Temporal Correlation in Last Passage Percolation with Flat Initial Condition via Brownian Comparison

Abstract

We consider directed last passage percolation on ${\mathbb {Z}}^2$ with exponential passage times on the vertices. A topic of great interest is the coupling structure of the weights of geodesics as the endpoints are varied spatially and temporally. A particular specialization is when one considers geodesics to points varying in the time direction starting from a given initial data. This paper considers the flat initial condition which corresponds to line-to-point last passage times. Settling a conjecture in [28], we show that for the passage times from the line $x+y=0$ to the points (r, r) and (n, n), denoted $X_{r}$ and $X_{n}$ respectively, as $n\rightarrow \infty$ and $\frac{r}{n}$ is small but bounded away from zero, the covariance satisfies

$$\begin{aligned} \text{ Cov }(X_{r},X_{n})=\Theta \left( (\frac{r}{n})^{4/3+o(1)} n^{2/3}\right) , \end{aligned}$$

thereby establishing $\frac{4}{3}$ as the temporal covariance exponent. This differs from the corresponding exponent for the droplet initial condition recently rigorously established in [3, 27] and requires novel arguments. Key ingredients include the understanding of geodesic geometry and recent advances in quantitative comparison of geodesic weight profiles to Brownian motion using the Brownian Gibbs property. The proof methods are expected to be applicable for a wider class of initial data.

Appendices

Appendix A. Brownian Calculations

As promised before, we provide the remaining details of some of the straightforward computations using Brownian motions that were omitted from the main text.

Proof of (4)

Recall that we need to show

$$\begin{aligned} {\mathbb {P}}\left[ \max _{x \in I} W(x) > \max _{x \in [-2M, 2M]} W(x)-\sqrt{\varepsilon }\right] \le C_1 \varepsilon . \end{aligned}$$

Let the end points of I be $-M \le x_1 < x_2 \le M$, and let $m:=x_2-x_1=|I|$. We have

$$\begin{aligned}&{\mathbb {P}}\left[ \max _{x \in I} W(x) > \max _{x \in [-2M, 2M]} W(x)-\sqrt{\varepsilon }\right] \\&\le \iint _{[0,\infty )^2} {\mathbb {P}}\left[ \max _{x \in [x_1, x_2]} W(x) - W(x_1) = h_1, \max _{x \in [x_1, x_2]} W(x) - W(x_2) = h_2\right] \\&\times {\mathbb {P}}\left[ \max _{x \in [x_1-M, x_1]} W(x) - W(x_1)< h_1 + \sqrt{\varepsilon }\right] \\&\quad {\mathbb {P}}\left[ \max _{x \in [x_2, x_2+M]} W(x) - W(x_2) < h_2 + \sqrt{\varepsilon }\right] dh_1 dh_2. \end{aligned}$$

(Above, the first term in the second line denotes the probability density). Using reflection principle, this equals to

$$\begin{aligned}&\iint _{[0,\infty )^2} \frac{(h_1+h_2)\exp (-(h_1+h_2)^2/4m)}{\sqrt{4\pi }m^{3/2}}\\&\times 2 {\mathbb {P}}\left[ |W(x_1-M) - W(x_1)|< h_1 + \sqrt{\varepsilon }\right] \cdot 2 {\mathbb {P}}\left[ |W(x_2+M) - W(x_2)| < h_2 + \sqrt{\varepsilon }\right] dh_1 dh_2\\&\le \iint _{[0,\infty )^2} \frac{2(h_1+h_2)\exp (-(h_1+h_2)^2/4m)}{\sqrt{4\pi }m^{3/2}} \cdot \frac{(h_1 + \sqrt{\varepsilon })(h_2 + \sqrt{\varepsilon })}{4\pi M} dh_1 dh_2\\&= \iint _{[0,\infty )^2} \frac{2(h_1+h_2)\exp (-(h_1+h_2)^2/4m)}{\sqrt{4\pi }m^{3/2}} \cdot \frac{h_1h_2 + \sqrt{\varepsilon }(h_1+h_2) + \varepsilon }{4\pi M} dh_1 dh_2 \end{aligned}$$

We note that (by change of variables)

$$\begin{aligned} \begin{aligned} \iint _{[0,\infty )^2} \frac{h_1h_2(h_1+h_2)\exp (-(h_1+h_2)^2/4m)}{m^{5/2}} dh_1 dh_2, \\ \iint _{[0,\infty )^2} \frac{(h_1+h_2)^2\exp (-(h_1+h_2)^2/4m)}{m^{2}} dh_1 dh_2, \\ \iint _{[0,\infty )^2} \frac{(h_1+h_2)\exp (-(h_1+h_2)^2/4m)}{m^{3/2}} dh_1 dh_2, \end{aligned} \end{aligned}$$

are finite and independent of m. This implies that for some constant $C_2$ depending on M,

$$\begin{aligned} {\mathbb {P}}\left[ \max _{x \in I} W(x) > \max _{x \in [-2M, 2M]} W(x)-\sqrt{\varepsilon }\right] \le C_2(m + \sqrt{\varepsilon }\sqrt{m} + {\varepsilon }), \end{aligned}$$

and our conclusion follows since $m \le \varepsilon$. $\quad \square$

Completion of Proof of Lemma 3.7

Denote

$$\begin{aligned} {\mathscr {C}}_*'&:= \left\{ \sup _{|x|\le \lambda } W(x)= \sup _{|x|\le \sqrt{2{\mathscr {M}}}} W(x) \le \sup _{2^{k-1}\lambda '\le |x|\le 2^{k}\lambda '} W(x)+2^{k(\frac{1}{2} - \tau )} \alpha \sqrt{\lambda '}~\text {for some}~k\le k_* \right\} , \text { and },\\ {\mathscr {C}}_{\#}'&:= \left\{ \sup _{|x|\le \lambda } W(x)= \sup _{|x|\le \sqrt{2{\mathscr {M}}}} W(x) \ge W(0) + \alpha ^{-1}\sqrt{\lambda '} \right\} . \end{aligned}$$

Recall the event ${\mathscr {C}}':= {\mathscr {C}}_*'\cup {\mathscr {C}}_{\#}'$, and that it was left to show that $\lim _{\alpha \rightarrow 0} \lambda ^{-1}{\mathbb {P}}({\mathscr {C}}')= 0$ uniformly in $0<\lambda<\lambda '<1$. We first study ${\mathscr {C}}_*'$. Take any $k\in {\mathbb {N}}$, $k \le k_*$. Let $y_1:=\sup _{-\sqrt{2{\mathscr {M}}} \le x \le -\lambda }W(x)-W(-\lambda )$, $y_2:=\sup _{|x| < \lambda }W(x)-W(-\lambda )$, and $y_3:=\sup _{|x| < \lambda }W(x)-W(\lambda )$. Then we have

$$\begin{aligned}&{\mathbb {P}}\left[ \sup _{x: |x| \le \lambda } W(x) = \sup _{x: |x| \le \sqrt{2{\mathscr {M}}}} W(x)< \sup _{2^{k-1}\lambda '\le x\le 2^{k}\lambda ' } W(x) + 2^{k(\frac{1}{2} - \tau )} \alpha \sqrt{\lambda '}\right] \nonumber \\&\quad = \int {\mathbb {P}}[y_1< y_2, y_3=t]{\mathbb {P}}\left[ \sup _{0\le x \le \sqrt{2{\mathscr {M}}}-\lambda } W(x)<t, \right. \nonumber \\&\quad \left. \sup _{2^{k-1}\lambda '-\lambda \le x\le 2^{k}\lambda '-\lambda } W(x) > t - 2^{k(\frac{1}{2} - \tau )} \alpha \sqrt{\lambda '}\right] dt. \end{aligned}$$

(51)

For any $t_1, t_2, t>0$, by reflection principle we have

$$\begin{aligned} {\mathbb {P}}[y_1=t_1]&= \frac{2\exp (-t_1^2/(4(\sqrt{2{\mathscr {M}}}-\lambda )))}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}};\\ {\mathbb {P}}[y_2=t_2, y_3=t]&= \frac{(t_2+t)\exp (-(t_2+t)^2/8\lambda )}{\sqrt{8\pi }\lambda ^{3/2}}, \text {and thus}\\ {\mathbb {P}}[y_1< y_2, y_3=t]&= \int _{0<t_1<t_2} \frac{2\exp (-t_1^2/(4(\sqrt{2{\mathscr {M}}}-\lambda )))}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}}\\&\quad \cdot \frac{(t_2+t)\exp (-(t_2+t)^2/8\lambda )}{\sqrt{8\pi }\lambda ^{3/2}} dt_1 dt_2\\&= \int _{0<t_1} \frac{2\exp (-t_1^2/(4(\sqrt{2{\mathscr {M}}}-\lambda )))}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}}\\&\quad \cdot \frac{4\exp (-(t_1+t)^2/8\lambda )}{\sqrt{8\pi \lambda }} dt_1\\&\le \frac{8}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}} \int _{0<t_1}\\&\quad \cdot \frac{\exp (-(t_1^2+t^2)/8\lambda )}{\sqrt{8\pi \lambda }} dt_1 = \frac{4\exp (-t^2/8\lambda )}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}}. \end{aligned}$$

For the other factor of the integrand in the RHS of (51), consider the stopping time:

$$\begin{aligned} x_* := \inf \{x \in [2^{k-1}\lambda '-\lambda , 2^k \lambda ' - \lambda ]: W(x)>t - 2^{k(\frac{1}{2} - \tau )} \alpha \sqrt{\lambda '}\}\bigcup \{\sqrt{2{\mathscr {M}}}-\lambda \}. \end{aligned}$$

Then we have that

$$\begin{aligned}&{\mathbb {P}}\left[ \sup _{0\le x \le \sqrt{2{\mathscr {M}}}-\lambda } W(x)<t, \sup _{2^{k-1}\lambda '-\lambda \le x\le 2^{k}\lambda '-\lambda } W(x) > t - 2^{k(\frac{1}{2} - \tau )} \alpha \sqrt{\lambda '}\right] \\&\quad = {\mathbb {P}}\left[ \sup _{0\le x \le \sqrt{2{\mathscr {M}}}-\lambda } W(x)<t, x_* < 8\right] . \end{aligned}$$

Using the fact that $x\mapsto W(x+x_*)-W(x_*)$ is again a Brownian motion and has the same law as W, we bound this by

$$\begin{aligned}&{\mathbb {P}}\left[ \sup _{0 \le x \le 2^{k-1}\lambda '-\lambda )} W(x)<t \right] {\mathbb {P}}\left[ \sup _{0 \le x \le \sqrt{2{\mathscr {M}}}-\lambda -8} W(x)<2^{k(\frac{1}{2} - \tau )} \alpha \sqrt{\lambda '} \right] \\&\quad \le \frac{2t}{\sqrt{4\pi \cdot 2^{k-1}\lambda '}} \cdot \frac{2\cdot 2^{k(\frac{1}{2} - \tau )}\alpha \sqrt{\lambda '}}{\sqrt{4\pi \cdot (\sqrt{2{\mathscr {M}}}-\lambda -8)}} \le 4t\alpha 2^{-k\tau }, \end{aligned}$$

where in the last inequality we assume that ${\mathscr {M}}$ is large enough. In conclusion, we have

$$\begin{aligned}&{\mathbb {P}}[{\mathscr {C}}_*']\le 2\sum _{k=1}^{k_*}\int _{t>0} \frac{4\exp (-t^2/8\lambda )}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}} \cdot 4t\alpha 2^{-k\tau } dt \nonumber \\&\quad \le \int _{t>0} \frac{8\exp (-t^2/8\lambda )}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}} \cdot \frac{4t\alpha }{1-2^{-\tau }} dt \nonumber \\&\quad = \frac{128\lambda \alpha }{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}(1-2^{-\tau })}, \end{aligned}$$

(52)

thus $\lambda ^{-1}{\mathbb {P}}[{\mathscr {C}}_*'] \rightarrow 0$ as $\alpha \rightarrow 0$, uniformly for $\lambda < \lambda ' \in (0, 1)$. We next consider ${\mathscr {C}}_{\#}'$. We let $z_1:= \sup _{-\sqrt{2{\mathscr {M}}}\le x \le 0}W(x) - W(0)$, $z_2:=\sup _{0< x \le \lambda }W(x) - W(0)$, and $z_3:=\sup _{0< x \le \lambda }W(x) - W(\lambda )$, $z_4:=\sup _{\lambda < x \le \sqrt{2{\mathscr {M}}}}W(x) - W(\lambda )$. By symmetry, we have

$$\begin{aligned}&{\mathbb {P}}\left[ \sup _{|x|\le \lambda } W(x)= \sup _{|x|\le \sqrt{2{\mathscr {M}}}} W(x) \ge W(0) + \alpha ^{-1}\sqrt{\lambda '}\right] \nonumber \\&= 2 {\mathbb {P}}\left[ \sup _{0\le x\le \lambda } W(x)= \sup _{|x|\le \sqrt{2{\mathscr {M}}}} W(x) \ge W(0) + \alpha ^{-1}\sqrt{\lambda '}\right] \nonumber \\&\quad =2{\mathbb {P}}[z_1< z_2, z_2> \alpha ^{-1}\sqrt{\lambda '}, z_3> z_4]\nonumber \\&=2\iint _{t_2> \alpha ^{-1}\sqrt{\lambda '}, t_3 > 0} {\mathbb {P}}[z_1< t_2]{\mathbb {P}}[z_4 < t_3]{\mathbb {P}}[z_2 = t_2, z_3 = t_3] dt_2 dt_3 \end{aligned}$$

(53)

For any $t_2, t_3 > 0$, by reflection principle we have

$$\begin{aligned} {\mathbb {P}}[z_1< t_2]&= \int _{0< t_1< t_2} \frac{2\exp (-t_1^2/(4\sqrt{2{\mathscr {M}}}))}{\sqrt{4\pi \sqrt{2{\mathscr {M}}}}} dt_1 \le \frac{2t_2}{\sqrt{4\pi \sqrt{2{\mathscr {M}}}}},\\ {\mathbb {P}}[z_4< t_3]&= \int _{0< t_4 < t_3} \frac{2\exp (-t_1^2/(4(\sqrt{2{\mathscr {M}}}-\lambda )))}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}} dt_4 \le \frac{2t_3}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}},\\ {\mathbb {P}}[z_2=t_2, z_3=t_3]&= \frac{(t_2+t_3)\exp (-(t_2+t_3)^2/4\lambda )}{\sqrt{\pi }\lambda ^{3/2}}, \end{aligned}$$

so (53) can be bounded by

$$\begin{aligned}&2\iint _{t_2> \alpha ^{-1}\sqrt{\lambda '}, t_3> 0} {\mathbb {P}}[z_1< t_2]{\mathbb {P}}[z_4 < t_3]{\mathbb {P}}[z_2 = t_2, z_3 = t_3] dt_2 dt_3\\&\le 2\iint _{t_2> \alpha ^{-1}\sqrt{\lambda '}, t_3> 0} \frac{2t_2}{\sqrt{4\pi \sqrt{2{\mathscr {M}}}}} \cdot \frac{2t_3}{\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}} \cdot \frac{(t_2+t_3)\exp (-(t_2+t_3)^2/4\lambda )}{\sqrt{\pi }\lambda ^{3/2}} dt_2 dt_3\\&= \frac{8\lambda }{\sqrt{4\pi \sqrt{2{\mathscr {M}}}}\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}}\iint _{t_2> \alpha ^{-1}\sqrt{\lambda '/\lambda }, t_3> 0} \frac{t_2t_3(t_2+t_3)\exp (-(t_2+t_3)^2/4)}{\sqrt{\pi }} dt_2 dt_3\\&\le \frac{8\lambda }{\sqrt{4\pi \sqrt{2{\mathscr {M}}}}\sqrt{4\pi (\sqrt{2{\mathscr {M}}}-\lambda )}}\iint _{t_2> \alpha ^{-1}, t_3 > 0} \frac{t_2t_3(t_2+t_3)\exp (-(t_2+t_3)^2/4)}{\sqrt{\pi }} dt_2 dt_3. \end{aligned}$$

Note that the integral is independent of $\lambda , \lambda '$, and converges to 0 as $\alpha \rightarrow 0$, so $\lambda ^{-1}{\mathbb {P}}[{\mathscr {C}}_{\#}']\rightarrow 0$ as $\alpha \rightarrow 0$, uniformly for $\lambda < \lambda ' \in (0, 1)$. Thus our conclusion follows.

$\square$

Appendix B. Convergence to Airy$_2$ Process and Consequences

We proceed to providing the previously omitted proofs of Theorem 3.8, Proposition 3.9 and Proposition 3.10. We start with the latter two results assuming the first one.

Proof of Proposition 3.9

Take $N \in {\mathbb {N}}$ such that $N\iota > 2$. We consider a finite collection of intervals:

$$\begin{aligned} \Theta := \left\{ \left[ \frac{N_1}{N}, \frac{N_2}{N}\right] : -MN \le N_1 < N_2 \le MN, N_1, N_2 \in {\mathbb {Z}}\right\} , \end{aligned}$$

and a finite collection of values:

$$\begin{aligned} \Upsilon := \left\{ 0 < \varepsilon \le 1 : \varepsilon N \in {\mathbb {Z}}\right\} . \end{aligned}$$

By Theorem 3.8, for any ${\mathcal {I}} \in \Theta$ and $\varepsilon \in \Upsilon$,

$$\begin{aligned}&\lim _{n\rightarrow \infty }{\mathbb {P}}\left[ \max _{u\in n^{2/3}{\mathcal {I}}\cap {\mathbb {Z}}} T_{(u,-u),{\mathbf {n}}}> \max _{u\in \llbracket -2Mn^{2/3}, 2Mn^{2/3}\rrbracket } T_{(u,-u),{\mathbf {n}}}-\sqrt{\varepsilon }n^{1/3}\right] \\&\quad = {\mathbb {P}}\left[ \sup _{x\in 2^{-2/3}{\mathcal {I}}} {\mathcal {L}}(x) > \sup _{x\in [-2^{-2/3}\cdot 2M, 2^{-2/3}\cdot 2M]} {\mathcal {L}}(x) - 2^{-4/3}\sqrt{\varepsilon }\right] . \end{aligned}$$

If $4|{\mathcal {I}}|< \varepsilon < \frac{1}{2}$, by Proposition 3.3 the right hand side is bounded by $C_1 \varepsilon \exp (C_1|\log (\varepsilon )|^{5/6})$, for some $C_1$ depending only on M. Hence by taking $n_0(\iota , M)$ large, we have that

$$\begin{aligned} {\mathbb {P}}\left[ \max _{u\in n^{2/3}{\mathcal {I}}\cap {\mathbb {Z}}} T_{(u,-u),{\mathbf {n}}} > \max _{u\in \llbracket -2Mn^{2/3}, 2Mn^{2/3}\rrbracket } T_{(u,-u),{\mathbf {n}}}-\sqrt{\varepsilon }n^{1/3}\right] \le C_1 \varepsilon \exp \left( C_1|\log (\varepsilon )|^{5/6}\right) , \end{aligned}$$

for any $n\ge n_0(\iota , M)$, and ${\mathcal {I}}\in \Theta$, $\varepsilon \in \Upsilon$ with $4|{\mathcal {I}}| < \varepsilon$. Now for any $I \subset \llbracket -Mn^{2/3},Mn^{2/3}\rrbracket$ and $\varepsilon \in \left( \iota , \frac{1}{18}\right)$, with $\iota n^{2/3}\le |I|\le \varepsilon n^{2/3}$, we can find ${\mathcal {I}} \in \Theta$, such that

$$\begin{aligned} I \subset n^{2/3}{\mathcal {I}},\,\, |{\mathcal {I}}| \le n^{-2/3}|I| + \frac{2}{N}; \text { and, } \varepsilon ' \in \Upsilon , \end{aligned}$$

such that $8\varepsilon < \varepsilon ' \le 8\varepsilon + \frac{1}{N}$. Then as $\frac{2}{N}<\iota \le n^{-2/3}|I|$, we have that $|{\mathcal {I}}| \le 2n^{-2/3}|I|$ and $\varepsilon ' \le 9 \varepsilon$, so $4|{\mathcal {I}}| \le 8n^{-2/3}|I| \le 8\varepsilon< \varepsilon ' < \frac{1}{2}$, and

$$\begin{aligned}&{\mathbb {P}}\left[ \max _{u\in I} T_{(u,-u),{\mathbf {n}}}> \max _{u\in \llbracket -2Mn^{2/3}, 2Mn^{2/3}\rrbracket } T_{(u,-u),{\mathbf {n}}}-\sqrt{\varepsilon }n^{1/3}\right] \\&\quad \le {\mathbb {P}}\left[ \max _{u\in n^{2/3}{\mathcal {I}}\cap {\mathbb {Z}}} T_{(u,-u),{\mathbf {n}}} > \max _{u\in \llbracket -2Mn^{2/3}, 2Mn^{2/3}\rrbracket } T_{(u,-u),{\mathbf {n}}}-\sqrt{\varepsilon }n^{1/3}\right] \\&\quad \le C_1 \varepsilon '\exp \left( C_1|\log (\varepsilon ')|^{5/6}\right) , \end{aligned}$$

and our conclusion follows by taking $C = 9C_1$. $\quad \square$

Proof of Proposition 3.10

Recall the event

$$\begin{aligned} H_j=\left\{ \max _{2^{j-1} r^{2/3} \le |u|< 2^{j} r^{2/3}} T_{(u,-u), {\mathbf {n}}}< \max _{|u| < r^{2/3}} T_{(u,-u), {\mathbf {n}}} - 2\alpha \cdot 2^{j\left( \frac{1}{2}-\tau \right) }r^{1/3}.\right\} , \end{aligned}$$

and let $C_{\delta , \theta } \in {\mathbb {N}}$ be chosen depending $\delta , \theta$ (to be chosen appropriately large later). We first claim that, for any n, r with $\delta n< r < n$, and n large enough, we have

$$\begin{aligned}&{\mathbb {P}}\left[ \bigcap _{j=0}^{C_{\delta , \theta }}H_j\right] > \frac{1}{2}{\mathbb {P}}\biggl [ \sup _{|x|<r^{2/3}(2n)^{-2/3}} {\mathcal {L}}(x) \nonumber \\&\quad = \sup _{|x|<\theta r^{2/3}(2n)^{-2/3}} {\mathcal {L}}(x){< {\mathcal {L}}(0) + 2\alpha ^{-1}r^{1/3}\cdot 2^{-4/3}n^{-1/3},}\nonumber \\&\quad \sup _{2^{j-1} r^{2/3}(2n)^{-2/3} \le |x|< 2^{j} r^{2/3}(2n)^{-2/3}} {\mathcal {L}}(x) \nonumber \\&\quad< \sup _{|x|<r^{2/3}(2n)^{-2/3}} {\mathcal {L}}(x)- 2\alpha \cdot 2^{j\left( \frac{1}{2}-\tau \right) }r^{1/3}\cdot 2^{-4/3}n^{-1/3}\nonumber \\&\quad \forall 1 \le j \le C_{\delta , \theta } \biggr ]. \end{aligned}$$

(54)

We argue by contradiction. Assume otherwise and hence there are sequences of integers $\{n_k\}_{k=1}^{\infty }$ and $\{r_k\}_{k=1}^{\infty }$, with $\lim _{k\rightarrow \infty }n_k = \infty$, such that for each k, $\delta n_k< r_k < n_k$, $n_k > n_0(\delta ,\theta )$, and (54) does not hold for each $n=n_k$, $r=r_k$. By taking a subsequence, we can assume that $\iota :=\lim _{k\rightarrow \infty }\frac{r_k}{n_k}$ exists. By Theorem 3.8, we have that by taking $n=n_k$, $r=r_k$ and $k\rightarrow \infty$, the left hand side of (54) converges to

$$\begin{aligned}&{\mathbb {P}}\left[ \sup _{|x|<2^{-2/3}\iota ^{2/3}} {\mathcal {L}}(x) = \sup _{|x|<2^{-2/3}\theta \iota ^{2/3}} {\mathcal {L}}(x) {< {\mathcal {L}}(0) + 2\alpha ^{-1}\cdot 2^{-4/3}\iota ^{1/3},} \right. \\&\quad \forall 1 \le j \le C_{\delta , \theta },\;\; \sup _{2^{j-1} 2^{-2/3}\iota ^{2/3} \le |x|< 2^{j} 2^{-2/3}\iota ^{2/3}} {\mathcal {L}}(x) \\&\quad \left.< \sup _{|x|<2^{-2/3}\iota ^{2/3}} {\mathcal {L}}(x)- 2\alpha \cdot 2^{j\left( \frac{1}{2}-\tau \right) } 2^{-4/3}\iota ^{1/3} \right] , \end{aligned}$$

while the right hand side of (54) converges to half of this. By Proposition 3.4, the right hand side of (54) is at least $\alpha \theta r^{2/3}(2n)^{-2/3}> 2^{-2/3}\alpha \theta \delta ^{2/3}>0$ and thus we get a contradiction.

Next we consider $H_j$ for $j > C_{\delta , \theta }$. If $H_j$ does not hold, clearly either

$$\begin{aligned} \max _{2^{j-1} r^{2/3} \le |u| < 2^{j} r^{2/3}} T_{(u,-u), {\mathbf {n}}} \ge 4n-2^jr^{1/3} \end{aligned}$$

(55)

or

$$\begin{aligned} \max _{|u| < r^{2/3}} T_{(u,-u), {\mathbf {n}}} \le 4n + 2\alpha \cdot 2^{j\left( \frac{1}{2}-\tau \right) }r^{1/3} -2^jr^{1/3} \end{aligned}$$

(56)

holds. By Theorem 4.1, and by lower bounding $\max _{|u| < r^{2/3}} T_{(u,-u), {\mathbf {n}}}$ by $T_{{\mathbf {0}}, {\mathbf {n}}}$, we have that the event (56) has probability at most $C \exp (-c 2^jr^{1/3}n^{-1/3})$, (in fact it provides a stronger probability bound which we do not use) for some universal constants c, C. For the event (55), we can divide the interval $2^{j-1} r^{2/3} \le |u| < 2^{j} r^{2/3}$ into $2^{j}r^{2/3}n^{-2/3}$ sub-intervals of length $n^{2/3}$. Provided $2^{j}r^{2/3} \le n/2$, for any $2^{j-1} r^{2/3} \le |u| < 2^{j} r^{2/3}$ we have using (12) that

$$\begin{aligned} {\mathbb {E}}T_{(u, -u), {\mathbf {n}}} \le 4n - 2^{2(j-1)}r^{4/3}n^{-1} + C'n^{1/3} < 4n - 2\cdot 2^jr^{1/3}, \end{aligned}$$

where the second inequality holds by choosing $C_{\delta , \theta }$ large enough. Then Theorem 4.2 applies again for each of the subintervals and we get an upper bound of $C 2^jr^{2/3}n^{-2/3} \exp (-c 2^jr^{1/3}n^{-1/3})$, for some universal constants c, C. On the other hand if $2^{j}r^{2/3}\ge n/2,$ the crude bound in (13) and an union bound over all points yield an upper bound of $C2^{j}r^{2/3}\exp (-c 2^jr^{1/3}n^{-1/2})$. Combining all of these and using $r>\delta n$ we get that

$$\begin{aligned}&\sum _{j> C_{\delta , \theta }} {\mathbb {P}}[H_j^c]< \sum _{j > C_{\delta , \theta }} 2C 2^j \exp (-c 2^j\delta ^{1/3})\\&\quad + \sum _{j:2^{j}r^{2/3}\ge n/2} C2^{j}r^{2/3}\exp (-c 2^jr^{1/3}n^{-1/2}) < \alpha \theta \delta ^{2/3}/2, \end{aligned}$$

where the last inequality holds by taking $C_{\delta , \theta }$ large enough and n sufficiently large. Thus our conclusion follows by letting $c_0 := (2^{-2/3}-2^{-1})\alpha$. $\quad \square$

We end with a discussion of the proof of Theorem 3.8. As this has appeared in the literature before, we shall not provide a complete proof, instead sketching how to obtain the finite dimensional convergence, and then the necessary equi-continuity to establish uniform convergence.

Proof of Theorem 3.8

Finite dimensional convergence of the (appropriately scaled) TASEP height functions is well known (see e.g. [11]). In the language of exponential LPP this translates to the following: for any $x_1, \cdots x_k, h_1, \cdots , h_k \in {\mathbb {R}}$, we have that as $n\rightarrow \infty$,

$$\begin{aligned} {\mathbb {P}}\left[ T_{(\lfloor x(2n)^{2/3}\rfloor +\lfloor 2^{-2/3}n^{1/3}h \rfloor , -\lfloor x(2n)^{2/3}\rfloor +\lfloor 2^{-2/3}n^{1/3}h \rfloor ) , {\mathbf {n}}}< 4n,\; \forall 1 \le i \le k \right] \rightarrow {\mathbb {P}}\left[ {\mathcal {L}}(x_i) < h_i, \forall 1 \le i \le k \right] . \end{aligned}$$

It is also standard that the above equation, using the phenomenon of so-called slow decorrelation can be used to establish the finite dimensional convergence of ${\mathcal {L}}_n$ to ${\mathcal {L}}$ (see, e.g. [26]). Indeed, it can be proved that for any fixed $h, x \in {\mathbb {R}}$, as $n\rightarrow \infty$,

$$\begin{aligned} \left| T_{(\lfloor x(2n)^{2/3}\rfloor +\lfloor 2^{-2/3}n^{1/3}h \rfloor , -\lfloor x(2n)^{2/3}\rfloor +\lfloor 2^{-2/3}n^{1/3}h \rfloor ) , {\mathbf {n}}} + 2^{4/3}n^{1/3}h - T_{(\lfloor x(2n)^{2/3}\rfloor , -\lfloor x(2n)^{2/3}\rfloor ), {\mathbf {n}}} \right| {\rightarrow } 0\nonumber \\ \end{aligned}$$

(57)

in probability. Clearly this suffices for the finite dimensional convergence. We shall omit the proof of (57).

To upgrade to weak convergence in the uniform convergence topology, it remains show equicontinuity of ${\mathcal {L}}_n$, i.e. given any $M, \varepsilon , \lambda >0$, there is $\delta >0$³ and $n_0 \in {\mathbb {Z}}_+$, such that

$$\begin{aligned} {\mathbb {P}}\left[ \sup _{|x_1|, |x_2|< M, |x_1-x_2|<\delta } |{\mathcal {L}}_n(x_1)-{\mathcal {L}}_n(x_2)| > \lambda \right] < \varepsilon \end{aligned}$$

(58)

for any $n>n_0$. To prove this we rely on the Brownian type fluctuation upper bounds of the weight profile in exponential LPP proved in [3]. To proceed, we divide $[-M, M]$ into intervals of length $c'\delta ^3$. For each such interval I, by [3, Theorem 3], we have

$$\begin{aligned} {\mathbb {P}}\left[ \sup _{x_1, x_2\in I} |{\mathcal {L}}_n(x_1)-{\mathcal {L}}_n(x_2)| > \lambda \right] \le C'\exp (-c'\lambda ^{4/9}\delta ^{-2/9}). \end{aligned}$$

Note that the above tail bounds are sub-optimal and one does expect Gaussian tail behavior as has been established for the pre-limiting model of Brownian LPP in [16]. However the above bound suffices for our purpose, since the total number of such intervals I is $\lceil M\delta ^{-3} \rceil$, the left hand side of (58) can be bounded by $C'\lceil M\delta ^{-3} \rceil \exp (-c'\lambda ^{4/9}\delta ^{-2/9})$, which is made less than $\varepsilon$ by taking $\delta$ small enough. The above equicontinuity, and by standard results (see e.g. [10, Theorem 7.1, 7.3]), the desired conclusion follows. $\quad \square$

Appendix C. Passage Times across Parallelograms and Transversal Fluctuation

As indicated before, in this appendix we provide the proofs of the estimates on last passage times across parallelograms (Theorem 4.2), and the proof of the fact that paths with large transversal fluctuation are likely to have significantly smaller weights than geodesics (Proposition 4.7). As pointed out before, a version of Theorem 4.2 for Poissonian LPP was obtained in [9] where [9, Proposition 10.1, 10.5, 12.2] are the analogous versions of Theorem 4.2 (i), (ii), and (iii) respectively. The proofs there use moderate deviation estimates for the passage time (a weaker version of Theorem 4.1 for Poissonian LPP) and we essentially repeat the arguments in the context of exponential LPP. However, we have tightened up the calculations therein using the optimal estimates in Theorem 4.1 and hence we get better exponents in our results (optimal ones for parts (i) and (ii)). Proposition 4.7 has not appeared before in the form stated, but the proof uses the same idea as in [9, Proposition 11.1] involving Theorem 4.2 (ii) and a chaining argument.

The basic structure of the section is as follows: In Sect. C.1 we prove Theorem 4.2 (i), while Sect. C.2 handles Theorem 4.2 (ii). Using the latter and Theorem 4.1, the proof of Proposition 4.7 is completed in Sect. C.3, and finally Theorem 4.2 (iii) is proved in Section C.4.

1.1 Minimum passage time in a parallelogram

In the course of proving Theorem 4.2(i), we shall first prove a weaker version. We start with a few definitions first. For m, h such that $|m|+h<\psi r^{1/3}$, let $U_0$ be the parallelogram whose one pair of opposite sides are parallel to the line $x+y=0$, have midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$ respectively and length $2hr^{2/3}$. Let ${\hat{U}}$ be the sub-parallelogram of $U_0$ restricted to the strip $\{0\le x+y\le \frac{r}{16}\}$. The main technical work goes into proving the following result.

Lemma C.1

For each $\psi <1$ and $h>0$, there exists $C,c>0$ depending only on $\psi , h$ such that for ${\hat{U}}$ as above with $|m|+h\le \psi r^{1/3}$, we have for all $x>0$ and all $r\ge 1$

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u\in {\hat{U}}} (T_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \le -xr^{1/3}\right) \le Ce^{-cx^3}. \end{aligned}$$

We shall come back to the proof of Lemma C.1 at the end of this subsection; first let us show how this leads to Theorem 4.2(i). Let $U_*, U_{*,1}$ and $U_{*,2}$ be the sub-parallelograms of $U_0$ restricted to the strips $\{0\le x+y\le \frac{9r}{5}\},$ $\{0\le x+y\le \frac{9r}{10}\},$ and $\{\frac{11r}{10}\le x+y\le 2r\}$ respectively. We upgrade Lemma C.1 to the following one.

Lemma C.2

For each $\psi <1$ and $h>0$, there exists $C,c>0$ depending only on $\psi , h$ such that for $U_*$ as above with $|m|+{10}h\le \psi r^{1/3}$, we have for all $x>0$ and all $r\ge 1$

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u\in U_*} (T_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \le -xr^{1/3}\right) \le Ce^{-cx^3}. \end{aligned}$$

The constraint $|m|+{10}h\le \psi r^{1/3}$ will allow us to apply Lemma C.1.

Proof

For each $0\le i \le 359,$ we denote ${\hat{U}}_i$ as the subparallelogram of $U_0$ restricted to the strip $\{\frac{ir}{200} \le x+y \le \frac{(i+1)r}{200}\}$ (the choice of 360 is guided by the fact that we are tiling $U_*$ by parallelograms of height $\frac{r}{200},$ and $360=(9/5)200,$ where the 9/5 factor appears in the definition of $U_*,$ while the latter choice of 200 is somewhat arbitrary and can be replaced by all large enough numbers).

Now to apply Lemma C.1 we define $U_i$ as the intersection of $U_0$ and the region $\{x+y\ge \frac{ir}{200}\}$. Then ${\hat{U}}_i$ is a subset of $U_i$, sharing the bottom face and having height at most 1/32th of the latter. Thus we could apply Lemma C.1, with $U_i$ in the place of $U_0$ and ${\hat{U}}_i$ in the place of ${\hat{U}}$. Also note that the necessary slope condition is satisfied by the hypothesis $|m|+10h\le \psi r^{1/3}$, since one pair of opposite sides of $U_i$ have length $2hr^{2/3}$ and midpoints $\frac{i}{400}{\mathbf {r}}+\frac{(400-i)}{400} (mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$ respectively. The proof is now completed by taking a union bound over i. $\quad \square$

Applying Lemma C.2 twice we get the following result.

Lemma C.3

For each $\psi <1$ and $h>0$, there exists $C,c>0$ depending only on $\psi , h$ such that for $U_{*,1}, U_{*,2}$ as above with $|m|+20h\le \psi r^{1/3}$, we have for all $x>0$ and $r\ge 1$

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u\in U_{*,1}, v\in U_{*,2}} (T_{u,v}-{\mathbb {E}}T_{u,v}) \le -xr^{1/3}\right) \le Ce^{-cx^3}. \end{aligned}$$

Proof

Let $w=(w_1,w_2)=\frac{{\mathbf {r}}+(mr^{2/3}, -mr^{2/3})}{2}$, which is the center of $U_0$. Consider the events

$$\begin{aligned} {\mathcal {A}}_1= & {} \left\{ \inf _{u\in U_{*,1}} (T_{u,w}-{\mathbb {E}}T_{u,w}) \ge -\frac{x}{4}r^{1/3}\right\} ;\\ {\mathcal {A}}_2= & {} \left\{ \inf _{v\in U_{*,2}} (T_{w,v}-{\mathbb {E}}T_{w,v}) \ge -\frac{x}{4}r^{1/3}\right\} . \end{aligned}$$

Simple algebra now shows that for all $u=(u_1,u_2)\in U_{*,1}$ and $v=(v_1,v_2)\in U_{*,2}$ we have

$$\begin{aligned}&0 \le (\sqrt{v_1-u_1}+\sqrt{v_2-u_2})^2 - (\sqrt{w_1-u_1}+\sqrt{w_2-u_2})^2 \\&\quad - (\sqrt{v_1-w_1}+\sqrt{v_2-w_2})^2 \le C_1 r^{1/3} \end{aligned}$$

for some constant $C_1$ depending on $\psi , h$. It follows from (12) that for x sufficiently large we have for all $u\in U_{*,1}$ and $v\in U_{*,2}$

$$\begin{aligned} |{\mathbb {E}}T_{u,w} + {\mathbb {E}}T_{w,v} - {\mathbb {E}}T_{u,v}|\le \frac{x}{2}r^{1/3} \end{aligned}$$

and hence on ${\mathcal {A}}_1\cap {\mathcal {A}}_2$ we have $\inf _{u\in U_{*,1}, v\in U_{*,2}} (T_{u,v}-{\mathbb {E}}T_{u,v}) \ge -xr^{1/3}$. The proof is completed by using a union bound and Lemma C.2 to upper bound ${\mathbb {P}}({\mathcal {A}}_1^c\cup {\mathcal {A}}_2^{c})$. $\quad \square$

Using Lemma C.3, we can now complete the proof of Theorem 4.2(i). Recall the parallelogram U from the statement of the theorem whose one pair of sides lie on ${\mathbb {L}}_0$ and ${\mathbb {L}}_r$ with length $2r^{2/3}$ and midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$ respectively. Let $U'$ be the parallelogram whose one pair of sides also lie on ${\mathbb {L}}_0$ and ${\mathbb {L}}_r$ with midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$ respectively, but the side length is now $20r^{2/3}$. For notational convenience, let us first define the following. Consider a parallelogram which is a translate (in ${\mathbb {Z}}^2$) of a parallelogram which has, for some $r\in {\mathbb {N}}$, a pair of opposite sides along ${\mathbb {L}}_0$ and ${\mathbb {L}}_r$ with length at most $2r^{2/3}$ and midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$ respectively. If $|m|+20<\psi r^{1/3}$ for some $\psi <1$ we say that such a parallelogram satisfied the $\psi$-slope condition.

Without loss of generality we shall assume that r and r/L are sufficiently large powers of 2, and omit the floor and ceiling signs in the following argument. For $i\in \llbracket 0, 2r \rrbracket$, let $A_{i}$ denote the line segment of the intersection of the line $x+y=i$ and $U'$. For $j\in {\mathbb {N}}$, and $i_1, i_2\in \llbracket 0, 2r \rrbracket$, with $i_1<i_2$, let us divide the lines segments $A_{i_1}$ and $A_{i_2}$ into $10\cdot 2^{2j/3}$ equal sub-segments of length $2(2^{-j}r)^{2/3}$. Consider all possible parallelograms formed by taking one pair of opposite sides from these sub-segments, one on $A_{i_1}$ the other on $A_{i_2}$. Call this family of parallelograms ${\mathcal {P}}_{i_1,i_2,j}$. We next record the following straightforward deterministic facts.

Observation C.4

We have the following:

• (i) If $|m|<\psi r^{1/3}$ for some $\psi <1$ then for any $\psi<\psi '<1,$ and each $j\in {\mathbb {N}}$ and each r sufficiently large (compared to $2^{j}$ and $\psi$), and each $i_1, i_2 \in \llbracket 0, 2r \rrbracket$ with $i_2-i_1 \ge 2^{-j}(2r)$, every parallelogram in ${\mathcal {P}}_{i_1,i_2,j}$ satisfies the $\psi '$-slope condition.

• (ii) For $j\in {\mathbb {N}}$, and $i_1,i_2\in \llbracket 0, 2r \rrbracket$ with $i_1<i_2$, let $u,v\in U$ be such that $i_1\le d(u)\le i_1+\frac{9(i_2-i_1)}{20}$ and $i_1+\frac{11 (i_2-i_1)}{20}\le d(v)\le i_2$. Then for r sufficiently large there exists a parallelogram in ${\mathcal {P}}_{i_1,i_2,j}$ containing both u and v.

While the first observation is rather simple to verify, the second observation follows from the fact that the straight line passing through such u, v intersects $A_{i_1}$ and $A_{i_2}$.

We can now complete the proof of Theorem 4.2(i).

Proof of Theorem 4.2(i)

For $j=1,2,\ldots \log _2 L + 1$, let us consider the family of parallelograms ${\mathcal {P}}_{i_1,i_2,j}$ for $i_1, i_2\in \{0, 2^{-j}(2r), 2\times 2^{-j}(2r), 3\times 2^{-j}(2r), \ldots , 2r\}$, satisfying $i_2-i_1 \in \{3\times 2^{-j}(2r), 4\times 2^{-j}(2r), 5\times 2^{-j}(2r)\}$. For a fixed j, let ${\mathcal {B}}_{j}$ denote the event that for any parallelogram $U_0$ as above we have

$$\begin{aligned} \inf _{u\in U_{*,1},v\in U_{*,2}} (T_{u,v}-{\mathbb {E}}T_{u,v}) \ge -xr^{1/3} \end{aligned}$$

where $U_{*,1}$ and $U_{*,2}$ are defined as follows: suppose $U_0\in {\mathcal {P}}_{i_1,i_2,j}$ for some $i_1,i_2$, then $U_{*,1}$ and $U_{*,2}$ are sub-parallelograms of $U_0$ restricted to the strips $\{i_1\le x+y \le \frac{11i_1+9i_2}{20}\}$ and $\{\frac{9i_1+11i_2}{20}\le x+y \le i_2\}$, respectively. By Observation C.4 (i), for r sufficiently large depending on L, Lemma C.3 applies; and since $|{\mathcal {P}}_{i_1,i_2,j}|=O(2^{4j/3})$ it follows that ${\mathbb {P}}({\mathcal {B}}_j^c)=O(2^{7j/3}e^{-c2^{j}x^3})$.

Now we consider any $u, v \in U$ with $d(v)-d(u)\ge \frac{r}{L}$. Suppose that $2^{-j}(2r)\le d(v)-d(u)<2^{-j+1}(2r)$ for some $j\le \log _2 L$. Then we can find some $i_1,i_2 \in \{0, 2^{-j-1}(2r), 2\times 2^{-j-1}(2r), 3\times 2^{-j-1}(2r), \ldots , 2r\}$, with $i_2-i_1 \in \{3\times 2^{-j-1}(2r), 4\times 2^{-j-1}(2r), 5\times 2^{-j-1}(2r)\}$, such that $i_1\le d(u)< i_1+ 2^{-j-1}(2r)< i_2- 2^{-j-1}(2r) \le d(v) < i_2$. By Observation C.4 (ii), there exists a parallelogram in $U_0\in {\mathcal {P}}_{i_1,i_2,j}$ such that $u\in U_{*,1}$ and $v\in U_{*,2}$. It follows that on $\cap _{j}{\mathcal {B}}_j$ we have

$$\begin{aligned} \inf _{d(v)-d(u)\ge \frac{r}{L}} (T_{u,v}-{\mathbb {E}}T_{u,v}) \ge -xr^{1/3}. \end{aligned}$$

The proof is completed by taking a union bound over ${\mathcal {B}}_j^c$. $\quad \square$

1.1.1 Proof of Lemma C.1

As already noted, the proof of Lemma C.1 is rather technical. For ease of exposition we first present a weaker version. For $|m|+h<\psi r^{1/3}$, let $A'$ denote the line segment of length $2hr^{2/3}$ on ${\mathbb {L}}_0$ with midpoint at $(mr^{2/3},-mr^{2/3})$ For $|m|+h<\psi r^{1/3}$. Then we have the following.

Lemma C.5

For each $\psi <1$ and $h>0$, there exists $C,c>0$ depending only on $\psi , h$ such that for $A'$ as above, we have for all $x>0$ and all $r\ge 1$

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u\in A'} (T_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \le -xr^{1/3}\right) \le Ce^{-cx^3}. \end{aligned}$$

We are going to present a full proof of Lemma C.1 and hence, for notational convenience, we shall write the proof of Lemma C.5 only for the special case $h=1$ and $m=0$. The reader will notice that the same proof will apply to the general case with minor adjustments. Before proceeding with the proof, let us present the basic idea. We shall construct a tree ${\mathcal {T}}$ whose vertices are a subset of vertices of ${\mathbb {Z}}^2$; in particular root of ${\mathcal {T}}$ will be the vertex ${\mathbf {r}}$ and the leaves of ${\mathcal {T}}$ are close to vertices on $A'$. The tree will be constructed such that if $T_{u,v}-{\mathbb {E}}T_{u,v}$ is not too small for every edge $(u,v)\in {\mathcal {T}}$ then we shall have $\inf _{u\in A'} (T_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \ge -xr^{1/3}$. Taking a union bound of the complements of the above events over all edges of ${\mathcal {T}}$ will then yield the result.

Let us now formally construct the tree ${\mathcal {T}}$. Let r be sufficiently large so that there exists J such that $r^{1/4}< 8^{-J}(2r)\le r^{1/3}$. For smaller r the lemma follows by taking C large and c small enough. We shall be ignoring the rounding issues for notational convenience. For $j=0,1,2,\ldots , J$, there will be $4^{j}$ vertices of ${\mathcal {T}}$ at level j (let us denote this set by ${\mathcal {T}}_j$) on the line $x+y=8^{-j}(2r)$, such that these $4^{j}$ vertices divide the line joining $8^{-j}{\mathbf {r}}+(-r^{2/3},r^{2/3})$ and $8^{-j}{\mathbf {r}}-(-r^{2/3},r^{2/3})$ into $4^{J}+1$ equal length intervals. Notice that, for each j, the vertices in ${\mathcal {T}}_j$ are ordered naturally from left to right. The vertex set of ${\mathcal {T}}$ is $\cup _{0\le j\le J} {\mathcal {T}}_j$, and the k-th vertex at level j from the left is connected to the four vertices in level $(j+1)$ which are labelled $4k-3,4k-2,4k-1$ and 4k from the left.

Recall that for any $u=(u_x,u_{y})\in {\mathbb {Z}}^2$, we denote $d(u)=u_{x}+u_{y}$. It would be also convenient to let ${\mathrm{ad}}(u)=u_{x}-u_{y}$ (where $\mathrm{ad}$ is used to denote the anti-diagonal deviation of u). By the construction of ${\mathcal {T}}$ we have for each $j\le J$

$$\begin{aligned} {d(u_{j})-d(u_{j+1})}&=\frac{14r}{8^{j+1}},\\ | {\mathrm{ad}}(u_{j+1})-{\mathrm{ad}}(u_j)|&\le C_1\frac{r^{2/3}}{4^{j}} \text { for some }C_1>0. \end{aligned}$$

(59)

Noticing that it suffices to prove Lemma C.5 for x sufficiently large, let ${\mathcal {A}}_j$ denote the event that for all $u\in {\mathcal {T}}_j$ and for all $v\in {\mathcal {T}}_{j+1}$ such that the edge (u, v) is present in ${\mathcal {T}},$ we have

$$\begin{aligned} T_{v,u}\ge {\mathbb {E}}T_{v,u}- x2^{-(9j/10+10)}r^{1/3}. \end{aligned}$$

We first have the following lemma.

Lemma C.6

In the above set-up, there exists $C,c>0$ such that for all x sufficiently large

$$\begin{aligned} {\mathbb {P}}(\cup _{j}{\mathcal {A}}_j^c) \le Ce^{-cx^3}. \end{aligned}$$

Proof

Notice that, by our construction of ${\mathcal {T}}$, for each edge between a vertex $u\in {\mathcal {T}}_j$ and a vertex $v\in {\mathcal {T}}_{j+1}$, Theorem 4.1 applies to $T_{v,u}$ (where the slope condition is satisfied for all large r by (59), (60)) and hence we have that

$$\begin{aligned} {\mathbb {P}}(T_{v,u}-{\mathbb {E}}T_{v,u}\le -y (8^{-j}r)^{1/3})\le Ce^{-cy^3} \end{aligned}$$

for some $C,c>0$ and all $y>0$. Applying this with $y=2^{j/10-10}x$ we obtain that

$$\begin{aligned} {\mathbb {P}}(T_{v,u}-{\mathbb {E}}T_{v,u}\le -x{2^{-(9j/10+10)}}r^{1/3})\le Ce^{-cx^32^{3j/10}} \end{aligned}$$

for some $C,c>0$. Now taking a union bound over all $4^{j+1}$ such edges gives that

$$\begin{aligned} {\mathbb {P}}({\mathcal {A}}_j^{c})\le Ce^{-cx^32^{j/10}} \end{aligned}$$

for $C,c>0$ and all $j=0,1,2,\ldots , J-1$. Taking another union bound over j completes the proof of the lemma. $\quad \square$

The proof of Lemma C.5 is completed by using the next lemma.

Lemma C.7

On $\cap _{0\le j\le J} {\mathcal {A}}_j$, we have $\inf _{u\in A'} (T_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \ge -xr^{1/3}$ for all x sufficiently large.

Proof

Let us fix $u\in A'$ and let $u_J$ be the vertex in ${\mathcal {T}}_J$ such that the difference between ${\mathrm{ad}}(u)$ and ${\mathrm{ad}}(u_J)$ is smallest. Let $u_J, u_{J-1},\ldots , u_0={\mathbf {r}}$ denote the path to ${\mathbf {r}}$ in ${\mathcal {T}}$. By our construction of ${\mathcal {T}}$, u is coordinate-wise smaller than $u_0$ and hence we have

$$\begin{aligned} T_{u,{\mathbf {r}}} \ge \sum _{j=0}^{J-1} T_{u_{j+1},u_j}. \end{aligned}$$

By definition, we have that on $\cap _{0\le j\le J} {\mathcal {A}}_j$

$$\begin{aligned} \sum _{j=0}^{J-1} T_{u_{j+1},u_j}- {\mathbb {E}}T_{u_{j+1},u_j} \ge -\frac{x}{2} r^{1/3}, \end{aligned}$$

for x sufficiently large. Observe also that by our definition of ${\mathcal {T}}$ and (12) we have

$$\begin{aligned} {\mathbb {E}}T_{u,u_{J}}\le \frac{x}{4}r^{1/3}, \end{aligned}$$

(60)

and hence it suffices to show that

$$\begin{aligned} \sum _{j=0}^{J} {\mathbb {E}}T_{u_{j+1},u_j} \ge {\mathbb {E}}T_{u,{\mathbf {r}}}-\frac{x}{4}r^{1/3}, \end{aligned}$$

where we write $u=u_{J+1}$ for convenience of writing.

Recalling (59), (60), and (12) (observe again that it applies to each $T_{u_{j+1},u_j}$) we get

$$\begin{aligned} {\mathbb {E}}T_{u_{j+1},u_{j}} \ge 2(d(u_j)-d(u_{j+1}))-C_22^{-j}r^{1/3} \end{aligned}$$

for each $j\le J$. Summing over j from 0 to J, along with the bound ${\mathbb {E}}T_{u,{\mathbf {r}}}\le 4r+O(r^{1/3})$ (which has been used several times already and in particular follows from (12)), and (61) we get that

$$\begin{aligned} \sum _{j=0}^{J} {\mathbb {E}}T_{u_{j+1},u_j} \ge {\mathbb {E}}T_{u,{\mathbf {r}}}-\frac{x}{4}r^{1/3} \end{aligned}$$

for x sufficiently large, as required. This completes the proof. $\quad \square$

In the above tree construction, each level of the tree made a deterministic progress along the diagonal direction with each vertex splitting into four offsprings spread in the anti-diagonal direction. To strengthen Lemma C.5 to Lemma C.1, we similarly construct a tree, which, in addition to branching in the anti-diagonal direction, also branches in the diagonal direction. For this tree, its leaves are dense in the parallelogram ${{\hat{U}}}$, allowing us to simultaneously lower bound the passage time from each vertex in ${{\hat{U}}}$ to ${\mathbf {r}}$.

Proof of Lemma C.1

We assume that r and x are large enough (depending on $\psi , h$), since otherwise the result follows by taking C large and c small enough. In this proof we use C and c to denote large and small constants depending on $\psi$ and h, and the specific values can change from line to line. Without loss of generality we also assume $m\ge 0$.

Similarly to the proof of Lemma C.5, we construct a tree ${\mathcal {T}}$ as follows. Take J such that $r^{1/4}< 8^{-J}(2r)\le r^{1/3}$. Let ${\mathcal {T}}_0=\{{\mathbf {r}}\}$. For each $j=1,2,\ldots , J$, there are $32^{j}$ vertices of ${\mathcal {T}}$ at level j (denoted as ${\mathcal {T}}_j$), given as follows. For each $i=1,2,\ldots , 8^j$, consider the intersection of the line $x+y=\frac{2i+1}{32}8^{-j}(2r)$ with $U_0$; this is also the line segment joining

$$\begin{aligned} \frac{2i+1}{32}8^{-j}{\mathbf {r}}+(m(1-\frac{2i+1}{32}8^{-j})+h)(r^{2/3},-r^{2/3}) \end{aligned}$$

and

$$\begin{aligned} \frac{2i+1}{32}8^{-j}{\mathbf {r}}+(m(1-\frac{2i+1}{32}8^{-j})-h)(r^{2/3},-r^{2/3}). \end{aligned}$$

On this line segment there are $4^j$ level j vertices, which divide this line segment into $4^{j}+1$ equal length intervals. From this construction we see that the sets ${\mathcal {T}}_0,\ldots ,{\mathcal {T}}_J$ are mutually disjoint, since the lines $x+y=\frac{2i+1}{32}8^{-j}(2r)$ are mutually different for different i, j. The vertex set of ${\mathcal {T}}$ is $\cup _{0\le j\le J} {\mathcal {T}}_j$. We can label the vertices in ${\mathcal {T}}_{j}$ using $\{(i,k):1\le i \le 8^j, 1\le k \le 4^j\}$, for i indexing the lines and k indexing vertices in a line from left to right. For $0\le j < J$, the vertex in ${\mathcal {T}}_{j}$ labelled (i, k) is connected to 32 vertices in level $(j+1)$, which are labelled $\{(8i-i',4k-k'):0\le i' \le 7, 0\le k' \le 3\}$. Then each vertex in ${\mathcal {T}}_{j+1}$ is connected to exactly one vertex in ${\mathcal {T}}_j$, and the graph we construct is a tree.

We record now statements analogous to (59), (60). Let us set $\rho =mr^{-1/3}$ which will parametrize the role of the slope in the subsequent calculations. For each $j< J$

$$\begin{aligned} d(u_j)-d(u_{j+1}) = 2r 8^{-j-1}(7+2i')/32,&\text { for some } i'=0,..., 7 \end{aligned}$$

(61)

$$\begin{aligned} |({\mathrm{ad}}(u_{j})-{\mathrm{ad}}(u_{j+1})) + \rho (d(u_{j})-d(u_{j+1}))|&\le C\frac{r^{2/3}}{4^{j}}. \end{aligned}$$

(62)

Since by choice $2r^{2/3}\le 8^{J}\le 2r^{3/4},$ as a consequence, we see that the slope of each tree edge is bounded away from 0 and $\infty$ uniformly in m,  for all large enough r which will allow us to apply Theorem 4.1.

Again, following the same strategy as in the proof of Lemma C.5, we let ${\mathcal {A}}_j$ denote the event that for all $u\in {\mathcal {T}}_j$ and for all $v\in {\mathcal {T}}_{j+1}$ such that the edge (u, v) is present in ${\mathcal {T}}$, we have $T_{v,u}\ge {\mathbb {E}}T_{v,u}- x2^{-(9j/10+10)}r^{1/3}$. As in Lemma C.6, we show that

$$\begin{aligned} {\mathbb {P}}(\cup _{j}{\mathcal {A}}_j^c) \le Ce^{-cx^3}. \end{aligned}$$

Indeed, by our construction of ${\mathcal {T}}$, for each edge between a vertex $u\in {\mathcal {T}}_j$ and a vertex $v\in {\mathcal {T}}_{j+1}$, by Theorem 4.1 applied to $T_{v,u}$ we have for all $y>0$,

$$\begin{aligned} {\mathbb {P}}(T_{v,u}-{\mathbb {E}}T_{v,u}\le -y (8^{-j}r)^{1/3})\le Ce^{-cy^3}. \end{aligned}$$

Using this with $y=2^{j/10-10}x$ we obtain that

$$\begin{aligned} {\mathbb {P}}(T_{v,u}-{\mathbb {E}}T_{v,u}\le -x{2^{-(9j/10+10)}}r^{1/3})\le Ce^{-cx^32^{3j/10}}. \end{aligned}$$

Now taking a union bound over all $32^{j+1}$ such edges, and then over all $j=0,1,2,\ldots , J-1$, we get ${\mathbb {P}}(\cup _{j}{\mathcal {A}}_j^c) \le Ce^{-cx^3}$.

It remains to show that on the event $\cap _{0\le j\le J} {\mathcal {A}}_j$, we must have $\inf _{u\in {\hat{U}}} (T_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \ge -xr^{1/3}$. Now let’s take any $u\in {\hat{U}}$, and take $u_J$ such that

$$\begin{aligned} d(u_J) - \frac{3\cdot 8^{-J}(2r)}{32} \le d(u) \le d(u_J) - \frac{8^{-J}(2r)}{32} \end{aligned}$$

and

$$\begin{aligned} | ({\mathrm{ad}}(u_J)-{\mathrm{ad}}(u)) + \rho (d(u_{J})-d(u))|\le C\frac{r^{2/3}}{4^{J}}. \end{aligned}$$

Such $u_J$ exists by our construction of the tree. It is evident from the above two displays that for r large enough, $|{\mathrm{ad}}(u_J)-{\mathrm{ad}}(u)| < d(u_{J})-d(u)$, so $u_J$ is coordinate-wise greater than u. Let $u_J, u_{J-1},\ldots ,u_0={\mathbf {r}}$ be the unique path from $u_J$ to ${\mathbf {r}}$ in ${\mathcal {T}}$. We have that $T_{u,{\mathbf {r}}} \ge \sum _{j=0}^{J-1} T_{u_{j+1},u_j}$. On $\cap _{0\le j\le J} {\mathcal {A}}_j$ we also have that

$$\begin{aligned} \sum _{j=0}^{J-1} T_{u_{j+1},u_j}- {\mathbb {E}}T_{u_{j+1},u_j} \ge -\frac{x}{2} r^{1/3}. \end{aligned}$$

So we have $T_{u,{\mathbf {r}}}\ge \sum _{j=0}^{J-1} {\mathbb {E}}T_{u_{j+1},u_j} -\frac{x}{2}r^{1/3}$, and we just need to show that

$$\begin{aligned} \sum _{j=0}^{J-1} {\mathbb {E}}T_{u_{j+1},u_j} \ge {\mathbb {E}}T_{u,{\mathbf {r}}}-\frac{x}{2}r^{1/3}. \end{aligned}$$

(63)

As in the the proof of Lemma C.7, we use (12) to estimate each expectation to establish (64). In particular, we will use the following bound.

$$\begin{aligned}&{\mathbb {E}}T_{u_{j+1},u_{j}} \ge (d(u_j)-d(u_{j+1}))(1+\sqrt{1-\rho ^2})\\&\quad +{\frac{\rho ( ({\mathrm{ad}}(u_{j})-{\mathrm{ad}}(u_{j+1})) + \rho (d(u_{j})-d(u_{j+1})) )}{\sqrt{1-\rho ^2}}}-C 2^{-j}r^{1/3}. \end{aligned}$$

This follows from (12) and the following calculation: For each $v\in {\mathbb {Z}}^2$ we have $v=\frac{1}{2}(d(v)+{\mathrm{ad}}(v), d(v)-{\mathrm{ad}}(v))$. When $d(v)>0$, and $\frac{|{\mathrm{ad}}(v)|}{d(v)}<(\psi +1)/2<1$, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2} \left( \sqrt{d(v)+{\mathrm{ad}}(v)}+\sqrt{d(v)-{\mathrm{ad}}(v)} \right) ^2 \\&\quad = d(v) + \sqrt{d(v)^2-{\mathrm{ad}}^2(v)}\\&\quad =d(v)(1+\sqrt{1-\rho ^2}) + \frac{\rho ({\mathrm{ad}}(v) +\rho d(v) )}{\sqrt{1-\rho ^2}} + E(v), \end{aligned} \end{aligned}$$

(64)

where E(v) is an error term satisfying $|E(v)|<C\frac{({\mathrm{ad}}(v)+\rho d(v) )^2}{d(v)}$. For $v=u_{j}-u_{j+1}$, and all large r, by (62) and (63) we get the bound on the error term which gives the sought lower bound.

Summing over j from 0 to $J-1$, we get

$$\begin{aligned} \sum _{j=0}^{J-1} {\mathbb {E}}T_{u_{j+1},u_j} \ge (2r-d(u_J))(1+\sqrt{1-\rho ^2})+\frac{\rho ( -{\mathrm{ad}}(u_{J}) + \rho (2r-d(u_J)))}{\sqrt{1-\rho ^2}} - C r^{1/3}. \end{aligned}$$

On the other hand, using (12) and (65) for $T_{u,{\mathbf {r}}}$, we have

$$\begin{aligned} {\mathbb {E}}T_{u,{\mathbf {r}}} \le (2r-d(u))(1+\sqrt{1-\rho ^2})+\frac{\rho (-{\mathrm{ad}}(u)+ \rho (2r-d(u)) )}{\sqrt{1-\rho ^2}}+C r^{1/3}. \end{aligned}$$

By our choice of $u_J$ we have $d(u_J)-d(u)\le Cr^{1/3}$ and $| ({\mathrm{ad}}(u_J)-{\mathrm{ad}}(u)) + \rho (d(u_{J})-d(u))|\le C\frac{r^{2/3}}{4^{J}}$. Thus we get (64) as x is large enough which completes the proof. $\quad \square$

1.2 Maximum Passage time in a parallelogram

Proof of Theorem 4.2(ii)

Observe first that it suffices to prove the result for x sufficiently large. Denote $u_- = -{\mathbf {r}}+ 2(mr^{2/3},-mr^{2/3})$ and $u_+ = 2{\mathbf {r}}- (mr^{2/3},-mr^{2/3})$, i.e., these are the points where the straight line joining ${\mathbf {r}}$ and $(mr^{2/3},-mr^{2/3})$ intersects ${\mathbb {L}}_{-r}$ and ${\mathbb {L}}_{2r}$ respectively. Consider the following events:

$$\begin{aligned} {\mathcal {A}}_1= & {} \{\inf _{u\in U_1} T_{u_-,u}-{\mathbb {E}}T_{u_-,u} \ge -\frac{xr^{1/3}}{10}\};\\ {\mathcal {A}}_2= & {} \{\inf _{v\in U_2} T_{v,u_+}-{\mathbb {E}}T_{v,u_+} \ge -\frac{xr^{1/3}}{10}\};\\ {\mathcal {A}}_3= & {} \{\sup _{u\in U_1,v\in U_2} T_{u,v}-{\mathbb {E}}T_{u,v} \ge xr^{1/3}\}. \end{aligned}$$

It follows from (12) that for x sufficiently large we have for any $u\in U_1$ and $v\in U_2$

$$\begin{aligned} {\mathbb {E}}T_{u_-,u}+{\mathbb {E}}T_{u,v}+ {\mathbb {E}}T_{v,u_+} \ge {\mathbb {E}}T_{u_-,u_+} -\frac{xr^{1/3}}{10}. \end{aligned}$$

(65)

It therefore follows that ${\mathcal {A}}\supseteq {\mathcal {A}}_1\cap {\mathcal {A}}_2 \cap {\mathcal {A}}_3$ where

$$\begin{aligned} {\mathcal {A}}=\{T_{u_-,u_+}-{\mathbb {E}}T_{u_-,u_+} \ge \frac{xr^{1/3}}{2}\}. \end{aligned}$$

Since ${\mathcal {A}}_1,{\mathcal {A}}_2,{\mathcal {A}}_3$ are all increasing events, it follows by the FKG inequality that

$$\begin{aligned} {\mathbb {P}}({\mathcal {A}})\ge {\mathbb {P}}({\mathcal {A}}_1\cap {\mathcal {A}}_2\cap {\mathcal {A}}_3)\ge {\mathbb {P}}({\mathcal {A}}_1){\mathbb {P}}({\mathcal {A}}_2){\mathbb {P}}({\mathcal {A}}_3) \end{aligned}$$

by the FKG inequality. The result follows by noting that we have ${\mathbb {P}}({\mathcal {A}}_1), {\mathbb {P}}({\mathcal {A}}_2)\ge \frac{1}{2}$ for x sufficiently large by Lemma C.2, and ${\mathbb {P}}({\mathcal {A}})\le Ce^{-c\min \{x^{3/2},xr^{1/3}\}}$ by Theorem 4.1. $\quad \square$

1.3 Transversal fluctuation estimates

Using Theorem 4.1 and Theorem 4.2(ii) one can show that paths with large transversal fluctuations are likely to have significantly smaller weights than geodesics and hence geodesics are unlikely to have large transversal fluctuation.

We state the following result, which is a stronger variant Proposition 4.7 dealing with general slopes. Let $A'$ be a segment of length $2r^{2/3}$ on ${\mathbb {L}}_0$ with midpoint $(mr^{2/3},-mr^{2/3})$. Let $U_{m,\phi }$ be the parallelogram whose one pair of opposite sides of length $\phi r^{2/3}$ lie on the lines ${\mathbb {L}}_0$ and ${\mathbb {L}}_r$ respectively with respective midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$. Denoted by $\mathrm {LargeTF}{(m, \phi , r)}$ the event where there exists a path $\gamma$ from $u\in A'$ to $v \in {\mathbb {L}}_{r,r^{2/3}}$ which exits $U_{m,\phi }$ and has $\ell (\gamma )\ge {\mathbb {E}}T_{u,v}-c_1\phi ^2 r^{1/3}$.

Proposition C.8

For each $\psi <1$, there exist constants $c_1,c_2>0$ such that for all $|m|\le \psi r^{1/3}$, sufficiently large $\phi$, and all $r\ge 1$,

$$\begin{aligned} {\mathbb {P}}(\mathrm {LargeTF}{(m, \phi , r)})\le e^{-c_2\phi ^3}. \end{aligned}$$

From this proposition together with Theorem 4.1, we can immediately deduce the following result, which is essentially [9, Proposition 11.1] and will be used in proving Theorem 4.2(iii).

Proposition C.9

Let ${\mathcal {A}}_\phi$ denote the event that the geodesic from $(mr^{2/3},-mr^{2/3})$ to ${\mathbf {r}}$ exits $U_{m,\phi }$. For each $\psi <1$, there exist $C,c>0$ such that for all $|m|\le \psi r^{1/3}$ and $\phi >0$, $r\ge 1$,

$$\begin{aligned} {\mathbb {P}}({\mathcal {A}}_{\phi })\le Ce^{-c\phi ^3}. \end{aligned}$$

Proof

It is immediate that for $c_1$ as in Proposition C.8 we have

$$\begin{aligned} {\mathbb {P}}({\mathcal {A}}_{\phi })\le {\mathbb {P}}(\mathrm {LargeTF}{(m, \phi , r)})+ {\mathbb {P}}\left( T_{(mr^{2/3},-mr^{2/3}), {\mathbf {r}}} \!\le \! {\mathbb {E}}T_{(mr^{2/3},-mr^{2/3}), {\mathbf {r}}}-c_1\phi ^2 r^{1/3}\right) . \end{aligned}$$

The result is then immediate from Proposition C.8 and Theorem 4.1. $\quad \square$

We now prove Proposition C.8 using a chaining argument introduced in [9].

Proof of Proposition C.8

We shall assume that r is large (depending on $\psi$), since otherwise the conclusion obviously holds. In this proof we use c to denote a small constant depending on $\psi$, and the value can change from line to line.

Let $\{a_j\}_{j\ge 2}$ be a sequence of positive real constants going to 0 satisfying $\prod _{j=2}^{\infty } (1+a_j) < \infty$, and to be chosen appropriately later. Let $h_1= \frac{1}{2\prod _{j=2}^{\infty } (1+a_j)}$, and for $j>1$ let us set $h_j=h_{j-1}(1+a_{j})$.

For $j\ge 1$, and for $\ell =1,2,\ldots , 2^{j}-1$, let ${\mathcal {B}}_{\ell ,j}$ denote the event that there exists a path $\gamma$ from $u \in A'$ to $v \in {\mathbb {L}}_{r,r^{2/3}}$ with $\ell (\gamma )\ge {\mathbb {E}}T_{u,v} -c_1\phi ^2 r^{1/3}$ which intersects line $x+y= \ell 2^{-j} (2r)$ outside the parallelogram $U_{m,h_j\phi }$. Let us set ${\mathcal {B}}_{j}=\cup _{\ell =1}^{2^j-1} {\mathcal {B}}_{\ell ,j}$. Let $j_0:=\log _{2}(r^{1/3}).$ To avoid rounding issues we will assume r is a power of 8. To treat a general $8^j\le r< 8^{j+1},$ one can instead use the result for $r'=8^{j+2},$ along with the event that no geodesic from ${\mathbf {r}}'$ to a point in ${\mathbb {L}}_{r,r^{2/3}}$ has a weight deficit from mean of order $\phi ^{2}r^{1/3}$, which occurs with probability $1-e^{-c\phi ^6}$ by Theorem 4.2 (i). It can be checked that the intersection of the latter event and $\mathrm {LargeTF}{(m, \phi , r)}$ with $c_1$ in the definition of the event replaced by $c_1/64$, which are independent, implies $\mathrm {LargeTF}{(m', \frac{\phi }{16}, r')}$ where $m'=m(\frac{r}{r'})^{2/3}$, allowing the upper bound on the latter to yield a similar upper bound for $\mathrm {LargeTF}{(m, \phi , r)}$ with $c_1/64$ in place of $c_1$. Above $r'$ is chosen to be $8^{j+2}$ and not $8^{j+1}$ to ensure enough room between ${\mathbf {r}}$ and ${\mathbf {r}}'$. Thus throughout the remaining proof, r will be a power of 8.

It is clear from our definition of $a_j$’s that $h_{j_0}\le \frac{1}{2}$. By the directed nature of the paths, for $\phi$ large, on the event $\cap _{j=1}^{j_0}{\mathcal {B}}^c_{j}$ every path $\gamma$ from $u\in A'$ to $v\in {\mathbb {L}}_{r,r^{2/3}}$ that exits $U_{m,\phi }$ satisfies $\ell (\gamma )< {\mathbb {E}}T_{u,v} -c_1\phi ^2 r^{1/3}$; i.e. we have $\mathrm {LargeTF}{(m, \phi , r)} \subset \cup _{j=1}^{j_0}{\mathcal {B}}_{j}.$ It is now immediate that Proposition C.8 will follow from the next two lemmas. $\quad \square$

Lemma C.10

In the above set-up we have ${\mathbb {P}}({\mathcal {B}}_1)\le e^{-c\phi ^3}$ for all $\phi$ sufficiently large.

Lemma C.11

In the above set up (for some appropriate choice of $a_j$), for $j_0\ge j\ge 2$, and $1\le \ell < 2^j$, we have ${\mathbb {P}}({\mathcal {B}}_{\ell ,j}\cap {\mathcal {B}}_{j-1}^{c})\le 4^{-j}e^{-c\phi ^3}$ when $\phi$ is sufficiently large.

Both Lemma C.10 and Lemma C.11 follow from the following general technical result. Let $A_-$ and $A_+$ be segments in ${\mathbb {L}}_{-r}, {\mathbb {L}}_r$ with length $2r^{2/3}$, and centers $-{\mathbf {r}}+ (mr^{2/3}, -mr^{2/3})$ and ${\mathbf {r}}+ (-mr^{2/3}, mr^{2/3})$. Let $B={\mathbb {L}}_0\setminus \{(u_1,-u_1): |u_1|<tr^{2/3}\}$. We have the following lemma.

Lemma C.12

For each $\psi <1$ there exist $C,c>0$ such that when $|m|\le \psi r^{1/3}$, $t>0$ and $r\ge 1$ we have

$$\begin{aligned} {\mathbb {P}}(\sup _{u\in A_-, v\in B, w\in A_+} T_{u,v} + T_{v,w} - {\mathbb {E}}T_{u,w} \ge -ct^{2}r^{1/3})\le Ce^{-ct^3}. \end{aligned}$$

For the proof we would need to assume a certain largeness of t,  which then can made into any t by choosing c small enough.

Proof

Let $B_{+}$ denote the part of B contained in the fourth quadrant. By the directed nature of the model and symmetry, it suffices to prove

$$\begin{aligned} {\mathbb {P}}(\sup _{u\in A_-, v\in B_+, w\in A_+} T_{u,v} + T_{v,w} - {\mathbb {E}}T_{u,w} \ge -ct^{2}r^{1/3})\le Ce^{-ct^3}. \end{aligned}$$

We will rely on Theorem 4.2(ii). To this end, we first prove the following bounds for the corresponding expectations.

For $j\ge 0$, let $B_{j}$ denote the line segment joining $((t+j)r^{2/3}, -(t+j)r^{2/3})$ and $((t+j+1)r^{2/3}, -(t+j+1)r^{2/3})$. It follows from (12) and (13) that for some $c'>0$ depending on $\psi$, we have

$$\begin{aligned} \sup _{u\in A_-, v\in B_j, w\in A_+} {\mathbb {E}}T_{u,v} + {\mathbb {E}}T_{v,w} - {\mathbb {E}}T_{u,w} \le - c'(t+j)^2r^{1/3}, \end{aligned}$$

for all $j\ge 0$ with $t+j+|m|\le \frac{(1+\psi )}{2} r^{1/3}$.

Note that for j such that $\frac{(1+\psi )}{2} r^{1/3} < t+j+|m| \le r^{1/3}+1,$ Theorem 4.2(ii) is not directly applicable, since the needed slope condition is violated by points in $A_-$ and $B_j$. While, given the slack, there are several ways to address this, what we do is simply translate the points in $A_{-}$ and $A_+$ by ${\mp } c_0{\mathbf {r}}$ for a small $\psi$ dependent constant $c_0.$ Note that this can only increase the passage times and their expectations. Now, while $c_0$ is large enough so that the points in $A_- - {c_0{\mathbf {r}}}$ and $B_j$ (and similarly for $A_+ +{c_0{\mathbf {r}}}$) satisfy the slope condition, the following bound on their expectation still holds,

$$\begin{aligned} \sup _{u\in A_-, v\in B_j, w\in A_+} {\mathbb {E}}T_{u-{{c_0{\mathbf {r}}}},v} + {\mathbb {E}}T_{v,w+{c_0{\mathbf {r}}}} - {\mathbb {E}}T_{u,w} \le - c'(t+j)^2r^{1/3}. \end{aligned}$$

Then it follows from Theorem 4.2(ii) (applied to $A_-,B_j$ and $B_j,A_+$, or $A_- - {c_0{\mathbf {r}}},B_j$ and $B_j,A_+ + {c_0{\mathbf {r}}}$) that for $j\ge 0$ we have

$$\begin{aligned} {\mathbb {P}}\left( \sup _{u\in A_-, v\in B_j, w\in A_+} T_{u,v} + T_{v,w} - {\mathbb {E}}T_{u,w} \ge -c''t^2r^{1/3}\right) \le e^{-c''(t+j)^3}, \end{aligned}$$

for some $c''>0$ depending on $\psi .$ Note that above, in the case $t+j+|m|> \frac{(1+\psi )}{2} r^{1/3}$ we use Theorem 4.2(ii) to $A_- - {c_0{\mathbf {r}}},B_j$ and $B_j,A_+ + {c_0{\mathbf {r}}}$ along with the deterministic bound that for all ${u\in A_-, v\in B_j, w\in A_+}$

$$\begin{aligned}T_{u,v} + T_{v,w} \le T_{u-{c_0{\mathbf {r}}},v} + {T_{v,w+{c_0{\mathbf {r}}}}}. \end{aligned}$$

Taking a union bound over j gives the desired result. $\quad \square$

Lemma C.10 is an immediate consequence of Lemma C.12. We now complete the proof of Lemma C.11.

Recall from before that $A'$ is a segment of length $2r^{2/3}$ on ${\mathbb {L}}_0$ with midpoint $(mr^{2/3},-mr^{2/3})$, and $U_{m,\phi }$ is the parallelogram whose one pair of opposite sides of length $\phi r^{2/3}$ lie on the lines ${\mathbb {L}}_0$ and ${\mathbb {L}}_r$ respectively with respective midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$.

Proof of Lemma C.11. Fix $2\le j\le j_0$. It is clear that for even $\ell$, ${\mathcal {B}}_{\ell ,j}\subseteq {\mathcal {B}}_{j-1}$ and hence it is only required to prove the lemma for odd $\ell$. Let us assume that $\ell =2s+1$. Let $B_1$ and $B_2$ be the line segments given by the intersection of the lines $x+y=2s 2^{-j}(2r)$ and $x+y=(2s+2) 2^{-j}(2r)$ with the parallelogram $U_{m,h_{j-1}\phi }$, respectively. For any u, v with $d(u)<(2s+1) 2^{-j} 2r < d(v)$, let $T^*_{u,v}$ denote the maximum passage time of a path from u to v that intersects the line $x+y=(2s+1) 2^{-j} 2r$ at a point outside the parallelogram $U_{m,h_{j}\phi }$. We will divide the journey from $A'$ to ${\mathbb {L}}_{r,r^{2/3}}$ into three parts and accordingly we define the following events where instead of centering by the expectation, we use centering by certain approximations of the expectation that will be convenient to work with.

For a fixed number $K>0$ depending on $\psi$, let

$$\begin{aligned} {\mathcal {D}}_1&:=\{\sup _{u\in A', v\in B_1} T_{u,v} - S_1(v-u) - S_2(v-u) \ge (c_1\phi ^2+Kj)r^{1/3}\};\\ {\mathcal {D}}_2&:=\{\sup _{u\in B_2, v\in {\mathbb {L}}_{r,r^{2/3}}} T_{u,v} - S_1(v-u) - S_2(v-u) \ge (c_1\phi ^2+Kj)r^{1/3}\};\\ {\mathcal {D}}_3&:= \{\sup _{u\in B_1, v\in B_2} T^*_{u,v} - S_1(v-u) - S_2(v-u) \ge -(3c_1\phi ^2+3Kj)r^{1/3}\}, \end{aligned}$$

where $S_1, S_2:{\mathbb {Z}}^2 \rightarrow {\mathbb {R}}$ are the following functions: for any $u=(u_1,u_2)\in {\mathbb {Z}}_{\ge 0}^2$ we have $S_1(u)=(\sqrt{u_1}+\sqrt{u_2})^2$; and $S_2(u)=c_1d(u)$ if $|u_1-u_2| \ge (\psi +1)d(u)/2$, and $S_2(u)=0$ otherwise; for $u \not \in {\mathbb {Z}}_{\ge 0}^2$ we set $S_1(u)=S_2(u)=0$. The idea is that in these definitions of ${\mathcal {D}}_1, {\mathcal {D}}_2, {\mathcal {D}}_3$, the $S_1(v-u)$ term is approximately ${\mathbb {E}}T_{u,v}$ due to (12), and $S_2(v-u)$ is the penalty term when the slope of the line connecting u, v is too large or too small. Observe that for any $u'\in A', u\in B_1, v\in B_2, v'\in {\mathbb {L}}_{r,r^{2/3}}$, when r is large (depending on $\psi$) the slope of the line connecting $u'$ and $v'$ is between $\frac{1-\psi }{2}$ and $\frac{2}{1-\psi }$. Thus when $c_1$ is small enough (depending on $\psi$) we have

$$\begin{aligned} S_1(u-u') \!+\! S_1(v-u) + S_1(v'-v) \!+\! S_2(u-u') \!+\! S_2(v-u) + S_2(v'-v) \!<\! S_1(v'-u'). \end{aligned}$$

This is a simple consequence of super-additivity of passage times and the concavity of the function $(\sqrt{x}+\sqrt{y})^2$ on the line $x+y=1.$

Now, by (12), $|S_1(v'-u')-{\mathbb {E}}(T_{u',v'})|\le C'r^{1/3}$ where $C'$ depends only on $\psi ,$ and hence for $\phi , K$ sufficiently large, we have ${\mathcal {B}}_{2s+1,j}\cap {\mathcal {B}}^c_{j-1}\subset {\mathcal {D}}_1 \cup {\mathcal {D}}_2 \cup {\mathcal {D}}_3$, so

$$\begin{aligned} {\mathbb {P}}({\mathcal {B}}_{2s+1,j}\cap {\mathcal {B}}^c_{j-1})\le {\mathbb {P}}({\mathcal {D}}_1)+{\mathbb {P}}({\mathcal {D}}_2)+{\mathbb {P}}({\mathcal {D}}_3). \end{aligned}$$

We next bound ${\mathbb {P}}({\mathcal {D}}_1)$. When $s=0$ we have ${\mathbb {P}}({\mathcal {D}}_1)=0$, so below we assume $s\ge 1$. We divide each of $A'$ and $B_1$ into $O(2^{2j/3})$ and $O(2^{2j/3}\phi )$ many intervals of length $(s2^{-j}r)^{2/3}$. At this point we again run into a similar issue as in the proof of Lemma C.12, namely having to deal with pairs of points who do not satisfy the slope condition in Theorem 4.2(ii). To address this we use a similar strategy of backing up to perturb the end points slightly to make the theorem applicable, and in the process only increasing the weights by a tolerable amount.

For each pair of the intervals $I_1\subset A'$ and $I_2\subset B_1$, if the slope of the line connecting the midpoints of $I_1, I_2$ is between $\frac{1-\psi }{4}$ and $\frac{4}{1-\psi }$, Theorem 4.2(ii) is applicable. If the slope of the line connecting the midpoints of $I_1, I_2$ is not between $\frac{1-\psi }{4}$ and $\frac{4}{1-\psi }$, we take $I_1'=I_1-cs2^{-j}{\mathbf {r}}$. This makes the slope of any line connecting any point of $I_1$ with any point of $I_2$ between $c_2$ and $c_2^{-1}$, where $c_2$ is a constant depending only on c,  allowing us to apply Theorem 4.2(ii). Now for any $u\in I_1, v\in I_2$, $T_{u-cs2^{-j}{\mathbf {r}}, v} \ge T_{u,v}$, and $S_1(v-u)+S_2(v-u)>S_1(v-u+cs2^{-j}{\mathbf {r}})$ when c is small enough (depending on $\psi , c_1$). Together we conclude that

$$\begin{aligned} {\mathbb {P}}\left( \sup _{u\in I_1, v\in I_2} T_{u,v} - S_1(v-u) - S_2(v-u) \ge (c_1\phi ^2+Kj)r^{1/3} \right) \le e^{-2c\phi ^3-2cKj}. \end{aligned}$$

Then by a union bound of all such intervals, we conclude that if K is sufficiently large, ${\mathbb {P}}({\mathcal {D}}_1)\le 8^{-j}e^{-c\phi ^3}$; and by symmetry we also have ${\mathbb {P}}({\mathcal {D}}_2)\le 8^{-j}e^{-c\phi ^3}$.

Fix such a K and it remains to bound ${\mathbb {P}}({\mathcal {D}}_3)$. Recall that by our definition of $h_j$, we have $(h_{j}-h_{j-1})\phi r^{2/3}=a_{j}h_{j-1}2^{2j/3} \phi (2^{-j}r)^{2/3}$. Notice that for small enough $c_1$ and large enough $\phi$ (depending on K) and setting $a_{j}=2^{-j/3}$,

$$\begin{aligned} \dfrac{a^2_{j}h^2_{j-1}2^{4j/3}\phi ^2}{(3c_1\phi ^2+3Kj)2^{j/3}} \end{aligned}$$

can be made arbitrarily large. We divide each of $B_1$ and $B_2$ into $O(2^{2j/3}\phi )$ many intervals of length $(2^{-j}r)^{2/3}$. Again as above, for a pair of such intervals $I_1\subset B_1, I_2\subset B_2$, if the slope of the line connecting their midpoints is between $\frac{1-\psi }{4}$ and $\frac{4}{1-\psi }$, we invoke Lemma C.12. If the slope of the line connecting the midpoints of $I_1, I_2$ is not between $\frac{1-\psi }{4}$ and $\frac{4}{1-\psi }$, we take $I_1'=I_1-c2^{-j}r(1,1)$ and $I_2'=I_2+c2^{-j}r(1,1)$ which ensures that the slope for the midpoints of $I_1'$ and $I_2'$ is bounded away from 0 and $\infty$; and for any $u\in I_1, v\in I_2$, we have $T_{u-c2^{-j}r(1,1), v+c2^{-j}r(1,1)} \ge T_{u,v}$, and $S_1(v-u)+S_2(v-u)>S_1(v-u+2c2^{-j}r(1,1))$ when c is small enough (depending on $\psi , c_1$). This allows us to again invoke Lemma C.12 for $I_1', I_2'$. In summary, we get that

$$\begin{aligned}&{\mathbb {P}}\left( \sup _{u\in I_1, v\in I_2}T^*_{u,v} - S_1(v-u)-S_2(v-u)\right. \\&\quad \left. \ge -(3c_1\phi ^2+3Kj)2^{j/3} (2^{-j}r)^{1/3}\right) \le 2^{-4j/3}8^{-j}e^{-2c\phi ^3}. \end{aligned}$$

Then by a union bound for all such intervals, we have ${\mathbb {P}}({\mathcal {D}}_3)\le 8^{-j}e^{-c\phi ^3}$ when $\phi$ is large. This completes the proof by summing up ${\mathbb {P}}({\mathcal {D}}_1)$, ${\mathbb {P}}({\mathcal {D}}_2)$ and ${\mathbb {P}}({\mathcal {D}}_3)$. $\quad \square$

Finally we deduce Proposition 4.7 from Proposition C.8.

Proof of Proposition 4.7

Let $\phi$ be large. Applying Proposition C.8 and union bound allows us to handle all $\{(u_1,-u_1): |u_1|<\frac{\phi }{8} r^{2/3}\}$. The remaining cases will be handled if we show that for some small constant c,

$$\begin{aligned} {\mathbb {P}}\left( \sup _{u\in {\mathbb {L}}_0\setminus \{(u_1,-u_1): |u_1|<\frac{\phi }{8} r^{2/3}\}, v\in {\mathbb {L}}_{r,r^{2/3}}} T_{u,v}\ge 4r-c\phi ^{2}r^{1/3}\right) \le e^{-c\phi ^3}. \end{aligned}$$

For $j\ge 0$, let $B_{j}$ denote the line segment joining $((\frac{\phi }{8}+j)r^{2/3}, -(\frac{\phi }{8}+j)r^{2/3})$ and $((\frac{\phi }{8}+j+1)r^{2/3}, -(\frac{\phi }{8}+j+1)r^{2/3})$. To handle slope issues, as multiple times done, when $\frac{\phi }{8}+j> 0.9 r^{1/3}$ we also denote $B_j'=B_j-(c_0r,0)$, for some small enough $c_0$. It follows from (12) that for some constant $c'>0$,

$$\begin{aligned} \sup _{u\in B_j, v\in {\mathbb {L}}_{r,r^{2/3}}} {\mathbb {E}}T_{u,v} \le 4r- c'(\frac{\phi }{8}+j)^2r^{1/3}, \end{aligned}$$

for all $j\ge 0$ with $\frac{\phi }{8}+j\le 0.9 r^{1/3}$, and

$$\begin{aligned} \sup _{u\in B_j', v\in {\mathbb {L}}_{r,r^{2/3}}} {\mathbb {E}}T_{u,v} \le 4r- c'(\frac{\phi }{8}+j)^2r^{1/3}, \end{aligned}$$

for all j with $0.9r^{1/3}< \frac{\phi }{8}+j \le 2r^{1/3}$. Then it follows from Theorem 4.2(ii) (applied to $B_j, {\mathbb {L}}_{r,r^{2/3}}$ or $B_j', {\mathbb {L}}_{r,r^{2/3}}$) that for any j and some constant $c''>0$ we have

$$\begin{aligned} {\mathbb {P}}\left( \sup _{u\in B_j, v\in {\mathbb {L}}_{r,r^{2/3}}} T_{u,v} \ge 4r-c''\phi ^2r^{1/3}\right) \le e^{-c''(\phi +2j)^3}. \end{aligned}$$

Taking a union bound over j, and using symmetry, we get the desired result. $\quad \square$

1.4 Paths constrained to be in a parallelogram

Proof of Theorem 4.2(iii) is similar to that of Theorem 4.2(i); the proof proceeds through the same steps except versions of Lemma C.1 and Lemma C.3 involving constrained passage times need to be used, which in turn are established using Proposition C.9. We first need a one point estimate for the constrained passage times which is presented next.

Let $U_{t}$ denote the parallelogram with one pair of opposite sides of length $2tr^{2/3}$ on the lines ${\mathbb {L}}_0$ and ${\mathbb {L}}_r$ respectively with respective midpoints $u_0:=(-mr^{2/3},mr^{2/3})$ and ${\mathbf {r}}$. We have the following lemma.

Lemma C.13

For each $\psi <1$ and $t>0$, there exists $C,c>0$ depending on them, such that for all $|m|\le \psi r^{1/3}$, we have for all $x>0$ and $r\ge 1$

$$\begin{aligned} {\mathbb {P}}\left( T^{U_t}_{u_0,{\mathbf {r}}}-{\mathbb {E}}T_{u_0,{\mathbf {r}}} \le -xr^{1/3}\right) \le Ce^{-cx^{3/2}}. \end{aligned}$$

A couple of remarks are in order. Although we have not stated explicitly the dependence of C and c on t,  the reader might observe that the arguments in Lemma 6.9 indicates that for small t, we have ${\mathbb {E}}T^{U_{t}}_{u_0,{\mathbf {r}}}={\mathbb {E}}T_{u_{0},{\mathbf {r}}}-\Theta (t^{-1}r^{1/3})$ and $\text{ Var } T^{U_{t}}_{u_0,{\mathbf {r}}}=\Theta (t^{-1/2}r^{2/3})$ (see also [3, Prop 7.5 arXiv Version 2]). Further, the tail exponent above, though suffices, is not optimal. Since the initial posting of this paper, optimal exponents for deviations of constrained geodesic weights have been derived by the second named author along with Milind Hegde in [29].

Proof of Lemma C.13

Without loss of generality, let us assume that x is sufficiently large and fix it. For J, to be chosen appropriately later, let us set $(-mr^{2/3},mr^{2/3})=u_0, u_1, \ldots , u_{J}={\mathbf {r}}$ to be $J+1$ equispaced points on the straight line joining $(-mr^{2/3},mr^{2/3})$ and ${\mathbf {r}}$. By ignoring rounding issues, we also assume that each $u_i\in {\mathbb {Z}}^2$. By Theorem 4.1 and Proposition C.9 it follows that for each i we have

$$\begin{aligned}&{\mathbb {P}}\left( T^{U_{t}}_{u_i,u_{i+1}}-{\mathbb {E}}T_{u_i,u_{i+1}} \le -2\frac{x}{J^{2/3}} (r/J)^{1/3} \right) \\&\quad \le {\mathbb {P}}\left( T_{u_i,u_{i+1}}-{\mathbb {E}}T_{u_i,u_{i+1}} \le -2\frac{x}{J^{2/3}} (r/J)^{1/3} \right) \\&\quad + {\mathbb {P}}\left( T^{U_{t}}_{u_i,u_{i+1}}\ne T_{u_i,u_{i+1}}\right) \\&\quad \le Ce^{-cx^3/J^{2}}+Ce^{-cJ^2}\le Ce^{-cx^3/J^2}=Ce^{-cx^{3/2}} \end{aligned}$$

for some $C,c>0$ depending on t, where the penultimate inequality and the final equality is guaranteed by choosing $J=\delta x^{3/4}$ for some $\delta >0$ small and x sufficiently large. For the term ${\mathbb {P}}\left( T^{U_{t}}_{u_i,u_{i+1}}\ne T_{u_i,u_{i+1}}\right)$, we invoked Proposition C.9 for the parallelogram such that the midpoints of one pair of edges are $u_i$, $u_{i+1}$ respectively which are $\frac{J-i}{J} (-mr^{2/3},mr^{2/3}) + \frac{i}{J} {\mathbf {r}}$, and $\frac{J-i-1}{J} (-mr^{2/3},mr^{2/3}) + \frac{i+1}{J} {\mathbf {r}}$ and side lengths $2tr^{2/3}$. Namely, this is just $U_t$ between the lines $x+y=2ir/J$ and $x+y=2(i+1)r/J$. Thus the $\phi$ in the application of Proposition C.9 would be $2tJ^{2/3}$.

By taking a union bound over all i and using that

$$\begin{aligned} T^{U_t}_{u_0,{\mathbf {r}}}\ge \sum _{i} T^{U_{t}}_{u_i,u_{i+1}} \end{aligned}$$

it follows that for x sufficiently large we have

$$\begin{aligned} {\mathbb {P}}\left( T^{U_t}_{u_0,{\mathbf {r}}}-\sum _{i} {\mathbb {E}}T_{u_i,u_{i+1}} \le -2xr^{1/3}\right) \le Ce^{-cx^{3/2}}. \end{aligned}$$

It follows from (12) that ${\mathbb {E}}T_{u_i,u_{i+1}}\ge J^{-1}{\mathbb {E}}T_{u_0,{\mathbf {r}}} - CJ^{-1/3}r^{1/3}$ and hence

$$\begin{aligned} \sum _{i} {\mathbb {E}}T_{u_i,u_{i+1}}- {\mathbb {E}}T_{u_0,{\mathbf {r}}}\ge -CJ^{2/3}r^{1/3} \ge -xr^{1/3} \end{aligned}$$

for x sufficiently large and our choice of J. The last two displays, combined together, complete the proof. $\quad \square$

Using Lemma C.13 we can now prove an analogue of Lemma C.2 for constrained passage times. Recall the set-up of Lemmas C.1 and C.2: in particular recall that for m, h such that $|m|+h<\psi r^{1/3}$, $U_0$ denotes the parallelogram whose one pair of opposite sides are parallel to the lines $x+y=0$, have midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$ respectively and length $2hr^{2/3}$; and ${\hat{U}}$ is the sub-parallelogram of $U_0$ restricted to the strip $\{0\le x+y\le \frac{r}{16}\}$. Finally $U_*, U_{*,1}$ and $U_{*,2}$ denote the sub-parallelograms of $U_0$ restricted to the strips $\{0\le x+y\le \frac{9r}{5}\},$ $\{0\le x+y\le \frac{9r}{10}\},$ and $\{\frac{11r}{10}\le x+y\le 2r\}$ respectively.

We have the following result.

Lemma C.14

For each $\psi <1$ and $h>0$, there exists $C,c>0$ depending only on $\psi , h$ such that for $U_*$ as above with $|m|+{10}h\le \psi r^{1/3}$, we have for all $x>0$ and all $r\ge 1$

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u\in U_*} (T^{U_0}_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \le -xr^{1/3}\right) \le Ce^{-cx^{3/2}}. \end{aligned}$$

This proof will be almost identical to the proof of Lemma C.2, except that we shall use Lemma C.13 instead of Theorem 4.1 for the lower tail of constrained passage times.

Proof of Lemma C.14. We assume that r and x are large enough, since otherwise the result follows by taking C large and c small enough. In this proof we also use C and c to denote large and small constants depending on $\psi$ and m, and the specific values can change from line to line.

We first prove that, when $|m|+h\le \psi r^{1/3}$, we have

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u\in {\hat{U}}} (T^{U_0}_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \le -xr^{1/3}\right) \le Ce^{-cx^{3/2}}. \end{aligned}$$

(66)

We consider the same tree ${\mathcal {T}}$ as in the proof of Lemma C.1. Let ${\mathcal {A}}_j$ now denote the event that for all $u\in {\mathcal {T}}_j$ and for all $v\in {\mathcal {T}}_{j+1}$ such that the edge (u, v) is present in ${\mathcal {T}}$, we have

$$\begin{aligned} T^{U_0}_{v,u}\ge {\mathbb {E}}T_{v,u}- x2^{-(9j/10+10)}r^{1/3}. \end{aligned}$$

We claim that there exists $C,c>0$ such that for all x sufficiently large

$$\begin{aligned} {\mathbb {P}}(\cup _{j}{\mathcal {A}}_j^c) \le Ce^{-cx^{3/2}}. \end{aligned}$$

Notice that we took care in constructing ${\mathcal {T}}$ so that none of the vertices are too close to the boundary of $U_0$. Indeed, by our construction of ${\mathcal {T}}$, it follows that for all $u_j\in {\mathcal {T}}_j$ and for all $u_{j+1}\in {\mathcal {T}}_{j+1}$ such that $(u_j,u_{j+1})$ is present in ${\mathcal {T}}$, the distance from $u_{j+1}$ and $u_j$ to the boundary of $U_0$ in the anti-diagonal direction is at least $h(4^{(j+1)}+1)^{-1}r^{2/3}$. Thus, the parallelogram with one pair of parallel sides parallel to ${\mathbb {L}}_0$ with length $2h r^{2/3}(4^{(j+1)}+1)^{-1}$ and midpoints $u_{j+1}$ and $u_j$ respectively is contained in $U_0$. Hence Lemma C.13 applies (the slope condition is satisfied for all large r by (62), (63)) showing that for each $(u_{j+1},u_{j})$ as above we have

$$\begin{aligned} {\mathbb {P}}(T^{U_0}_{u_{j+1},u_{j}}-{\mathbb {E}}T_{u_{j+1},u_j}\le -y (8^{-j}r)^{1/3})\le Ce^{-cy^{3/2}} \end{aligned}$$

for some $C,c>0$ all $y>0$ and all r sufficiently large. Using this with $y=2^{j/10-10}x$ we obtain that

$$\begin{aligned} {\mathbb {P}}(T^{U_0}_{u_{j+1},u_j}-{\mathbb {E}}T_{u_{j+1},u_{j}}\le -x{2^{-(9j/10+10)}}r^{1/3})\le Ce^{-cx^{3/2}2^{3j/20}}. \end{aligned}$$

Now taking a union bound over all $32^{j+1}$ such edges, and then over all $j=0,1,2,\ldots , J-1$, we get ${\mathbb {P}}(\cup _{j}{\mathcal {A}}_j^c) \le Ce^{-cx^{3/2}}$.

To establish (67), it remains to show that on $\cap _{0\le j\le J} {\mathcal {A}}_j$, $\inf _{u\in {\hat{U}}} (T^{U_0}_{u,{\mathbf {r}}}-{\mathbb {E}}T_{u,{\mathbf {r}}}) \ge -xr^{1/3}$. Let’s take any $u\in {\hat{U}}$. Let $u_{J}$ be as in the proof of Lemma C.1 and $u_J, u_{J-1},\ldots ,u_0={\mathbf {r}}$ be the path from $u_J$ to ${\mathbf {r}}$ in ${\mathcal {T}}$. We have that $T^{U_0}_{u,{\mathbf {r}}} \ge \sum _{j=0}^{J-1} T^{U_0}_{u_{j+1},u_j}$. On $\cap _{0\le j\le J} {\mathcal {A}}_j$ we also have that

$$\begin{aligned} \sum _{j=0}^{J-1} \left( T^{U_0}_{u_{j+1},u_j}- {\mathbb {E}}T_{u_{j+1},u_j} \right) \ge -\frac{x}{2} r^{1/3}. \end{aligned}$$

It was already shown in (64) that

$$\begin{aligned} \sum _{j=0}^{J-1} {\mathbb {E}}T_{u_{j+1},u_j} \ge {\mathbb {E}}T_{u,{\mathbf {r}}}-\frac{x}{2}r^{1/3}. \end{aligned}$$

Adding these two inequalities up we get (67).

Finally, as in the proof of Lemma C.2, for each $0\le i \le 359,$ we consider the same $U_i$ and ${\hat{U}}_i$ and then the conclusion follows by applying the above result to each ${\hat{U}}_i$ and ${\mathbf {r}}$, and taking a union bound. $\quad \square$

Using the above, we have the constrained version of Lemma C.3. Recall the set-up of Lemma C.3 and in particular the parallelograms $U_0, U_{*,1}, U_{*,2}$. We have the following result.

Lemma C.15

For each $\psi <1$ and $h>0$, there exists $C,c>0$ depending only on $\psi , h$ such that for $U_0, U_{*,1}, U_{*,2}$ as above with $|m|+20h\le \psi r^{1/3}$, we have for all $x>0$ and $r\ge 1$

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u\in U_{*,1}, v\in U_{*,2}} (T^{U_0}_{u,v}-{\mathbb {E}}T_{u,v}) \le -xr^{1/3}\right) \le Ce^{-cx^{3/2}}. \end{aligned}$$

Proof

The proof of Lemma C.15 is identical to that of Lemma C.3 except that all passage times are now replaced by the passage times constrained in the parallelogram $U_0$ (e.g., in the definition of the events ${\mathcal {A}}_1$, ${\mathcal {A}}_2$) and the application of Lemma C.2 is replaced by that of Lemma C.14. The weaker probability bound of $Ce^{-cx^{3/2}}$ in the input (Lemma C.14) as opposed to $Ce^{-cx^{3}}$ in the former (Lemma C.2) is the reason why we obtain the same weaker bound here compared to Lemma C.3. $\quad \square$

We can now provide the proof of Theorem 4.2(iii). We first need an estimate about constrained passage times between point on the sides of the parallelogram U, whose one pair of sides lie on ${\mathbb {L}}_0$ and ${\mathbb {L}}_r$ with length $2r^{2/3}$ and midpoints $(mr^{2/3},-mr^{2/3})$ and ${\mathbf {r}}$ respectively. Let $A_l, A_r$ be the taller ‘left and right sides’, specifically, let $A_l$ be the collection of all vertices u, such that $u\in U$ while $u+(-1,1) \not \in U$; and let $A_r$ be the collection of all vertices u, such that $u\in U$ while $u+(1,-1) \not \in U$. We have the following result concerning passage times between vertices in $A_l$ or $A_r$.

Lemma C.16

For each $\psi <1$, there exists $C,c>0$ depending only on $\psi$ such that for all x, r sufficiently large, $|m|<\psi r^{1/3}$ and U as above we have

$$\begin{aligned} {\mathbb {P}}\left( \inf _{u,v\in A_*, d(v)>d(u)} (T_{u,v}^U - T_{u,v}) \le -xr^{1/3} \right) \le Ce^{-cx}. \end{aligned}$$

Here $A_*$ is $A_l$ or $A_r$.

Postponing the proof of the above, we first finish the proof Theorem 4.2(iii) using Theorem 4.2(i) and Lemma C.16.

Proof of Theorem 4.2(iii)

For $A_l, A_r$ as in Lemma C.16, we consider the intersection of the events

$$\begin{aligned} \inf _{u,v\in A_l, d(v)>d(u)} (T_{u,v}^U - T_{u,v})> -\frac{xr^{1/3}}{3},\quad \inf _{u,v\in A_r, d(v)>d(u)} (T_{u,v}^U - T_{u,v}) > -\frac{xr^{1/3}}{3}, \end{aligned}$$

and

$$\begin{aligned} \inf _{u,v\in U: d(v)-d(u)\ge \frac{r}{L}} (T_{u,v}-{\mathbb {E}}T_{u,v}) > -\frac{xr^{1/3}}{3}. \end{aligned}$$

By Theorem 4.2(i) and Lemma C.16, the probability of the intersection is at least $1-Ce^{-cx}$.

Now on the intersection of these events, we consider any $u, v\in U$ with $d(v)-d(u)\ge \frac{r}{L}$. Let $\gamma$ be the geodesic from u to v. If $\gamma \cap A_l=\emptyset$ let $\gamma '=\gamma$; otherwise, let $u_{l,-}, u_{l,+} \in \gamma \cap A_{\ell }$ be such that $d(u_{l,-})\le d(u') \le d(u_{l,+})$ for any $u'\in \gamma \cap A_l$. Consider the maximum weight path constrained within U from $u_{l,-}$ to $u_{l,+}$, and replace the part of $\gamma$ between $u_{l,-}, u_{l,+}$ by it, and we denote this new path from u to v by $\gamma '$. Next, from $\gamma '$ we construct $\gamma ''$ similarly, this time replacing the excursion outside $A_r$. Specifically, if $\gamma '\cap A_r=\emptyset$ let $\gamma ''=\gamma '$; otherwise, let $u_{r,-}, u_{r,+} \in \gamma '\cap A_{r}$ such that $d(u_{r,-})\le d(u') \le d(u_{r,+})$ for any $u'\in \gamma '\cap A_r$. Consider the maximum weight path constrained within U from $u_{r,-}$ to $u_{r,+}$, and replace the part of $\gamma '$ between $u_{r,-}, u_{r,+}$ by it, and we denote this new path from u to v as $\gamma ''$.

From this construction we have that $\gamma ''$ is constrained in U, so we have

$$\begin{aligned} \begin{aligned} T^U_{u,v} - {\mathbb {E}}T_{u,v}&\ge T^U_{u,v} - T_{u,v} + T_{u,v} - {\mathbb {E}}T_{u,v} \\&= (T^U_{u_{l,-}, u_{l,+}} - T_{u_{l,-}, u_{l,+}}) + (T^U_{u_{r,-}, u_{r,+}} - T_{u_{r,-}, u_{r,+}}) + (T_{u,v} - {\mathbb {E}}T_{u,v})\\&> -xr^{1/3}. \end{aligned} \end{aligned}$$

with probability at least $1-Ce^{-cx}$. Thus the conclusion follows. $\quad \square$

We finish with the proof of Lemma C.16.

Proof of Lemma C.16

We prove for $A_*=A_l$, and by symmetry the other case also follows. In this proof we let $C,c>0$ be constants depending on $\psi$, and their values can change from line to line.

We start by defining parallelograms “supported" on $A_l.$ Namely, for $i_1, i_2\in \llbracket 0, 2r \rrbracket$, $i_1<i_2$, we denote by $P_{i_1,i_2}$ the following parallelogram: it is contained in U, with one pair of sides on $x+y=i_1$, $x+y=i_2$ respectively with length $2^{1/3}(i_2-i_1)^{2/3}$; and it contains $i_2-i_1+1$ vertices in $A_l$ i.e., it is the set of all lattice points inside the $(i_2-i_1+1)$ by $2^{1/3}(i_2-i_1)^{2/3}$ continuous parallelogram supported on an interval of size $(i_2-i_1+1)$ on the taller side of $U_0$ thought of as a real interval (whose lattice version is $A_l$).

We next have a comparison estimate between constrained and unconstrained passage times between points in a family of such parallelograms defined in the following way.

Let $j_0$ be the maximum integer satisfying $2^{j_0}<r^{3/4}$. For each $j=1,2,\ldots , j_0$, consider parallelograms $P_{i_1,i_2}$ for $i_1, i_2\in \{0, 2^{-j}(2r), 2\times 2^{-j}(2r), 3\times 2^{-j}(2r), \ldots , 2r\}$, satisfying $i_2-i_1 \in \{3\times 2^{-j}(2r), 4\times 2^{-j}(2r), 5\times 2^{-j}(2r)\}$. Let ${\mathcal {B}}_{j}$ denote the event that for a fixed j and any parallelogram $U_0$ in this family, we have

$$\begin{aligned} \inf _{u\in U_{1},v\in U_{2}} (T_{u,v}^U-T_{u,v}) \ge -xr^{1/3} \end{aligned}$$

where $U_{1}$ and $U_{2}$ are defined as follows: suppose $U_0= P_{i_1,i_2}$ for some $i_1,i_2$, then $U_{1}$ and $U_{2}$ are $U_0$ restricted to the strips $\{i_1\le x+y \le \frac{2i_1+i_2}{3}\}$ and $\{\frac{i_1+2i_2}{3}\le x+y \le i_2\}$, respectively. Notice that for $u\in U_1, v\in U_2$ we have

$$\begin{aligned} \left\{ (T_{u,v}^U-T_{u,v}) \le -xr^{1/3}\right\} \subseteq \left\{ (T_{u,v}^U-{\mathbb {E}}T_{u,v}) \le -\frac{xr^{1/3}}{2}\right\} \cup \left\{ (T_{u,v}-{\mathbb {E}}T_{u,v}) \ge \frac{xr^{1/3}}{2}\right\} . \end{aligned}$$

For r sufficiently large, Theorem 4.2(ii) and Lemma C.15 apply (while the slope conditions are satisfied for a slightly larger $\psi$), and it follows that ${\mathbb {P}}({\mathcal {B}}_j^c)=O(2^{j}e^{-c2^{j}x})$, and ${\mathbb {P}}(\cup _j{\mathcal {B}}_j^c)=O(e^{-cx})$.

Given the above, consider any $u, v \in A_l$ with $d(u)<d(v)$. We now have the two following cases:

• We first assume that $d(v)-d(u)>2^{-j_0+1}(2r)$. Suppose that $2^{-j}(2r)\le d(v)-d(u)<2^{-j+1}(2r)$ for some $j\le j_0-1$. Then we can find some $i_1,i_2 \in \{0, 2^{-j-1}(2r), 2\times 2^{-j-1}(2r), 3\times 2^{-j-1}(2r), \ldots , 2r\}$, with $i_2-i_1 \in \{3\times 2^{-j-1}(2r), 4\times 2^{-j-1}(2r), 5\times 2^{-j-1}(2r)\}$, such that $i_1\le d(u)< i_1+ 2^{-j-1}(2r)< i_2- 2^{-j-1}(2r) \le d(v) < i_2$. Thus we can apply the above estimate to conclude that on $\cap _{j}{\mathcal {B}}_j$ we have

  $$\begin{aligned} \inf _{u,v\in A_l, d(v)-d(u)>2^{-j_0+1}(2r)} (T_{u,v}^U - T_{u,v}) \ge -xr^{1/3}. \end{aligned}$$

• Finally we consider those $u,v \in A_l$ with $0<d(v)-d(u)\le 2^{-j_0+1}(2r)\le 8r^{1/4}$. We have ${\mathbb {P}}(T_{u,v}^U-T_{u,v} \le -xr^{1/3})< {\mathbb {P}}(T_{u,v}>xr^{1/3}) <Ce^{-cxr^{1/4}}$ by Theorem 4.1. The number of such u, v is at most $O(r^{5/4})$ since $A_l$ contains O(r) points. By taking a union bound the conclusion follows.

$\quad \square$