points when the points are each allowed to vary over an interval. We call A the interval over which the starting point varies, B the interval over which the ending point varies, and U the region between A and B (i.e., the set of all points covered by some up-right path from A to B). A and B were called $\mathbb L_{\textrm{low}}$ and $\mathbb L_{\textrm{up}}$ in Lemma 3.5 and I and J in Lemma 4.5.

By considering the event that Z 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; see Fig. 5. To define these events, first define points $\phi _{\textrm{low}}$ and $\phi _{\textrm{up}}$ on either side of the lower and upper intervals as follows, where $\delta = \frac{1}{2}\left( \frac{\lambda }{\lambda '}-1\right) > 0$:

$$\begin{aligned} \phi _{\textrm{low}}&:= \left( -\delta r, -\delta r\right) \\ \phi _{\textrm{up}}&:= \left( (1+\delta )r-w, (1+\delta )r + w\right) . \end{aligned}$$

Let $u^*$ and $v^*$ be the points on A and B where the suprema in the definition of Z are attained, and let the events $E_{\textrm{low}}$ and $E_{\textrm{up}}$ be defined as

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

Let ${\tilde{r}} = \frac{\lambda }{\lambda '}r = (1+2\delta )r$, and observe that the diagonal distance of $\phi _{\textrm{low}}$ and $\phi _{\textrm{up}}$ is ${\tilde{r}}$. Also note

$$\begin{aligned}X_{\phi _{\textrm{low}},\phi _{\textrm{up}}} \ge X_{\phi _{\textrm{low}}, u*-(1,0)} + Z + X_{v^*+(1,0),\phi _{\textrm{up}}}.\end{aligned}$$

Then we have the following:

$$\begin{aligned}&\mathbb {P}\bigg (Z > \mu r -\lambda '\frac{Gw^2}{r}+t r^{1/3}, E_{\textrm{low}}, E_{\textrm{up}}\bigg )\nonumber \\&\quad \le \mathbb {P}\left( X_{\phi _{\textrm{low}}, \phi _{\textrm{up}}} \ge \mu (1+2\delta ) r - \lambda ' \frac{Gw^2}{r}+ \frac{t}{3} r^{1/3}\right) \nonumber \\&\quad = \mathbb {P}\left( X_{\phi _{\textrm{low}}, \phi _{\textrm{up}}} \ge \mu {\tilde{r}} - \lambda \frac{Gw^2}{{\tilde{r}}} + \frac{t}{3}\cdot \left( \frac{\lambda '}{\lambda }\right) ^{1/3}\cdot ({\tilde{r}})^{1/3}\right) \nonumber \\&\quad \le {\left\{ \begin{array}{ll} \exp (-{\tilde{c}}t^\beta ) &{} t_0< t < r^{\zeta }\\ \exp (-{\tilde{c}}t^\alpha ) &{} t \ge r^{\zeta }. \end{array}\right. } \end{aligned}$$

(38)

The final inequality uses the hypothesis (15) on the point-to-point tail, which is applicable since the antidiagonal separation of $\phi _{\textrm{low}}$ and $\phi _{\textrm{up}}$ is w while the diagonal separation is $(1+2\delta )r$, and clearly $|w|\le r^{5/6}$ implies $|w| \le (1+2\delta )^{5/6}r^{5/6}$. We applied (15) with $\theta = t(\lambda '/\lambda )^{1/3}/3$, which is required to be greater than $\theta _0$. This translates to $t\ge t_0$ for a $t_0$ depending on $\theta _0$ and $\lambda '/\lambda$. Similarly, we absorbed the $\lambda '/\lambda$ dependency in the tail into the value of ${\tilde{c}}$, which thus depends on the original tail coefficient c in (15) and $\lambda '/\lambda$.

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 upper and lower sides A and B.

Then we see

$$\begin{aligned}&\mathbb {P}\bigg (Z> \mu r - \lambda '\frac{Gw^2}{r}+t r^{1/3}, E_{\textrm{low}}, E_{\textrm{up}}\mid U\bigg )\\&\quad = \mathbb {P}\left( Z > \mu r - \lambda '\frac{Gw^2}{r}+t r^{1/3}\mid U\right) \cdot \mathbb {P}\left( E_{\textrm{low}}\mid U\right) \cdot \mathbb {P}\left( E_{\textrm{up}}\mid U\right) . \end{aligned}$$

So with (38), all we need is a lower bound on $\mathbb {P}\left( E_{\textrm{low}}\mid U\right)$ and $\mathbb {P}\left( E_{\textrm{up}}\mid U\right)$. This is straightforward using independence of the environment below and above U from U:

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

(39)

for large enough t (independent of $\delta$) and r (depending on $\delta$), using Assumption 3b.

A similar upper bound holds for $\mathbb {P}\left( E^c_{\textrm{upper}}\mid U\right)$. Together this gives

$$\begin{aligned} \mathbb {P}\bigg (Z> \mu r - \lambda '\frac{Gw^2}{r}+t r^{1/3}, E_{\textrm{low}}, E_{\textrm{up}}\mid U\bigg )&\ge \frac{1}{4}\cdot \mathbb {P}\bigg (Z >\mu r - \lambda '\frac{Gw^2}{r}+ t r^{1/3} \mid U \bigg ), \end{aligned}$$

and taking expectation on both sides, combined with (38), gives Lemma 3.5. The fact that $\lambda '/\lambda$ depends only on j and the previously mentioned dependencies gives the claimed dependencies of ${\tilde{t}}_0, {\tilde{r}}_0,$ and ${\tilde{c}}$.

To prove Lemma 4.5, we take $\delta = 1$, which is equivalent to $\lambda ' = \lambda /3$. Then in (38) the final bound is done with the hypothesized bound on $X_{\phi _{\textrm{low}}, \phi _{\textrm{up}}}$, i.e.,

$$\begin{aligned}\mathbb {P}\left( X_{\phi _{\textrm{low}}, \phi _{\textrm{up}}} > \mu {\tilde{r}} - \lambda \frac{Gw^2}{{\tilde{r}}} + tr^{1/3}\right) \le \exp (-{\tilde{c}}t^\alpha ).\end{aligned}$$

Applying this bound requires $|w|\le {\tilde{r}}/2$. Since $|w|\le r$ and ${\tilde{r}} = \lambda r/\lambda ' = 3r$, this is valid. $\square$

Next we prove Lemma 4.2, on a constant probability lower bound on the lower tail, based on Assumptions 2 and 3b.

Proof of Lemma 4.2

Let ${\tilde{X}}_r^z = X_r^z - \mu r + Gz^2/r$. We know from Assumption 2 that $\mathbb {E}[{\tilde{X}}_r^z] \le -g_2 r^{1/3}$. Let E be the event

$$\begin{aligned}E = E(\theta ) = \left\{ X_r^z < \mu r - \frac{Gz^2}{r} -\theta r^{1/3}\right\} ,\end{aligned}$$

so that $\mathbb {P}(E) \le \exp (-c\theta ^\alpha )$ for $\theta > \theta _0$, by Assumption 3b.

Observe that $-{\tilde{X}}_r^z\mathbb {1}_E$ is a positive random variable and so, by Assumption 3b,

$$\begin{aligned}&\mathbb {E}[-{\tilde{X}}_r^z\mathbb {1}_E] = r^{1/3}\int _0^\infty \mathbb {P}\left( {\tilde{X}}_r^z\mathbb {1}_E< -tr^{1/3}\right) \, \mathrm dt\\&\quad = r^{1/3}\left[ \theta \cdot \mathbb {P}\left( X_r^z< \mu r - \frac{Gz^2}{r}-\theta r^{1/3}\right) + \int _\theta ^\infty \mathbb {P}\left( X_r^z <\mu r- \frac{Gz^2}{r} -tr^{1/3}\right) \, \mathrm dt\right] \\&\quad \le r^{1/3}\left[ \theta \exp (-c\theta ^\alpha )+ \int _\theta ^\infty \exp (-ct^\alpha )\, \mathrm dt\right] ; \end{aligned}$$

this may be made smaller than $0.5g_2r^{1/3}$ by taking $\theta$ large enough. We now set $\theta$ to such a value.

We also have $\mathbb {E}[{\tilde{X}}_r^z] = \mathbb {E}[{\tilde{X}}_r^z(\mathbb {1}_E + \mathbb {1}_{E^c})].$ Combining this, the above lower bound on $\mathbb {E}[{\tilde{X}}_r^z\mathbb {1}_E]$, and the upper bound on $\mathbb {E}[{\tilde{X}}_r^z]$, gives that

$$\begin{aligned} \mathbb {E}[{\tilde{X}}_r^z\mathbb {1}_{E^c}] \le -\frac{1}{2}g_2r^{1/3}. \end{aligned}$$

(40)

The fact that the distribution of ${\tilde{X}}_r^z\mathbb {1}_{E^c}$ is supported on $[-\theta r^{1/3}, \infty )$ implies that

$$\begin{aligned} \mathbb {P}\left( X_r^z\mathbb {1}_{E^c} < \mu r - \frac{Gz^2}{r} - \frac{1}{4}g_2r^{1/3}\right) \ge \frac{g_2}{4\theta }; \end{aligned}$$

(41)

this follows from (40) and since

$$\begin{aligned} \mathbb {E}[{\tilde{X}}_r^z\mathbb {1}_{E^c}]&\ge -\theta r^{1/3}\cdot \mathbb {P}\left( {\tilde{X}}_r^z\mathbb {1}_{E^c}< -\frac{1}{4}g_2 r^{1/3}\right) -\frac{1}{4}g_2r^{1/3}\mathbb {P}\left( {\tilde{X}}_r^z\mathbb {1}_{E^c} \ge -\frac{1}{4}g_2 r^{1/3}\right) \\&\ge -\theta r^{1/3}\cdot \mathbb {P}\left( {\tilde{X}}_r^z\mathbb {1}_{E^c} < -\frac{1}{4}g_2 r^{1/3}\right) -\frac{1}{4}g_2r^{1/3}. \end{aligned}$$

Since ${\tilde{X}}_r^z \mathbb {1}_E < 0$, it follows that ${\tilde{X}}_r^z \le {\tilde{X}}_r^z\mathbb {1}_{E^c}$, so (41) gives a lower bound on the lower tail of $X_r^z$, as desired, with $C = \frac{1}{4}g_2$ and $\delta = g_2/4\theta$. $\square$

1.1 A.1 Proof of transversal fluctuation bound, Proposition 1.10

In this section we prove Proposition 1.10 on the tail (with exponent $2\alpha$) of the transversal fluctuation of the geodesic path on scale $r^{2/3}$; we closely follow the proof of Theorem 11.1 of the preprint [16], but adapted to our setting and assumptions. We give the argument for the left-most geodesic $\Gamma _r^z$ from (1, 1) to $(r-z,r+z)$; the argument is symmetric for the right-most geodesic. (Note that these are well-defined by the planarity and the weight-maximizing properties of all geodesics).

We start with a similar bound at the midpoint of the geodesic, which needs some notation. For $x\in \llbracket 1,r \rrbracket$, let $\Gamma _r^z(x)$ be the unique point y such that $(x-y,x+y) \in \Gamma _r^z$.

Proposition A.1

Under Assumption 2 and 3, there exist positive $c = c(\alpha )$, $r_0$, and $s_0$ such that, for $r>r_0$, $s>s_0$, and $|z|\le {r^{5/6}}$,

$$\begin{aligned}\mathbb {P}\left( |\Gamma _r^z(r/2)| > z/2+sr^{2/3}\right) \le 2\exp (-cs^{2\alpha }).\end{aligned}$$

To prove this we will need a bound on the maximum, ${\widetilde{Z}}$, of fluctuations of the point-to-point weight as the endpoint varies over an interval, i.e.,

$$\begin{aligned}{\widetilde{Z}} = \sup _{v\in \mathbb L_{\textrm{up}}}\Big (X_{v} - \mathbb {E}[X_{v}]\Big ),\end{aligned}$$

where $\mathbb L_{\textrm{up}}$ is the interval of width $2r^{2/3}$ around $(r-w,r+w)$. Note that this is not the same as the point-to-interval weight.

Lemma A.2

Let $K>0$ and $|w|\le K r^{5/6}$. Under Assumptions 2 and 3, there exist positive c, $\theta _0 = \theta _0(K)$, and $r_0$, such that, for $\theta >\theta _0$ and $r>r_0$,

$$\begin{aligned}\mathbb {P}\left( {\widetilde{Z}} > \theta r^{1/3}\right) \le \exp (-c\theta ^{\alpha }).\end{aligned}$$

Proof

The proof is very similar to that of Lemma 3.5 above.

We take $\phi _{\textrm{up}}= (2(r-w), 2(r+w))$ to be the backed up point. Let $v^*\in \mathbb L_{\textrm{up}}$ be the maximizing point in the definition of ${\widetilde{Z}}$. For clarity, define the lower and upper mean weight functions $M_\textrm{low}$ and $M_\textrm{up}$ by $M_\textrm{low}(v) = \mathbb {E}[X_{v}]$ and $M_\textrm{up}(v) = \mathbb {E}[X_{v,\phi _{\textrm{up}}}]$; this is to use the unambiguous notation $M_\textrm{low}(v^*)$ (which is a function of $v^*$) instead of $\mathbb {E}[X_{v^*}]$. We also define

$$\begin{aligned} E_{\textrm{up}}&= \left\{ X_{v^*+(1,0), \phi _{\textrm{up}}} - M_\textrm{up}(v^*+(1,0)) > - \frac{\theta }{2}r^{1/3}\right\} . \end{aligned}$$

Now observe that

$$\begin{aligned} \begin{aligned}&X_{v^*} - M_\textrm{low}(v^*) + X_{v^*+(1,0), \phi _{\textrm{up}}} - M_\textrm{up}(v^*+(1,0))\\&\quad \le X_{\phi _{\textrm{up}}} - \inf _{v\in \mathbb L_{\textrm{up}}}\left( M_\textrm{low}(v) + M_\textrm{up}(v+(1,0))\right) . \end{aligned} \end{aligned}$$

(42)

We want to replace the infimum on the right hand side by $\mathbb {E}[X_{\phi _{\textrm{up}}}]$. The latter is at most $2\mu r - 2Gw^2/r$. We need to show that the infimum term is at least something which is within $O(r^{1/3})$ of this expression. For this we do the following calculation using Assumption 2. Parametrize $v\in \mathbb L_{\textrm{up}}$ as $(r-w-tr^{2/3}, r+w+tr^{2/3})$ for $t\in [-1,1]$. Then, for all $t\in [-1,1]$,

$$\begin{aligned} M_\textrm{low}(v) + M_\textrm{up}(v+(1,0))&\ge \left[ \mu r - \frac{G(w+tr^{2/3})^2}{r} - H\frac{(w+tr^{2/3})^4}{r^3}\right] \\&\qquad + \left[ \mu r - \frac{G(w-tr^{2/3})^2}{r} - H\frac{(w-tr^{2/3})^4}{r^3}\right] \\&\ge 2\mu r - \frac{2G w^2}{r}-2Gt^2r^{1/3} - 32HK^4r^{1/3}, \end{aligned}$$

the last inequality since $|w\pm tr^{2/3}|\le 2Kr^{5/6}$. Since $t\in [-1,1]$, $2Gt^2r^{1/3}\le 2Gr^{1/3}$, and so the right hand side of (42) is at most $X_{\phi _{\textrm{up}}} - \mathbb {E}[X_{\phi _{\textrm{up}}}] + \frac{\theta }{4}r^{1/3}$ for all large enough $\theta$ (depending on K). Thus, recalling the definition of $E_{\textrm{up}}$,

$$\begin{aligned} \mathbb {P}\left( {\widetilde{Z}}> \theta r^{1/3}, E_{\textrm{up}}\right) \le \mathbb {P}\left( X_{\phi _{\textrm{up}}} - \mathbb {E}[X_{\phi _{\textrm{up}}}] > \frac{\theta }{4}r^{1/3})\right) \le \exp (-c\theta ^\alpha ). \end{aligned}$$

We now claim that, conditionally on $v^*$, $E_{\textrm{up}}$ almost surely has probability at least 1/2; since $E_{\textrm{up}}$ is conditionally independent, given $v^*$, of ${\widetilde{Z}}$, this will imply with the previous display that ${\mathbb {P}({\widetilde{Z}} > \theta r^{1/3})\le 2\exp (-c\theta ^\alpha )}$. The proof of the claim is straightforward using the independence of $v^*$ with the environment above $\mathbb L_{\textrm{up}}$ and Assumption 3b, for

$$\begin{aligned} \mathbb {P}\left( E_{\textrm{up}}^c \mid v^*\right) \le \sup _{v\in \mathbb L_{\textrm{up}}}\mathbb {P}\left( X_{v+(1,0)} - M_\textrm{up}(v+(1,0)) \le -\frac{\theta }{2}r^{1/3}\right) \le 1/2, \end{aligned}$$

for all $\theta$ larger than an absolute constant. $\square$

Proof of Proposition A.1

We will prove the bound for the event that $\Gamma _r^z(r/2) > z/2 + sr^{2/3}$, as the event that it is less than $-z/2-sr^{2/3}$ is symmetric.

For $j\in \llbracket 0,r^{1/3} \rrbracket$, let $I_j$ be the interval

$$\begin{aligned} \left( \frac{r}{2}-\frac{z}{2}-sr^{2/3}, \frac{r}{2}+\frac{z}{2}+sr^{2/3}\right) - [j,j+1]\cdot (r^{2/3}, -r^{2/3}). \end{aligned}$$

Let $A_j$ be the event that $\Gamma _r^z$ passes through $I_j$, for $j\in \llbracket 0,r^{1/3} \rrbracket$. Observe that

$$\begin{aligned} \left\{ \Gamma _r^z(r/2) > z/2 + sr^{2/3}\right\} \subseteq \bigcup _{j=0}^{r^{1/3}} A_j. \end{aligned}$$

(43)

We claim that $\mathbb {P}(A_j) \le \exp (-c(s+j)^{2\alpha })$ for each such j; this will imply Proposition A.1 by a union bound which we perform at the end.

Let $Z_j^{(1)} = X_{(1,1), I_j}$ and $Z^{(2)}_j = X_{I_j, (r-z,r+z)}$. Also, let ${\widetilde{Z}}_j^{(1)} = \sup _{v\in I_j} (X_{v} - \mathbb {E}[X_v])$, and define ${\widetilde{Z}}_j^{(2)}$ analogously.

We have to bound the probability of $A_j$. The basic idea is to show that any path from (1, 1) to $(r-z,r+z)$ which passes through $I_j$ suffers a weight loss greater than that which $X_r^z$ typically suffers (which is of order $Gz^2/r$), and so such paths are not competitive. When j is very large, it is possible to show this even if we do not have the sharp coefficient of G for the parabolic loss; but for smaller values of j, we will need to be very tight with the coefficient of the parabolic loss. So we divide into two cases, depending on the size of j, and first address the case when j is large (in a sense to be specified more precisely shortly). Observe that, for a $c_2>0$ to be fixed,

$$\begin{aligned}&\mathbb {P}\left( A_j\right) \le \mathbb {P}\left( X_r^z < \mathbb {E}[X_r^z] - c_2 (s+j)^2r^{1/3}\right) \\ {}&+ \mathbb {P}\left( Z_j^{(1)} + Z_j^{(2)} > \mathbb {E}[X_r^z] - c_2(s+j)^2r^{1/3}\right) ; \end{aligned}$$

the first term is bounded by $\exp (-c(s+j)^{2\alpha })$ by Assumption 3b for a c depending on $c_2$, and we must show a similar bound for the second. Note that the second term is bounded by

$$\begin{aligned} \mathbb {P}\left( Z_j^{(1)} + Z_j^{(2)} > \mu r - \frac{Gz^2}{r} - Hr^{1/3} - c_2(s+j)^2r^{1/3}\right) , \end{aligned}$$

(44)

using Assumption 2 and since $|z|\le r^{5/6}$.

Recall from (36) and Lemma 4.5 that there exists a $\lambda \in (0,1)$ such that, for $|z/2+(s+j)r^{2/3}|\le r$, and $i=1$ and 2,

$$\begin{aligned} \mathbb {P}\left( Z_j^{(i)} > \nu _{i,j} + \theta r^{1/3}\right) \le \exp (-c\theta ^\alpha ), \end{aligned}$$

(45)

where $\nu _{i,j} = \tfrac{1}{2}\mu r -\lambda \cdot \frac{G}{r/2}\cdot (\tfrac{1}{2}z\pm (s+j)r^{2/3})^2$ with the $+$ for $i=1$ and − for $i=2$; $\nu _{i,j}$ captures the typical weight of these paths. Note that we are very crude with the parabolic coefficient, but the bound (45) holds for all j; and also that we measure the deviation from the same expression $\nu _{i,j}$ (which is obtained by evaluating (36) at one endpoint) for all points in the interval. As we will see, comparing the full interval to a single point will not work for the second case of small j.

We want to show that the typical weight $\nu _{1,j} + \nu _{2,j}$ is much lower than $\mu r - Gz^2/r$. Simple algebraic manipulations show that, if $(s+j)r^{2/3} > (\lambda ^{-1}-1)^{1/2} r^{5/6}$ (which is the largeness condition on j defining the first case),

$$\begin{aligned} \sum _{i=1}^2\nu _{i,j}< & {} \mu r- \lambda \frac{Gz^2}{r}- (1-\lambda )Gr^{2/3} - 3\lambda G(s+j)^2r^{1/3} \\ {}< & {} \mu r- \frac{Gz^2}{r} - 3\lambda G(s+j)^2r^{1/3}, \end{aligned}$$

the final inequality since $|z|\le r^{5/6}$. We have to bound (44) with some value of $c_2$, and we take it to be $2\lambda G$; note that any bound we prove on (44) will still be true if we later further lower $c_2$. The previous displayed bound shows that, for $(s+j)r^{2/3} > (\lambda ^{-1}-1)^{1/2}r^{5/6}$,

$$\begin{aligned}&\mathbb {P}\left( Z_j^{(1)} + Z_j^{(2)}> \mu r - \frac{Gz^2}{r} - Hr^{1/3} - c_2(s+j)^2r^{1/3}\right) \\&\quad \le \mathbb {P}\left( Z_j^{(1)} + Z_j^{(2)} > \nu _{1,j} + \nu _{2,j} +\tfrac{1}{2}\lambda G(s+j)^2r^{1/3}\right) . \end{aligned}$$

In the inequality we absorbed $-Hr^{1/3}$ into the last term by imposing that s is large enough, depending on $\lambda$, G, and H. Now by a union bound and (45), the last display, and hence (44), is bounded by $2\exp (-c(s+j)^{2\alpha })$.

Now we address the other case that $(s+j)r^{2/3} \le (\lambda ^{-1}-1)^{1/2} r^{5/6}$. Thus $I_j$ is close to the interpolating line, and we need a bound on the interval-to-interval weight with a much sharper parabolic term than in the previous case. Here above approach of the first case faces an issue. Since the gradient of $Gz^2/r$ at z is 2Gz/r, the weight difference across an interval of length $r^{2/3}$ at antidiagonal displacement z is of order $z/r^{1/3}$, which is much larger than the bearable error of $O(r^{1/3})$ when z is, say, $r^{5/6}$; so the crude approach of using the same expression (which we need to be less than $\mu r - Gz^2/r$) for the typical weight of all points in the interval, as we did in the first case, is insufficient—to have a single expression for which a tail bound exists for all points in the interval, we must necessarily include the linear gain of moving across the interval in the expression, and this will force it above $\mu r - Gz^2/r$. So, for this case, we will use Lemma A.2, which avoids the problem by taking the supremum after centering by the point-specific expectation.

Let $X'_v = X_{v, (r-z,r+z)}$. Now we observe

$$\begin{aligned}&\mathbb {P}\left( A_j\right) \le \mathbb {P}\left( X_r^z < \mathbb {E}[X_r^z] - c_2 (s+j)^2r^{1/3}\right) \\&\quad + \mathbb {P}\left( \sup _{v\in I_j} (X_v + X'_{v}) > \mathbb {E}[X_r^z] - c_2(s+j)^2r^{1/3}\right) ; \end{aligned}$$

note that $X_v+X_v'$ counts the weight of v twice, but this is acceptable as this sum dominates the weight of the best path through v. The first term is at most $\exp (-c(s+j)^{2\alpha })$ for a $c>0$ depending on $c_2$. We bound the second term as follows. First we note that $\mathbb {E}[X_r^z] \ge \mu r - Gz^2/r - Hr^{1/3}$ and that $\sup _{v\in I_j}\left( \mathbb {E}[X_v + X'_v]\right) \le \mu r - Gz^2/r -G(s+j)^2r^{1/3}$ by a simple calculation with Assumption 2, and so

$$\begin{aligned}&\mathbb {P}\left( \sup _{v\in I_j} (X_v + X'_{v})> \mathbb {E}[X_r^z] - c_2(s+j)^2r^{1/3}\right) \\&\quad \le \mathbb {P}\left( \sup _{v\in I_j} (X_v - \mathbb {E}[X_v] + X'_{v} - \mathbb {E}[X'_v]) > -Hr^{1/3} + (G - c_2)(s+j)^2r^{1/3}\right) . \end{aligned}$$

We lower $c_2$ (if required) from its earlier value to be less than G/2. Now, we need to absorb the $-Hr^{1/3}$ term above into the $(s+j)^2 r^{1/3}$ term, which we can do for $s>s_0$ by setting $s_0$ large enough depending on G and H. So for such s, by a union bound we see that the previous display is at most

$$\begin{aligned}&\mathbb {P}\left( \sup _{v\in I_j} (X_v - \mathbb {E}[X_v])> \tfrac{1}{6}G(s+j)^2r^{1/3}\right) \\&+ \mathbb {P}\left( \sup _{v\in I_j} (X'_v - \mathbb {E}[X'_v]) > \tfrac{1}{6}G(s+j)^2r^{1/3}\right) . \end{aligned}$$

We bound this by applying Lemma A.2, with $K = (\lambda ^{-1}-1)^{1/2}$ and $\theta = \tfrac{1}{6}G(s+j)^2$. Recall that the bound of Lemma A.2 holds for $\theta >\theta _0(K)$. Thus we raise $s_0$ further if necessary so that $(s+j)^2 > \theta _0(K)$ for all $s>s_0$ and $j\ge 0$. Then we see that, for s and j such that $s>s_0$ and $(s+j)r^{2/3}\le (\lambda ^{-1}-1)r^{5/6}$, the last display is at most $2\exp (-c(s+j)^{2\alpha })$.

Returning to the inclusion (43) and the bound of $\exp (-c(s+j)^{2\alpha })$ of $\mathbb {P}(A_j)$ for the two cases, we see that

$$\begin{aligned}\mathbb {P}\left( \Gamma _r^z(r/2) > z/2 + sr^{2/3}\right) \le \sum _{j=1}^{r^{1/3}} \exp (-c(s+j)^{2\alpha })\le C\exp (-cs^{2\alpha })\end{aligned}$$

for some absolute constant $C<\infty$ and $c>0$ depending on $\alpha$. Here we used that, if $\alpha \in (0,1/2)$, then $(s+j)^{2\alpha } \ge 2^{2\alpha -1}(s^{2\alpha }+j^{2\alpha })$, while if $\alpha \ge 1/2$, then $(s+j)^{2\alpha } \ge s^{2\alpha }+j^{2\alpha }$; and finally $\exp (-cj^{2\alpha })$ is summable over j. This completes the proof of Proposition A.1. $\square$

To extend the transversal fluctuation bound from the midpoint (as in Proposition A.1) to anywhere along the geodesic (as in Proposition 1.10), we follow very closely a multiscale argument previously employed in [16, Theorem 11.1] and [7, Theorem 3.3] for similar purposes. For this reason, we will not write a detailed proof but only outline the idea.

Proof sketch of Proposition 1.10

First, the interpolating line is divided up into dyadic scales, indexed by j. The j^th scale consists of $2^j+1$ anti-diagonal intervals, placed at separation $2^{-j}r$, of length of order $s_jr^{2/3}:=\prod _{i=1}^j(1+2^{-i/3})sr^{2/3}$. By choosing the maximum j for which this is done large enough, it can be shown that, on the event that $\mathop {\textrm{TF}}\limits (\Gamma _r^z) > sr^{2/3}$, there must be a j such that there is a pair $(I_1, I_3)$ of consecutive intervals on the j^th scale, and the interval $I_2$ of the $(j+1)$^th scale in between such that the following holds: the geodesic passes through $I_1$ and $I_3$, but fluctuates enough that it avoids $I_2$, say by passing to its left.

Planarity and that the geodesic is a weight-maximising path then implies that the geodesic from the left endpoint of $I_1$ to that of $I_3$ is to the left of the geodesic $\Gamma _r^z$ (this observation is often called geodesic or polymer ordering), and so must have midpoint transversal fluctuation at least of order $(s_{j+1}-s_{j})r^{2/3} = 2^{-(j+1)/3}sr^{2/3}$. But since this transversal fluctuation happens across a scale of length $r' = 2^{-j}r$, in scaled coordinates it is of order $2^{j/3}s(r')^{2/3}$. Applying Proposition A.1 says that this probability is at most $\exp (-c2^{2\alpha j/3}s^{2\alpha })$. Now it remains to take a union bound over all the scales and the intervals within each scale. Since the number of intervals in the j^th scale is $2^j$, and since $2^j\exp (-c2^{2\alpha j/3}s^{2\alpha })\le 2^{-j}\exp (-cs^{2\alpha })$ for all $s\ge s_0$ (by setting $s_0$ large enough) and $j\ge 1$, we obtain the overall probability bound of $\exp (-cs^{2\alpha })$ of Proposition 1.10. $\square$