We start with the positive temperature analog of Lemma 5.2.

######  Lemma 5.4.

For all $K>0$, $L\geq 2$ and $s\in(0,1)$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left[|\Gamma_{0}(s)|\geq
K(s(1-s))^{1/2}\right]>\exp(-\tfrac{1}{2}K^{2})\ \Big{|}\
\mathcal{L}^{\beta}(0,0;0,1)>L\right)<C\exp(-cK^{2}L^{1/2}).$

######  Remark 5.5.

Unlike the zero-temperature case, here the concentration is on an $O(1)$
scale. Some more work is needed to obtain an $L^{-1/4}$ scale concentration,
and we will turn to that shortly.

###### Proof of Lemma 5.4.

By the convolution formula, for any $K>0$ and $\varepsilon>0$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left[|\Gamma_{0}(s)|\geq
K\right]>\varepsilon\mid\mathcal{L}^{\beta}(0,0;0,1)>L\right)$
$\displaystyle=\mathbb{P}\left(\int_{|x|\geq
K}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;0,1)\,\mathrm{d}x>\varepsilon\mathcal{Z}(0,0;0,1)\
\Big{|}\ \mathcal{L}^{\beta}(0,0;0,1)>L\right)$
$\displaystyle\leq\frac{\mathbb{P}\left(\int_{|x|\geq
K}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;0,1)\,\mathrm{d}x>\varepsilon
e^{L-1/12}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)}$
$\displaystyle\leq\frac{2\cdot\mathbb{P}\left(\int_{x\geq
K}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;0,1)\,\mathrm{d}x>\tfrac{1}{2}\varepsilon
e^{L-1/12}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)},$
(5.3)

where the last inequality uses the reflection symmetry of
$\mathcal{Z}(0,0;\cdot,s)$ and $\mathcal{Z}(\cdot,s;0,1)$ and their
independence. By the shear invariance and independence of the same two
processes, we have that the following distributional equality holds as
processes in $x$ for every fixed $s$:

$\displaystyle\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;0,1)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\mathcal{Z}(0,0;x-K,s)\mathcal{Z}(x-K,s;0,1)e^{(s(1-s))^{-1}[(x-K)^{2}-x^{2}]}$
$\displaystyle\leq\mathcal{Z}(0,0;x-K,s)\mathcal{Z}(x-K,s;0,1)e^{-(s(1-s))^{-1}K^{2}},$

where the inequality is due to that $x\geq K$. Substituting this into (5.3)
yields that

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left[|\Gamma_{0}(s)|\geq
K\right]>\varepsilon\mid\mathcal{L}^{\beta}(0,0;0,1)>L\right)$
$\displaystyle\leq\frac{2\cdot\mathbb{P}\left(\int_{x\geq
0}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;0,1)\,\mathrm{d}x>\tfrac{1}{2}\varepsilon
e^{L-1/12+(s(1-s))^{-1}K^{2}}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)}$
$\displaystyle\leq\frac{2\cdot\mathbb{P}\left(\int_{\mathbb{R}}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;0,1)\,\mathrm{d}x>\tfrac{1}{2}\varepsilon
e^{L-1/12+(s(1-s))^{-1}K^{2}}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)}$
$\displaystyle=\frac{2\cdot\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L+(s(1-s))^{-1}K^{2}+\log(\varepsilon/2)\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)}$

We set $\varepsilon=\exp\left(-(s(1-s))^{-1}K^{2}/2\right)$ and invoke Theorem
4.1 to obtain that the previous display is upper bounded by

$\displaystyle C\exp\left(-cL^{1/2}(s(1-s))^{-1}K^{2}\right).$

Replacing $K$ by $K(s(1-s))^{1/2}$ completes the proof. ∎

We next derive a two-point estimate.

######  Lemma 5.6.

For all $K>0$, $L\geq 2$, and $0<s<t<1$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left[|\Gamma_{0}(s)-\Gamma_{0}(t)|>K(t-s)^{1/2}\right]>\exp(-\tfrac{1}{2}K^{2})\
\Big{|}\ \mathcal{L}^{\beta}(0,0;0,1)>L\right)<C\exp(-cK^{2}L^{1/2}).$

###### Proof.

We give the proof under the assumption that $t-s\in(0,\frac{1}{2})$, since
case where $t-s\in[\frac{1}{2},1]$ follows from Lemma 5.4 easily. Observe
that, for any $\varepsilon>0$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\Gamma_{0}(t)|>K\right)\geq\varepsilon\
\Big{|}\ \mathcal{L}^{\beta}(0,0;0,1)>L\right)$
$\displaystyle=\mathbb{P}\left(\int_{|x-y|\geq
K}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;y,t)\mathcal{Z}(y,t;0,1)\,\mathrm{d}x\,\mathrm{d}y>\varepsilon\mathcal{Z}(0,0;0,1)\
\Big{|}\ \mathcal{L}^{\beta}(0,0;0,1)>L\right)$
$\displaystyle\leq\frac{\mathbb{P}\left(\int_{|x-y|\geq
K}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;y,t)\mathcal{Z}(y,t;0,1)\,\mathrm{d}x\,\mathrm{d}y>\varepsilon
e^{L-1/12}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)}$
$\displaystyle\leq\frac{2\cdot\mathbb{P}\left(\int_{x-y\geq
K}\mathcal{Z}(0,0;x,s)\mathcal{Z}(x,s;y,t)\mathcal{Z}(y,t;0,1)\,\mathrm{d}x\,\mathrm{d}y>\tfrac{1}{2}\varepsilon
e^{L-1/12}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)},$
(5.4)

where the factor of 2 comes from removing the absolute value in the condition
under the supremum by a union bound and using that
$\mathcal{L}^{\beta}(x,s;y,t)\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}^{\beta}(-x,s;-y,t)$,
$\mathcal{L}^{\beta}(0,0;x,s)\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}^{\beta}(0,0;-x,s)$, and
$\mathcal{L}^{\beta}(y,t;0,1)\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}^{\beta}(-y,t;0,1)$ each as processes in the relevant
spatial variables, as well as the independence of all three processes on the
LHS of the equalities.

Now, using the stationarity (and independence) properties of
$\mathcal{L}^{\beta}$,

$\displaystyle\mathcal{L}^{\beta}(0,0;x,s)+\mathcal{L}^{\beta}(x,s;y,t)+\mathcal{L}^{\beta}(y,t;0,1)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}^{\beta}(-K,0;x-K,s)+\mathcal{L}^{\beta}(x-K,s;y,t)+\mathcal{L}^{\beta}(y,t;0,1)$
$\displaystyle\qquad+(t-s)^{-1}\left[(x-y-K)^{2}-(x-y)^{2}\right]$

as a process in $(x,y)$. Now since $x-y>K$, we see that

$\displaystyle(t-s)^{-1}\left[(x-y-K)^{2}-(x-y)^{2}\right]<-(t-s)^{-1}K^{2}.$

Thus the RHS of (5.4) is upper bounded by

$\displaystyle
2\cdot\frac{\mathbb{P}\left(\int_{x-y>K}\mathcal{Z}(-K,0;x-K,s)\mathcal{Z}(x-K,s;y,t)\mathcal{Z}(y,t;0,1)\,\mathrm{d}x\,\mathrm{d}y>\frac{1}{2}\varepsilon
e^{L-1/12+(t-s)^{-1}(K)^{2}}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L\right)}.$
(5.5)

Now using that

$\mathcal{L}^{\beta}(-K,0;0,1)-1/12=\log\int_{x,y\in\mathbb{R}}\mathcal{Z}(-K,0;x-K,s)\mathcal{Z}(x-K,s;y,t)\mathcal{Z}(y,t;0,1)\,\mathrm{d}x\,\mathrm{d}y,$

and that $\mathcal{L}^{\beta}(-K,0;0,1)\stackrel{{\scriptstyle
d}}{{=}}\mathcal{L}^{\beta}(0,0;0,1)-K^{2}$, it follows that (5.5) is upper
bounded by

$\displaystyle
2\cdot\frac{\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;0,1)>L+((t-s)^{-1}+1)K^{2}+\log(\varepsilon/2)\right)}{\mathbb{P}(\mathcal{L}^{\beta}(0,0;0,1)>L)}.$

Set $\varepsilon=\exp(-(t-s)^{-1}K^{2}/2)$. Applying Theorem 4.1 gives that
the previous display is upper bounded by

$\displaystyle C\exp\left(-c(t-s)^{-1}K^{2}L^{1/2}\right).$

Replacing $K$ by $K(t-s)^{1/2}$ completes the proof. ∎

##  6\. Proportionality and estimates on sums

In this section, we record a number of estimates on the sum of passage times
or free energies. Consider
$(s_{1},\cdots,s_{k})\in\mathring{\Lambda}_{k}([0,\infty))$ and
$\vec{y}\in\mathbb{R}^{k}$ for $k\in\mathbb{N}$. For the convenience of
notations we denote $s_{0}=y_{0}=0$, and adopt the shorthand
$\mathcal{L}^{\beta}=\mathcal{L}^{\beta}(0,0;0,s_{k})$ and
$\mathcal{L}^{\beta}_{i}=\mathcal{L}^{\beta}(s_{i-1},y_{i-1};s_{i},y_{i})$ for
each $1\leq i\leq k$. Estimates in this section provide control on tail
probabilities for the sum $\sum_{i}\mathcal{L}^{\beta}_{i}$ (such as Lemma
6.1, Lemma 6.4) or on the deviation of each $\mathcal{L}^{\beta}_{i}$ from
$(s_{i}-s_{i-1})L$ conditional on $\sum_{i}\mathcal{L}^{\beta}_{i}>L$ (Lemma
6.3).

We assume that $s_{k}<C_{0}$ for a large $C_{0}$. All the constants (within
this section) can depend on $k$ and $C_{0}$.

Our first statement bounds the upper tail of
$\sum_{i}\mathcal{L}^{\beta}_{i}$, in terms of the upper tail of
$\mathcal{L}^{\beta}$.

######  Lemma 6.1.

For any $L\geq 2$, any $M$, and any $\vec{z}\in\mathbb{R}^{k}$, if
$\min_{1\leq i\leq k}s_{i}-s_{i-1}>L^{-1}$, we have

$\mathbb{P}\left(\sup_{\vec{y}:\|\vec{y}-\vec{z}\|_{\infty}\leq
L^{-2}}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>M,\;\mathcal{L}^{\beta}(0,0;z_{k},s_{k})<M-C\log
L\right)<C\exp(-cL^{2}).$

###### Proof.

This is immediate in the case of $\beta=\infty$ by subadditivity, i.e.,
$\mathcal{L}^{\beta}(0,0;y_{k},s_{k})\geq\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}$
and the unconditional local fluctuation bound (Lemma 3.1) to obtain that
$|\mathcal{L}^{\beta}(0,0;y_{k},s_{k})-\mathcal{L}^{\beta}(0,0;z_{k},s_{k})|\leq
1$ with probability at least $1-\exp(-cL^{2})$.

We next turn to $\beta=1$. We note from the unconditional fluctuation bound
(Lemma 3.1) that, for any fixed intervals $I_{1},\ldots,I_{k}$, each with
length $L^{-2}$, and each $z_{i}\in I_{i}$,

$\inf_{\vec{y}\in\prod_{i=1}^{k}I_{i}}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>\sup_{\vec{y}:\|\vec{y}-\vec{z}\|_{\infty}\leq
cL^{-2}}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}-1,$

with probability at least $1-C\exp(-cL^{2})$. Under this event, and assuming
that $\sup_{\vec{y}:\|\vec{y}-\vec{z}\|_{\infty}\leq
L^{-2}}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>M$, we have (with $y_{k}=z_{k}$)

$\displaystyle\mathcal{L}^{\beta}(0,0;z_{k},s_{k})$
$\displaystyle\geq\log\int_{\prod_{i=1}^{k-1}I_{i}}\exp\left(\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}\right)\,\prod_{y=1}^{k-1}\,\mathrm{d}y_{i}$
$\displaystyle>M-1+(k-1)\log(L^{-2}).$

This completes the proof. ∎

By taking $L=M$ and using Theorem 2.13, we get the following.

######  Corollary 6.2.

For any large enough $L$, and any $\vec{z}\in\mathbb{R}^{k}$, if $\min_{1\leq
i\leq k}s_{i}-s_{i-1}>L^{-0.99}$,

$\mathbb{P}\left(\sup_{\vec{y}:\|\vec{y}-\vec{z}\|_{\infty}\leq
L^{-2}}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>L\right)<\exp\left(-\frac{4}{3}s_{k}^{-1/2}L^{3/2}+Cs_{k}^{-1/4}L^{3/4}\right).$

We note that the constraint $\min_{1\leq i\leq k}s_{i}-s_{i-1}>L^{-0.99}$ is
to ensure that the bound is much larger than $\exp(-cL^{2})$, the upper bound
in Lemma 6.1.

The following statement asserts that, conditional on the sum of independent
free energies being large, the individual terms are with high probability
proportionate to the total, up to a certain scale of fluctuation.

######  Lemma 6.3.

Fix each $y_{i}=0$. For any $L$ large enough, $K>C_{1}L^{3/8}$ for a constant
$C_{1}>0$ (so that $KL^{1/4}>C_{1}L^{5/8}$), if $\min_{1\leq i\leq
k}s_{i}-s_{i-1}>L^{-0.99}$, then for each $j=1,\ldots,k$,

$\displaystyle\mathbb{P}\left(\mathcal{L}^{\beta}_{j}<(s_{j}-s_{j-1})L-K(s_{j}-s_{j-1})^{1/2}L^{1/4}\
\Big{|}\ \sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>s_{k}L\right)<\exp(-cK^{2}).$

The above bound is optimal except for the fact that we require $K>CL^{3/8}$,
while it should hold for $K>C$; the loss is due to the non-optimal error term
in our tail bound Theorem 2.13. The constraint $\min_{1\leq i\leq
k}s_{i}-s_{i-1}>L^{-0.99}$ is due to applying Theorem 2.13 and Corollary 6.2.
We also mention the related result [Liu22b, Theorem 1.1] that implies that,
conditional on $\mathcal{L}(0,0;0,1)=L$, the weight of the geodesic up to
height $s$ is $sL$ plus a random term on scale $L^{1/4}$ which, when scaled to
be unit order, converges to a Gaussian random variable.

###### Proof of Lemma 6.3.

By the independence of these $\mathcal{L}_{j}^{\beta}$, without loss of
generality, we prove the estimate for $j=k$.

Let $Y=\sum_{i=1}^{k-1}\mathcal{L}^{\beta}_{i}$, and
$X=\mathcal{L}^{\beta}_{k}-(s_{k}-s_{k-1})L$. Then the probability in the LHS
equals

$\displaystyle\sum_{\ell=K(s_{k}-s_{k-1})^{1/2}L^{1/4}}^{\infty}\mathbb{P}\left(X\in-\ell+[-1,0]\
\Big{|}\ \sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>s_{k}L\right)$
$\displaystyle=\frac{\sum_{\ell=K(s_{k}-s_{k-1})^{1/2}L^{1/4}}^{\infty}\mathbb{P}\left(\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>s_{k}L\
\Big{|}\
X\in-\ell+[-1,0]\right)\cdot\mathbb{P}\left(X\in-\ell+[-1,0]\right)}{\mathbb{P}\left(\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>s_{k}L\right)}$
$\displaystyle\leq\frac{\sum_{\ell=K(s_{k}-s_{k-1})^{1/2}L^{1/4}}^{\infty}\mathbb{P}\left(Y>s_{k-1}L+\ell\
\Big{|}\
X\in-\ell+[-1,0]\right)\cdot\mathbb{P}\left(X\in-\ell+[-1,0]\right)}{\mathbb{P}\left(\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>s_{k}L\right)}.$
(6.1)

We can lower bound the denominator by
$\prod_{i=1}^{k}\mathbb{P}(\mathcal{L}^{\beta}_{i}>(s_{i}-s_{i-1})L)$, which,
by Theorem 2.13, is
$>\exp\left(-\frac{4}{3}s_{k}L^{3/2}-Cs_{k}^{1/2}L^{3/4}\right)$. For the
numerator, note that $Y$ is independent of $X$, so the summand indexed by
$\ell$ in (6.1) can be upper bounded, using Corollary 6.2 and Theorem 2.13 by

$\displaystyle\exp\left(-\tfrac{4}{3}s_{k-1}^{-1/2}\left(s_{k-1}L+\ell\right)^{3/2}-\tfrac{4}{3}(s_{k}-s_{k-1})^{-1/2}\left((s_{k}-s_{k-1})L-\ell\right)^{3/2}+Cs_{k}^{1/2}L^{3/4}\right)$
$\displaystyle\leq\exp\left(-\tfrac{4}{3}s_{k}L^{3/2}-c\ell^{2}(s_{k}-s_{k-1})^{-1}L^{-1/2}+Cs_{k}^{1/2}L^{3/4}\right).$

Substituting this into (6.1) and using that $K>C_{1}L^{3/8}$ for a large
enough constant $C_{1}$ (so that $cK^{2}$ is much larger than
$s_{k}^{1/2}L^{3/4}$) yields the claim. ∎

The following lemma provides control on the lower tail of the sum
$\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}$, conditional on the upper tail of
$\mathcal{L}^{\beta}$.

######  Lemma 6.4.

Assume that $\min_{1\leq i\leq k}s_{i}-s_{i-1}>t_{0}$ for some $t_{0}>0$. For
any $L>0$,

$\mathbb{P}\left(\inf_{\vec{y}:\|\vec{y}\|_{\infty}\leq L^{-1/4}\log
L}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}<L-L^{5/8}\log L\ \Big{|}\
\mathcal{L}^{\beta}>L\right)\\\ <C\exp(-c(\log L)^{2}),$

where the constants $C,c$ may depend on $t_{0}$.

###### Proof.

We assume that $L$ is large enough since otherwise the conclusion follows
obviously.

We start with $\beta=1$. We first claim that, conditional on
$\mathcal{L}^{\beta}>L$, it holds with probability at least $1-C\exp(-c(\log
L)^{2})$ that $\sup_{\vec{y}:\|\vec{y}\|_{\infty}\leq L^{-1/4}\log
L}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}>L-C\log L$. Indeed, suppose that this
inequality does not hold. In the $\beta=1$ case, we know from the quenched
transversal fluctuation estimate Lemma 5.4 that with conditional probability
at least $1-C\exp(-c(\log L)^{2})$,

$\displaystyle\exp(L)$ $\displaystyle<\left(1-e^{-cL^{-1/2}(\log
L)^{2}}\right)^{-1}\int_{[-L^{-1/4}\log L,L^{-1/4}\log
L]^{k-1}}\exp\left(\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}\right)\prod_{i=1}^{k-1}\,\mathrm{d}y_{i},$

where $y_{k}=0$ in the integral. If $\sup_{\vec{y}:\|\vec{y}\|_{\infty}\leq
L^{-1/4}\log L}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}<L-C\log L$, the RHS is
upper bounded by

$\displaystyle CL^{1/2}(\log L)^{-2}(2L^{-1/4}\log L)^{k}\exp\left(L-C\log
L\right)\ll\exp(L),$

which is a contradiction.

In the $\beta=\infty$ case, by Lemma 5.2, with conditional probability at
least $1-C\exp(-c(\log L)^{2})$ it holds that
$\mathcal{L}^{\beta}=\sup_{\vec{y}:\|\vec{y}\|_{\infty}\leq L^{-1/4}\log
L}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}$, which implies our claim since we
have conditioned on $\mathcal{L}^{\beta}>L$.

Next, we know from Lemma 3.1 that, with (unconditional) probability at least
$1-C\exp(-cL^{3/2}\log L)$,

$\sup_{\|\vec{y}\|_{\infty}\leq L^{-1/4}\log
L}\sum_{i=1}^{k}\left|\mathcal{L}^{\beta}_{i}-\mathcal{L}^{\beta}(0,s_{i-1};0,s_{i})\right|\leq\tfrac{1}{2}L^{3/4}(\log
L)^{1/2}\cdot(L^{-1/4}\log L)^{1/2}=\tfrac{1}{2}L^{5/8}\log L.$

By Theorem 2.13 we know $\mathbb{P}(\mathcal{L}^{\beta}>L)>\exp(-CL^{3/2})$,
the previous bound also holds conditionally on $\mathcal{L}^{\beta}>L$ with
probability at least $1-C\exp(-cL^{3/2}\log L)$. This completes the proof. ∎

##  7\. Concentration of polymers

In this section, we mainly work with polymers, i.e., set $\beta=1$. We prove
the following fact: the polymer measure at a given height $s$ is a Dirac mass
spread out over an $L^{-1/2}$ interval around a random location. More
specifically, we define

$\pi(s)=\operatorname*{argmax}_{x\in\mathbb{R}}\mathcal{L}^{\beta}(0,0;x,s)+\mathcal{L}^{\beta}(x,s;0,1),$

for any $s\in(0,1)$. Note that this is a function purely of the environment.
The main result of this section asserts that under the polymer measure
$\boldsymbol{\mathscr{P}}$, the polymer path at each height in
$[L^{-5/8},1-L^{-5/8}]$ stays within $O(L^{-1/2}\log L)$ of $\pi$ with high
probability.

######  Proposition 7.1.

There exists $M_{0}>0$ such that for any $L\geq 2$, $M>M_{0}$ and
$s\in[L^{-5/8},1-L^{-5/8}]$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\pi(s)|>ML^{-1/2}\log
L\right)>L^{-2M}\ \Big{|}\
\mathcal{L}^{\beta}(0,0;0,1)>L\right)<C\exp(-c(\log(L))^{2}).$

In the rest of this section, we shall always take $s\in[L^{-5/8},1-L^{-5/8}]$.
For the convenience of notations, we adopt the notation $s_{1}=s$ and
$s_{2}=1-s$, $\mathcal{L}^{\beta}_{1}(x)=\mathcal{L}^{\beta}(0,0;x,s)$ and
$\mathcal{L}^{\beta}_{2}(x)=\mathcal{L}^{\beta}(x,s;0,1)$, and
$\mathcal{L}^{\beta}=\mathcal{L}^{\beta}(0,0;0,1)$.

###  7.1. Global to two-segments conditioning

In the upcoming proof of Proposition 7.1, it will be useful to go from
conditioning on $\mathcal{L}^{\beta}>L$ to conditioning on
$\mathcal{L}^{\beta}_{i}(x)\in(h_{i},h_{i}+\,\mathrm{d}h_{i})$ for each
$i=1,2$, with $x,h_{1},h_{2}$ belonging to a set of nice values. The following
is the general statement that allows us to do this.

Let $\mathrm{Val}\subseteq\mathbb{R}^{2}$ be defined by

$\displaystyle\mathrm{Val}=\left\\{(h_{1},h_{2})\in\mathbb{R}^{2}:L-(\log
L)^{2}<h_{1}+h_{2}<L+(\log L)^{2},h_{i}>s_{i}L-s_{i}^{1/2}L^{5/8}\log L\text{
for }i=1,2\right\\}.$

######  Lemma 7.2 (Global to two-segments conditioning).

For any $L>0$, $0<K\leq\log L$, and any event $A$,

$\mathbb{P}\left(A\mid\mathcal{L}^{\beta}>L\right)<\max_{\begin{subarray}{c}x\in
L^{-2}\mathbb{Z},|x|\leq
KL^{-1/4}\end{subarray}}\sup_{(h_{1},h_{2})\in\mathrm{Val}}\mathbb{P}\left(A\
\Big{|}\
\mathcal{L}^{\beta}_{i}(x)\in(h_{i},h_{i}+\,\mathrm{d}h_{i}),i=1,2\right)\\\
\times C\exp(C(\log L)^{2}L^{1/2})+C\exp(-cK^{2}).$

###### Proof.

We assume that $K>1$ since otherwise the conclusion follows trivially by
setting $C$ large.

We observe from Lemma 5.4 that, with probability at least $1-C\exp(-cK^{2})$,
conditionally on $\mathcal{L}^{\beta}>L$,

$\displaystyle\int_{[-KL^{-1/4},KL^{-1/4}]}\exp\left((\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)\right)\,\mathrm{d}x$
$\displaystyle\geq\left(1-\exp\left(-\tfrac{1}{2}K^{2}L^{-1/2}\right)\right)\exp(L)$
$\displaystyle\geq\exp\left(L+\log\left(\tfrac{1}{2}K^{2}L^{-1/2}\right)\right),$

which implies that

$\displaystyle\mathbb{P}\left(\max_{[-KL^{-1/4},KL^{-1/4}]}\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}\geq
L-\log L\ \Big{|}\ \mathcal{L}^{\beta}>L\right)>1-C\exp(-cK^{2}).$

This with unconditional local fluctuation estimates (Proposition 2.16) implies
that

$\mathbb{P}\left(\max_{x\in L^{-2}\mathbb{Z},|x|\leq
KL^{-1/4}}(\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)\leq L-(\log
L)^{2}\ \Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-cL^{2}(\log
L)^{4})+C\exp(-cK^{2}).$

Also, for each $|x|\leq KL^{-1/4}$, by Theorem 4.1 and Lemma 6.1,

$\mathbb{P}\left((\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)\geq
L+(\log L)^{2}\mid\mathcal{L}^{\beta}>L\right)<C\exp(-c(\log L)^{2}L^{1/2});$

and by proportionality (Lemma 6.3), we have

$\mathbb{P}\left(\mathcal{L}^{\beta}_{i}(x)\leq s_{i}L-s_{i}^{1/2}L^{5/8}\log
L,\;(\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)>L-(\log L)^{2}\
\Big{|}\ \mathcal{L}^{\beta}>L\right)\\\
<\mathbb{P}\left(\mathcal{L}^{\beta}_{i}(x)\leq s_{i}L-s_{i}^{1/2}L^{5/8}\log
L\ \Big{|}\ (\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)>L-(\log
L)^{2}\right)\frac{\mathbb{P}\left((\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)>L-(\log
L)^{2}\right)}{\mathbb{P}(\mathcal{L}^{\beta}>L)}\\\ <C\exp(-c(\log
L)^{2}L^{3/4}),$

for each $i=1,2$, where the ratio is bounded using Corollary 6.2 and Theorem
2.13. Combining the above three estimates, we see that with probability
$>1-C\exp(-cK^{2})$, there exists one $x\in L^{-2}\mathbb{Z},|x|\leq
KL^{-1/4}$, such that $\mathrm{ValueCtrl}_{x}$ holds; where

$\displaystyle\mathrm{ValueCtrl}_{x}=\left\\{L-(\log
L)^{2}<(\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)<L+(\log
L)^{2},\mathcal{L}^{\beta}_{i}(x)>s_{i}L-s_{i}^{1/2}L^{5/8}\log L\text{ for
}i=1,2\right\\}.$

In other words, we have

$\mathbb{P}(\mathrm{ValueCtrl}^{c}\mid\mathcal{L}^{\beta}>L)<C\exp(-cK^{2}),$

where $\mathrm{ValueCtrl}=\cup_{x\in L^{-2}\mathbb{Z},|x|\leq
KL^{-1/4}}\mathrm{ValueCtrl}_{x}$. Now we have

$\mathbb{P}(A\mid\mathcal{L}^{\beta}>L)<\mathbb{P}(A\cap\mathrm{ValueCtrl}\mid\mathcal{L}^{\beta}>L)+C\exp(-cK^{2}).$

Note that the first term in the RHS is bounded by

$\max_{\begin{subarray}{c}x\in L^{-2}\mathbb{Z},|x|\leq
KL^{-1/4}\end{subarray}}\sup_{(h_{1},h_{2})\in\mathrm{Val}}\mathbb{P}\left(A\
\Big{|}\
\mathcal{L}^{\beta}_{i}(x)\in(h_{i},h_{i}+\,\mathrm{d}h_{i}),i=1,2\right)\frac{\mathbb{P}(\mathrm{ValueCtrl})}{\mathbb{P}(\mathcal{L}^{\beta}>L)}.$

By Lemma 6.1 and Theorem 4.1, we have

$\mathbb{P}(\mathrm{ValueCtrl})\leq\sum_{x\in L^{-2}\mathbb{Z},|x|\leq
KL^{-1/4}}\mathbb{P}(\mathrm{ValueCtrl}_{x})\\\ \leq\sum_{x\in
L^{-2}\mathbb{Z},|x|\leq
KL^{-1/4}}\mathbb{P}((\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)>L-(\log
L)^{2})<C\exp(C(\log L)^{2}L^{1/2})\mathbb{P}(\mathcal{L}^{\beta}>L).$

Combining the last three displays leads to the conclusion. ∎

###  7.2. Random location and concentration

We next use Lemma 7.2 to control the location of $\pi(s)$, as well as prove
Proposition 7.1.

The next result asserts that $\pi(s)$ is of order $L^{-1/4}$.

######  Proposition 7.3.

For any $L$ large enough, $s\in[L^{-5/8},1-L^{-5/8}]$, and $0<K\leq\log(L)$,

$\displaystyle\mathbb{P}\left(|\pi(s)|>KL^{-1/4}\mid\mathcal{L}^{\beta}>L\right)<C\exp(-cK^{2}).$

The general idea to prove Proposition 7.3 is to (1) upper bound
$\max_{(-KL^{-1/4},KL^{-1/4})^{c}}\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}$
conditional on $\mathcal{L}^{\beta}_{1}(x)$ and $\mathcal{L}^{\beta}_{2}(x)$
for some $x=O(KL^{-1/4})$, using Corollary 2.21; and (2) connect the
conditioning $\mathcal{L}^{\beta}>L$ and the conditioning on the values of
$\mathcal{L}^{\beta}_{1}(x)$ and $\mathcal{L}^{\beta}_{2}(x)$, using Lemma
7.2.

###### Proof of Proposition 7.3.

We assume that $K>1$ since otherwise the conclusion follows trivially by
setting $C$ large.

As in the proof of Lemma 7.2, we have

$\displaystyle\mathbb{P}\left(\max\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}\geq
L-\log L\ \Big{|}\ \mathcal{L}^{\beta}>L\right)>1-C\exp(-c(\log L)^{2}).$
(7.1)

We also have

$\mathbb{P}\left(\max_{(-(s_{1}\wedge s_{2})L^{1/2}/2,(s_{1}\wedge
s_{2})L^{1/2}/2)^{c}}\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}\geq(1-\tfrac{1}{10}(s_{1}\wedge
s_{2}))L\right)\\\ <C\exp(-c(s_{1}\wedge
s_{2})L^{3/2})\mathbb{P}(\mathcal{L}^{\beta}>L),$ (7.2)

by upper bounding the LHS using Corollary 6.2 plus shear invariance and a
union bound, and lower bounding $\mathbb{P}(\mathcal{L}^{\beta}>L)$ using
Theorem 2.13. Combining (7.1) and (7.2) implies that

$\mathbb{P}\left(|\pi(s)|>KL^{-1/4}\mid\mathcal{L}^{\beta}>L\right)<C\exp(-c(\log
L)^{2})\\\ +\mathbb{P}\left(\max_{[-\frac{1}{2}(s_{1}\wedge
s_{2})L^{1/2},-KL^{-1/4}]\cup[KL^{-1/4},\frac{1}{2}(s_{1}\wedge
s_{2})L^{1/2}]}\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}\geq L-\log L\
\Big{|}\ \mathcal{L}^{\beta}>L\right).$ (7.3)

By Corollary 2.21 with $a=KL^{-1/4}$, for any $|x|\leq KL^{-1/4}/2$, and
$(h_{1},h_{2})\in\mathrm{Val}$, we have

$\mathbb{P}\Bigg{(}\max_{[-\frac{1}{2}(s_{1}\wedge
s_{2})L^{1/2},-KL^{-1/4}]\cup[KL^{-1/4},\frac{1}{2}(s_{1}\wedge
s_{2})L^{1/2}]}\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}\geq L-\log L\\\
\ \Big{|}\
\mathcal{L}^{\beta}_{i}(x)\in(h_{i},h_{i}+\,\mathrm{d}h_{i}),i=1,2\Bigg{)}<C\exp(-cKL^{3/4})$

Thus by Lemma 7.2, the second term in the RHS of (7.3) can be bounded by
$C\exp(-cK^{2})$, and the conclusion follows. ∎

We next finish proving the polymer concentration result.

###### Proof of Proposition 7.1.

Consider the event $\mathrm{Tent}$ defined by

$\mathrm{Tent}:=\left\\{\max_{|z-\pi(s)|\geq ML^{-1/2}\log
L}\mathcal{L}^{\beta}_{1}(z)+\mathcal{L}^{\beta}_{2}(z)-(L-2L^{1/2}|z-\pi(s)|)<0\right\\}$

To understand this definition, recall that we expect each of
$\mathcal{L}^{\beta}_{1}$ and $\mathcal{L}^{\beta}_{2}$ to essentially adopt
tent shapes under the conditioning $\mathcal{L}^{\beta}>L$, where the tents
each have slope approximately $\pm 2L^{1/2}$. Thus the sum
$\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}$ can be expected to be a line
of slope $\pm 4L^{1/2}$ up to random fluctuations; in the definition of the
event, we have reduced the slope magnitude by $1/2$ for the benefit of
ignoring the random fluctuation.

Now, on $\mathrm{Tent}$,

$\displaystyle\int_{|x-\pi(s)|\geq ML^{-1/2}\log
L}\exp\left((\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2})(x)\right)\,\,\mathrm{d}x\leq\int_{|x-\pi(s)|\geq
ML^{-1/2}\log L}\exp\left(L-2L^{1/2}|x-\pi(s)|\right)\,\,\mathrm{d}x$
$\displaystyle=2\int_{ML^{-1/2}\log
L}^{\infty}\exp\left(L-2L^{1/2}x\right)\,\,\mathrm{d}x=L^{-1/2}\exp\left(L-2M\log
L\right),$

which implies that, on $\mathrm{Tent}$ and $\mathcal{L}^{\beta}>L$,

$\displaystyle\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\pi(s)|>ML^{-1/2}\log
L\right)<L^{-2M}.$

Thus our task is now to upper bound
$\mathbb{P}(\mathrm{Tent}^{c}\mid\mathcal{L}^{\beta}>L)$. We note that by
Proposition 7.3,

$\mathbb{P}\left(|\pi(s)|>\log(L)L^{-1/4}\mid\mathcal{L}^{\beta}>L\right)<C\exp(-c(\log
L)^{2}).$ (7.4)

Then it remains to upper bound
$\mathbb{P}(\mathrm{Tent}^{c}\cap\\{|\pi(s)|\leq\log(L)L^{-1/4}\\}\mid\mathcal{L}^{\beta}>L)$.

We define the event $\mathrm{Tent}_{+}$ by

$\mathrm{Tent}_{+}:=\left\\{\max_{|z|\geq(s_{1}\wedge
s_{2})L^{1/2}/3}\mathcal{L}^{\beta}_{1}(z)+\mathcal{L}^{\beta}_{2}(z)-(L-3L^{1/2}|z|)<0\right\\}.$

Similar to (7.2) in the proof of Proposition 7.3, we have

$\mathbb{P}(\mathrm{Tent}_{+}^{c})<C\exp(-c(s_{1}\wedge
s_{2})L^{3/2})\mathbb{P}(\mathcal{L}^{\beta}>L),$

by upper bounding the LHS using Corollary 6.2 plus shear invariance and a
union bound, and lower bounding $\mathbb{P}(\mathcal{L}^{\beta}>L)$ using
Theorem 2.13. Then we have
$\mathbb{P}(\mathrm{Tent}_{+}^{c}\mid\mathcal{L}^{\beta}>L)<C\exp(-c(s_{1}\wedge
s_{2})L^{3/2})$. It now suffices to upper bound
$\mathbb{P}(\mathrm{Tent}^{c}\cap\mathrm{Tent}_{+}\cap\\{|\pi(s)|\leq\log(L)L^{-1/4}\\}\mid\mathcal{L}^{\beta}>L)$,
and we apply Lemma 7.2 for this.

Take $x\in L^{-2}\mathbb{Z}$, $|x|\leq\log(L)L^{-1/4}$ and
$(h_{1},h_{2})\in\mathrm{Val}$, and consider

$\mathbb{P}\left(\mathrm{Tent}^{c}\cap\mathrm{Tent}_{+}\cap\\{|\pi(s)|\leq\log(L)L^{-1/4}\\}\
\Big{|}\
\mathcal{L}^{\beta}_{i}(x)\in(h_{i},h_{i}+\,\mathrm{d}h_{i}),i=1,2\right).$
(7.5)

Assuming that for each $i=1,2$,
$\mathcal{L}^{\beta}_{i}(x)\in(h_{i},h_{i}+\,\mathrm{d}h_{i})$, and

$\max_{ML^{-1/2}\log(L)/5\leq|y|\leq(s_{1}\wedge
s_{2})L^{1/2}/2}\mathcal{L}^{\beta}_{i}(x+y)+\tfrac{3}{2}|y|(h_{i}/s_{i})^{1/2}\leq
h_{i},$

and $\mathrm{Tent}_{+}\cap\\{|\pi(s)|\leq\log(L)L^{-1/4}\\}$, we must have
$|\pi(s)-x|<ML^{-1/2}\log(L)/5$, and $\mathrm{Tent}$ holds. Therefore by
Corollary 2.21 (with $a=ML^{-1/2}\log L$), we have that (7.5) is bounded by
$C\exp(-cML^{1/2}\log L)+C\exp(-cL^{3/2})$. Then by Lemma 7.2, and taking
$M_{0}$ large enough, we have
$\mathbb{P}(\mathrm{Tent}^{c}\mid\mathcal{L}^{\beta}>L)<C\exp(-cL^{1/2}\log
L)+C\exp(-c(\log L)^{2})$. Thus the conclusion follows. ∎

##  8\. Tightness for polymers

We prove the $\beta=1$ case of Proposition 5.1 in this section. We also denote
$\mathcal{L}^{\beta}=\mathcal{L}^{\beta}(0,0;0,1)$ for simplicity of
notations.

The main task is to prove the following two points estimate, which refines
Lemma 5.6.

######  Proposition 8.1.

For all $L\geq 2$, $0<s<t<1$ and $K>0$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\Gamma_{0}(t)|>K(t-s)^{1/11}L^{-1/4}\right)>L^{-K}\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-c(K\wedge\log L)^{2}).$

Compared to the zero temperature setting (i.e., Proposition 5.3), we weaken
our demand to a Hölder $\frac{1}{11}-$ bound instead of $\frac{1}{2}-$, due to
technical reasons which will be clear from its proof.

Proposition 8.1 immediately implies the following.

######  Corollary 8.2.

For all $L\geq 2$, $K>0$ and $0<s<t<1$,

$\displaystyle\mathbb{E}\left[\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\Gamma_{0}(t)|>K(t-s)^{1/11}L^{-1/4}\right)\
\Big{|}\ \mathcal{L}^{\beta}>L\right]<C\exp(-c(K\wedge\log L)^{2}).$

######  Remark 8.3.

As indicated in the introduction, from these tightness results and the polymer
concentration of Proposition 7.1, one can define a random backbone
$\tilde{\pi}$, by e.g., taking $\tilde{\pi}(s)=\pi(s)$ for each $s\in[0,1]\cap
L^{-10}\mathbb{Z}$, and linearly interpolate between them. Then with
probability $>1-C\exp(-c(\log L)^{2})$, $\Gamma_{0}$ is within distance
$L^{-1/2}(\log L)^{2}$ from $\tilde{\pi}$ at each $s\in[0,1]\cap
L^{-10}\mathbb{Z}$, by Proposition 7.1; and between any two points,
$\Gamma_{0}$ can deviate at most $L^{-1}$, by Corollary 8.2. Therefore (with
the same probability) $\Gamma_{0}$ is within distance $L^{-1/2}(\log L)^{2}$
from the backbone $\tilde{\pi}$ throughout $[0,1]$.

###### Proof of the $\beta=1$ case of Proposition 5.1.

We recall that tightness on $\mathcal{C}([0,1],\mathbb{R})$ follows by
establishing a uniform modulus of continuity as well as one-point tightness.
Since our processes are fixed at 0 at heights 0 and 1, the former also implies
the latter, so we only need to establish a uniform modulus of continuity
bound.

With Corollary 8.2, via a union bound over all $s=2^{-i}j$, $t=2^{-i}(j+1)$,
$0<s<t<1$, with $i,j\in\mathbb{Z}$, we have the following. For any $K>0$, with
probability $>1-C\exp(-c(K\wedge\log L)^{2})$ conditional on
$\mathcal{L}^{\beta}>L$,

$\sup_{t_{0}\leq s<t\leq
1-t_{0}}|\Gamma_{0}(s)-\Gamma_{0}(t)|(t-s)^{-1/12}\leq KL^{-1/4}.$

Note that the exponent is $\frac{1}{12}$ in contrast to the $\frac{1}{11}$
present in Corollary 8.2; this is simply so that the union bound over all the
scales can be performed. This completes the proof. ∎

To prove Proposition 8.1, the key idea is to upgrade Lemma 5.6 using the
concentration result Proposition 7.1. It allows us to essentially say that
(with high probability under $\mathbb{P}$) if the law of $\Gamma_{0}(s)$ under
$\boldsymbol{\mathscr{P}}$ assigns a probability greater than $\varepsilon$
(for a carefully chosen small $\varepsilon$) to the event of large transversal
fluctuation, it must assign a close to $1$ probability to the event of having
at least say half of the same transversal fluctuation. The latter event’s
probability is bounded by Lemma 5.6. Note that in this argument it is actually
not important that the interval around which most of the mass is spread is
centered around $\pi(s)$, only that the interval is small.

We start by first upgrading the one-point estimate, at least $L^{-5/8}$ away
from the ends.

######  Proposition 8.4.

For all $K>0$, $L\geq 2$, and $s\in[L^{-5/8},1-L^{-5/8}]$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)|\geq
KL^{-1/4}\right)>L^{-K}\ \Big{|}\
\mathcal{L}^{\beta}>L\right)<C\exp(-c(K\wedge\log L)^{2}).$

###### Proof.

We assume that $K$ is large enough since otherwise we just choose $C$ large to
make the estimate hold. We also assume that $L$ is large enough since
otherwise the conclusion follows from Lemma 5.4.

Let $X(s,K)=\boldsymbol{\mathscr{P}}(|\Gamma_{0}(s)|\geq KL^{-1/4})$. By
Proposition 7.1, with probability $\geq 1-C\exp(-c(\log L)^{2})$ conditional
on $\mathcal{L}^{\beta}>L$ we have

$\displaystyle X(s,K)$
$\displaystyle\leq\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\pi(s)|\geq
KL^{-1/2}\log L\right)+\mathbbm{1}_{|\pi(s)|\geq\frac{1}{2}KL^{-1/4}}$
$\displaystyle\leq L^{-2K}+\mathbbm{1}_{|\pi(s)|\geq\frac{1}{2}KL^{-1/4}},$

and

$\displaystyle X(s,\tfrac{1}{4}K)$
$\displaystyle\geq\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\pi(s)|\leq
KL^{-1/2}\log L\right)\mathbbm{1}_{|\pi(s)|\geq\frac{1}{2}KL^{-1/4}}$
$\displaystyle\geq(1-L^{-2K})\mathbbm{1}_{|\pi(s)|\geq\frac{1}{2}KL^{-1/4}}.$

Here we used that $KL^{-1/2}\log L<\frac{1}{4}KL^{-1/4}$. These two bounds
show that

$\displaystyle X(s,K)>L^{-2K}\implies|\pi(s)|\geq\tfrac{1}{2}KL^{-1/4}\implies
X(s,\tfrac{1}{4}K)\geq 1-L^{-2K},$

and thus

$\mathbb{P}\left(X(s,K)>L^{-2K}\ \Big{|}\
\mathcal{L}^{\beta}>L\right)<\mathbb{P}\left(X(s,\tfrac{1}{4}K)\geq 1-L^{-2K}\
\Big{|}\ \mathcal{L}^{\beta}>L\right)+C\exp(-c(\log L)^{2}).$

By Lemma 5.4, $\mathbb{P}\left(X(s,\tfrac{1}{4}K)>\exp(-K^{2}L^{-1/2}/32)\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-cK^{2})$. This completes the
proof. ∎

Next, we turn to the two-point estimates. The proof strategies are analogous
to what we just saw for the one-point: we combine a cruder estimate coming
from shear invariance with the information that the polymer measure is
localized on a smaller scale. We initially get the following two-point
estimate, which is under the additional constraints that the two points are at
least $L^{-5/8}$ away from each other, and $L^{-5/8}$ away from the ends. To
get Proposition 8.1 from it, it turns out that the cruder estimate Lemma 5.6
is sufficient, since we just prove an Hölder $\frac{1}{10}-$ bound instead of
$\frac{1}{2}-$.

######  Proposition 8.5.

For all $K>0$, $L\geq 2$, and $L^{-5/8}\leq s<t\leq 1-L^{-5/8}$, $t-s\geq
L^{-5/8}$,

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\Gamma_{0}(t)|>K(t-s)^{1/3}L^{-1/4}\right)>L^{-K}\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-c(K\wedge\log L)^{2}).$

We note that the exponent $1/3$ can be replaced by any number $<2/5$, due to
the exponent of $5/8$ (we only need that their product is $<1/4$).

###### Proof of Proposition 8.5.

This proof is very similar to that of Proposition 8.4. Again, we can assume
that $L,K$ are large enough. Let
$X(s,t,K)=\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\Gamma_{0}(t)|>K(t-s)^{1/3}L^{-1/4}\right)$.
By Proposition 7.1, with probability $\geq 1-C\exp(-c(\log L)^{2})$
conditional on $\mathcal{L}^{\beta}>L$,

$\displaystyle X(s,t,K)$
$\displaystyle\leq\boldsymbol{\mathscr{P}}\left(\max_{r\in\\{s,t\\}}|\Gamma_{0}(r)-\pi(r)|\geq
KL^{-1/2}\log
L\right)+\mathbbm{1}_{|\pi(s)-\pi(t)|\geq\frac{1}{2}K(t-s)^{1/3}L^{-1/4}}$
$\displaystyle\leq
2L^{-2K}+\mathbbm{1}_{|\pi(s)-\pi(t)|\geq\frac{1}{2}K(t-s)^{1/3}L^{-1/4}}$

and

$\displaystyle X(s,t,\tfrac{1}{4}K)$
$\displaystyle\geq\boldsymbol{\mathscr{P}}\left(\max_{r\in\\{s,t\\}}|\Gamma_{0}(r)-\pi(r)|\leq
KL^{-1/2}\log
L\right)\mathbbm{1}_{|\pi(s)-\pi(t)|\geq\frac{1}{2}(t-s)^{1/3}KL^{-1/4}}$
$\displaystyle\geq(1-2L^{-2K})\mathbbm{1}_{|\pi(s)-\pi(t)|\geq\frac{1}{2}(t-s)^{1/3}KL^{-1/4}}.$

Here we used that $KL^{-1/2}\log L<\frac{1}{4}K(t-s)^{1/3}L^{-1/4}$ since
$t-s>L^{-5/8}$. These imply that

$\displaystyle
X(s,t,K)>2L^{-2K}\implies|\pi(s)-\pi(t)|\geq\tfrac{1}{2}K(t-s)^{1/3}L^{-1/4}\implies
X(s,t,\tfrac{1}{4}K)\geq 1-2L^{-2K}.$

Besides, by Lemma 5.6 we have
$\mathbb{P}\left(X(s,t,\frac{1}{4}K)>\exp(-K^{2}L^{-1/2}/32)\ \Big{|}\
\mathcal{L}^{\beta}>L\right)<C\exp(-cK^{2})$. These complete the proof. ∎

###### Proof of Proposition 8.1.

We assume that $K$ is large enough by taking $C$ large (if necessary), and
assume that $L$ is large enough by applying Lemma 5.6 otherwise.

For the case where $t-s<2L^{-5/8}$, by Lemma 5.6, we can bound the conditional
probability in the statement of Proposition 8.1 by

$C\exp(-cK^{2}(t-s)^{-9/11})<C\exp(-cK^{2}),$

since
$\exp(-\frac{1}{2}K^{2}(t-s)^{-9/11}L^{-1/2})\leq\exp(-cK^{2}L^{1/88})<L^{-K}$.

For the case where $t-s\geq 2L^{-5/8}$, if $s<L^{-5/8}$, for $s_{*}=s$ or
$L^{-5/8}$, we apply Lemma 5.4 to get that

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left[|\Gamma_{0}(s_{*})|\geq
K(s_{*}(1-s_{*}))^{1/11}L^{-1/4}\right]>L^{-3K}\ \Big{|}\
\mathcal{L}^{\beta}>L\right)<C\exp(-cK^{2}s_{*}^{-9/11}),$

since $\exp(-\frac{1}{2}K^{2}(s_{*}(1-s_{*}))^{-9/11}L^{-1/2})<L^{-3K}$.
Therefore we have

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(s)-\Gamma_{0}(L^{-5/8})|>K(L^{-5/8}-s)^{1/11}L^{-1/4}\right)>L^{-2K}\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-cK^{2}).$

Similarly, if $t>1-L^{-5/8}$, we have

$\displaystyle\mathbb{P}\left(\boldsymbol{\mathscr{P}}\left(|\Gamma_{0}(t)-\Gamma_{0}(1-L^{-5/8})|>K(t-1+L^{-5/8})^{1/11}L^{-1/4}\right)>L^{-2K}\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-cK^{2}).$

Thus we have reduced the problem of $s,t$ to the same problem of $s\vee
L^{-5/8},t\wedge(1-L^{-5/8})$, and that follows from Proposition 8.5. ∎

The remaining sections are devoted to proving finite dimensional convergence.

##  9\. Estimates on free energies under conditionings

As indicated in Section 1.2, our strategy of deducing finite dimensional limit
heavily relies on realizing the conditioning on $\mathcal{L}^{\beta}(0,0;0,1)$
as conditioning on the existence of peaks at certain heights at certain
locations. As such, we will often need estimates on the probability of the
existence of such peaks given the global conditioning
$\mathcal{L}^{\beta}(0,0;0,1)$, or on the probability of the latter
conditioned on the former. One such estimate (Lemma 6.4) has appeared before,
and in this section we provide some more refined ones.

Let us introduce the setup, and some notations needed to state these
estimates. We will work under a setting similar to that in Section 6. Namely,
we consider $(s_{1},\cdots,s_{k-1})\in\mathring{\Lambda}_{k-1}([0,1])$ and
$\vec{x}\in\mathbb{R}^{k-1}$ for $k\in\mathbb{N}$, and denote
$s_{0}=x_{0}=x_{k}=0$ and $s_{k}=1$. All the constants within this section can
depend on $k$ and $(s_{1},\cdots,s_{k-1})$. For the convenience of notations
we adopt the shorthand $\mathcal{L}^{\beta}=\mathcal{L}^{\beta}(0,0;0,1)$ and
$\mathcal{L}^{\beta}_{i}=\mathcal{L}^{\beta}(s_{i-1},x_{i-1};s_{i},x_{i})$ for
each $1\leq i\leq k$. We also write $\vec{\mathcal{L}}^{\beta}$ for the vector
$\\{\mathcal{L}^{\beta}_{i}\\}_{i=1}^{k}$.

Below we use $x\approx y$ to denote that $x\in y+[0,e^{-L}]$; and for any
vector $\vec{h}\in\mathbb{R}^{k}$,
$\smash{\vec{\mathcal{L}}^{\beta}\approx\vec{h}}$ is the event where
$\mathcal{L}^{\beta}_{i}\approx h_{i}$ for each $i$. A main reason for
introducing this notation is that we will need to invoke coalescence or
Brownian comparison statements (Proposition 3.4 and Proposition 3.5) which do
not allow conditioning on exact values.

Take $\vec{a}=(a_{0},\ldots,a_{k})\in[-L^{5/16}\log L,L^{5/16}\log L]^{k-1}$.
For each $i=1,\ldots,k-1$, we write

$\pi^{*}(s_{i})=\operatorname*{argmax}_{x}\mathcal{L}^{\beta}(a_{i-1},s_{i-1};x,s_{i})+\mathcal{L}^{\beta}(x,s_{i};a_{i+1},s_{i+1}).$

This differs from $\pi$ because $\pi(s_{i})$ is defined as the maximizer of
$\mathcal{L}^{\beta}(0,0;x,s_{i})+\mathcal{L}^{\beta}(x,s_{i};0,1)$, while
$\pi^{*}(s_{i})$ is the maximizer of profiles within certain time strips which
are disjoint for different $i$; this disjointness gives some independence
which will be useful for the arguments.

We next introduce useful notations generalizing the maximum to positive
temperature: for $f:\mathbb{R}^{k-1}\to\mathbb{R}$ and a set
$I\subseteq\mathbb{R}^{k-1}$,

$\displaystyle\emax[(\beta)]_{\vec{x}\in
I}f=\begin{cases}\displaystyle\log\int_{I}\exp\bigl{(}f(x_{1},\ldots,x_{k-1})\bigr{)}\,\,\mathrm{d}x_{1}\cdots\,\mathrm{d}x_{k-1}&\beta=1\\\
\displaystyle\max_{\vec{x}\in
I}f(x_{1}\cdots\,\mathrm{d}x_{k-1})&\beta=\infty.\end{cases}$ (9.1)

We further define the restricted free energy $\mathcal{L}^{\beta}[\vec{y},R]$
for $R>0$ and $\vec{y}\in\mathbb{R}^{k-1}$ by

$\mathcal{L}^{\beta}[\vec{y},R]=\emax[(\beta)]_{\|\vec{x}-\vec{y}\|_{\infty}\leq
R}\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}.$

Let $r_{\beta=1}=1$ and $r_{\beta=\infty}=L^{-1/2}$. These are the fluctuation
scales of the total free energy conditional on the peak heights and locations.
Let $w_{\beta=1}=L^{-1/2}$ and $w_{\beta=\infty}=L^{-1}$. These are the window
sizes around each $\pi^{*}(s_{i})$ that would affect $\mathcal{L}^{\beta}$,
under the upper tail.

The next two statements record the above-mentioned complementary estimates and
will be the goal of this section.

######  Proposition 9.1.

Take any large enough $L,M$, and $\vec{h}\in\mathbb{R}^{k}$,
$\vec{a}\in[-L^{5/16}\log L,L^{5/16}\log L]^{k+1}$, $\vec{x}\in[-L^{5/16}\log
L,L^{5/16}\log L]^{k-1}$. Denote $H=\sum_{i=1}^{k}h_{i}$ and assume that
$H>L/2$ and each $\left|h_{i}-(s_{i}-s_{i-1})H\right|<L^{8/9}$. Then we have

$\mathbb{P}\bigg{(}\mathcal{L}^{\beta}>H-(k-1)\beta^{-1}\log(2H^{1/2})+Mr_{\beta},\\\
\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}|\leq w_{\beta}\ \Big{|}\
\vec{\mathcal{L}}^{\beta}\approx\vec{h}\bigg{)}<C\exp\left(-cM^{2}L^{1/2}r_{\beta}\right)+C\exp(-cL^{3/2}),$
(9.2) $\mathbb{P}\bigg{(}\mathcal{L}^{\beta}[\vec{x},L^{-1/2}(\log
L)^{2}]>H-(k-1)\beta^{-1}\log(2H^{1/2})+Mr_{\beta},\\\
\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}|\leq w_{\beta}\ \Big{|}\
\vec{\mathcal{L}}^{\beta}\approx\vec{h}\bigg{)}>c\exp\left(-CM^{2}L^{1/2}r_{\beta}\right)-C\exp(-cL^{3/2}).$
(9.3)

Moreover, we also have

$\mathbb{P}\left(\mathcal{L}^{\beta}>H-(k-1)\beta^{-1}\log(2H^{1/2})+M\
\Big{|}\
\vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)<C\exp\left(-cML^{1/2}\right).$
(9.4)

The term $(k-1)\log(2H^{1/2})$ is meant to be present in the case $\beta=1$
and absent in the case $\beta=\infty$, and multiplying the term by
$\beta^{-1}$ is a convenient notational tool to this effect (though in fact if
one were to work out the arguments in the case of general $\beta$, the term
would be $(k-1)\beta^{-1}\log(2\beta H^{1/2})$). The source of the log term
for $\beta=1$ comes from the fact that
$\int_{-\infty}^{\infty}\exp(-4H^{1/2}|x|)\,\mathrm{d}x=(2H^{1/2})^{-1}$,
which itself is a result that the dominant contribution to the integral is
from an interval of scale $L^{-1/2}$ around zero. Since on
$\vec{\mathcal{L}}^{\beta}\approx\vec{h}$ the terms in the exponential in the
convolution formula for $\exp(\mathcal{L}^{\beta})$ are essentially sums of
two Brownian bridges with slope $-2H^{1/2}$ each, heuristically taking
logarithms will yield that $\mathcal{L}^{\beta}$ loses $(k-1)\log(2H^{1/2})$
compared to the peak height of $H$.

We next give an estimate on the probability of
$\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}$ being much smaller than $L$,
conditional on $\mathcal{L}^{\beta}>L$ and $\pi^{*}$. It can be viewed as a
refinement of Lemma 6.4.

######  Proposition 9.2.

For all large enough $L,M$ with $M<L^{0.01}$, and $\vec{x}\in[-L^{5/16}\log
L,L^{5/16}\log L]^{k-1}$, $\vec{x}\in[-L^{5/16}\log L,L^{5/16}\log L]^{k-1}$,
we have

$\displaystyle\mathbb{P}\left(E_{k,M,w,L}\ \Big{|}\
\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\right)\leq\exp\left(-cM^{2}L^{1/2}r_{\beta}\right),$

where

$E_{k,M,w,L}=\left\\{\sum_{i=1}^{k}\mathcal{L}^{\beta}_{i}\in
L+(k-1)\beta^{-1}\log(2L^{1/2})+[-L^{8/9},-Mr_{\beta}]\right\\}\\\
\cap\bigcap_{i=1}^{k}\left\\{\mathcal{L}^{\beta}_{i}>(s_{i}-s_{i-1})L-k^{-1}L^{8/9}\right\\}.$
(9.5)

Proposition 9.2 is proved by invoking Bayes’ theorem and Proposition 9.1, and
we give its proof now; we will return to proving Proposition 9.1 later. For
any vector $\vec{h}\in\mathbb{R}^{k}$, denote

$\displaystyle A_{\vec{h}}=\bigcap_{i=1}^{k}\left\\{\mathcal{L}^{\beta}_{i}\in
h_{i}+[0,r_{\beta}]\right\\}.$

###### Proof of Proposition 9.2.

We assume without loss of generality that $M\in\mathbb{N}$. Define
$\mathrm{Val}_{M}$ by

$\mathrm{Val}_{M}=\left\\{\vec{h}\in(r_{\beta}\mathbb{Z})^{k}:\parbox[c]{289.07999pt}{\centering$\sum_{i=1}^{k}h_{i}\in
L+(k-1)\beta^{-1}\log(2L^{1/2})+[-L^{8/9}-r_{\beta},-Mr_{\beta}]$ \\\
$h_{i}\in(s_{i}-s_{i-1})L+[-L^{8/9},L^{8/9}],\quad i=1,\ldots,k$
\<EMAIL_ADDRESS>
By doing a disjoint decomposition and applying Bayes’ theorem,

$\displaystyle\mathbb{P}\left(E_{k,M,w,L}\ \Big{|}\
\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\right)$
$\displaystyle\leq\frac{\sum_{\vec{h}\in\mathrm{Val}_{M}}\mathbb{P}\left(\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\ \Big{|}\
A_{\vec{h}}\right)\cdot\mathbb{P}\left(A_{\vec{h}}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\right)}.$

We can also decompose the denominator in a similar fashion, but with the sum
over all $\vec{h}$. Now for each $\vec{h}\in\mathrm{Val}_{M}$, we wish to
bound the ratio

$\frac{\mathbb{P}\left(\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\ \Big{|}\
A_{\vec{h}}\right)\cdot\mathbb{P}(A_{\vec{h}})}{\mathbb{P}\left(\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\ \Big{|}\
A_{\vec{h}^{\prime}}\right)\cdot\mathbb{P}(A_{\vec{h}^{\prime}})},$

where $\vec{h}^{\prime}\in(r_{\beta}\mathbb{Z})^{k}$ is defined as follows:
for each $i=2,\ldots,k$, $h_{i}^{\prime}=h_{i}$, while
$\sum_{i=1}^{k}h_{i}^{\prime}\in L+(k-1)\beta^{-1}\log(2L^{1/2})+[0,1]$,
$h_{1}^{\prime}-h_{1}\in\lfloor h_{1}^{\prime}-h_{1}\rfloor+[0,r_{\beta})$.

By Theorem 4.1 we have

$\frac{\mathbb{P}(A_{\vec{h}})}{\mathbb{P}(A_{\vec{h}^{\prime}})}<C\exp(C(h_{1}^{\prime}-h_{1})L^{1/2}),$

and by Proposition 9.1 we have

$\frac{\mathbb{P}\left(\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\ \Big{|}\
A_{\vec{h}}\right)}{\mathbb{P}\left(\mathcal{L}^{\beta}>L,\max_{i=1,\ldots,k}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta}\ \Big{|}\
A_{\vec{h}^{\prime}}\right)}<C\exp(-c(h_{1}^{\prime}-h_{1})^{2}L^{1/2}r_{\beta})+C\exp(-cL^{3/2}).$

By combining these two estimates, we can now bound the ratio as desired. Then
by summing over all $\vec{h}\in\mathrm{Val}_{M}$ the conclusion follows. ∎

We next give the proof of Proposition 9.1.

###  9.1. $k=2$ setting

The basic step is to prove the following $k=2$ version: from this, we can
obtain the general $k$-version by invoking coalescence (Proposition 3.4) to
break down the case of general $k$ to a collection of $k=2$ cases.

The notations within this subsection are slightly different, and we setup now.
We take large enough $L>0$, $h_{1}$ and $h_{2}=\Theta(L)$, $0<s_{1},s_{2}<1$,
$|x_{*}|\leq 2L^{5/16}\log L$. All the constants in this section can depend on
$s_{1}\wedge s_{2}$. We write
$\vec{\mathcal{L}}^{\beta}=(\mathcal{L}^{\beta}(0,0;x_{*},s_{1}),\mathcal{L}^{\beta}(x_{*},s_{1};0,s_{1}+s_{2}))$.
For any $R>0$ we denote

$\mathcal{L}^{\beta}[x_{*},R]=\emax[(\beta)]_{|x-x_{*}|\leq
R}\mathcal{L}^{\beta}(0,0;x,s_{1})+\mathcal{L}^{\beta}(x,s_{1};0,s_{1}+s_{2})$

and we write $\mathcal{L}^{\beta}[R]=\mathcal{L}^{\beta}[0,R]$.

We also denote $\lambda=(s_{1}^{-1/2}h_{1}^{1/2}+s_{2}^{-1/2}h_{2}^{1/2})$ to
be the first order of the slope of the sum
$\mathcal{L}^{\beta}(0,0;x,s_{1})+\mathcal{L}^{\beta}(x,s_{1};0,s_{1}+s_{2})$
under the conditioning that $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$. We take
some $x_{-}$, $x_{+}$ with $|x_{-}|,|x_{+}|\leq 2L^{5/16}\log L$ and denote

$\pi^{*}(s_{1})=\operatorname*{argmax}\mathcal{L}^{\beta}(x_{-},0;\cdot,s_{1})+\mathcal{L}^{\beta}(\cdot,s_{1};x_{+},s_{1}+s_{2}).$

######  Lemma 9.3.

Denote $W=10^{-6}(h_{1}s_{1})^{1/2}\wedge(h_{2}s_{2})^{1/2}$ and assume
$h_{1},h_{2}=\Theta(L)$. In the case of $\beta=1$, for any $M>L^{1/16}\log L$
we have

$\mathbb{P}\Big{(}\mathcal{L}^{\beta}[W]>h_{1}+h_{2}-\beta^{-1}\log(\lambda)+ML^{-1/4},|\pi^{*}(s_{1})-x_{*}|\leq
w_{\beta}\ \Big{|}\ \vec{\mathcal{L}}^{\beta}\approx\vec{h}\Big{)}\\\
<C\exp(-cM^{2})+C\exp(-cL^{3/2}),$ (9.6)

and for any $M>0$,

$\mathbb{P}\Big{(}\mathcal{L}^{\beta}[x_{*},L^{-1/2}(\log
L)^{2}]>h_{1}+h_{2}-\beta^{-1}\log(\lambda)+ML^{-1/4},|\pi^{*}(s_{1})-x_{*}|\leq
w_{\beta}\ \Big{|}\ \vec{\mathcal{L}}^{\beta}\approx\vec{h}\Big{)}\\\
>c\exp(-CM^{2})-C\exp(-cL^{3/2}).$ (9.7)

In the case of $\beta=\infty$, the first two bounds hold for any $M>0$, after
(1) replacing $ML^{-1/4}$ with $ML^{-1/2}$ in both; (2) in the lower bound,
replacing $\mathcal{L}^{\beta}[x_{*},L^{-1/2}(\log L)^{2}]$ with
$\mathcal{L}^{\beta}[x_{*},L^{-1}]$.

Moreover, for both $\beta=1$ and $\beta=\infty$, and $M$ large enough, we have

$\mathbb{P}\Big{(}\mathcal{L}^{\beta}[W]>h_{1}+h_{2}-\beta^{-1}\log(\lambda)+M\
\Big{|}\ \vec{\mathcal{L}}^{\beta}\approx\vec{h}\Big{)}<C\exp(-cML^{1/2}).$
(9.8)

###### Proof.

The general idea of this proof is to invoke Lemma 2.19, then do computations
of Brownian motions.

We can assume that $M$ is large enough since otherwise the estimates trivially
hold. For the convenience of notations, we denote
$S(x)=\mathcal{L}^{\beta}(0,0;x,s_{1})+\mathcal{L}^{\beta}(x,s_{1};0,s_{1}+s_{2})$,
and write $B(x)=S(x_{*}+x)-S(x_{*})+2\lambda|x|$. In light of Lemma 2.19, we
shall think of $B(x)$ as a Brownian motion in the interval $[-CL^{-1/2}\log
L,CL^{-1/2}\log L]$.

Positive temperature. We first consider the case of $\beta=1$ (where
$w_{\beta=1}=L^{-1/2}$), and turn to the zero temperature ($\beta=\infty$)
case later.

Upper bound. By the convolution formula,

$\mathcal{L}^{\beta}[W]=\log\int_{-W}^{W}\exp(S(x))\,\mathrm{d}x.$ (9.9)

Since $h_{1},h_{2}=\Theta(L)$ and $|x_{*}|,|x_{-}|,|x_{+}|\leq 2L^{5/16}\log
L$, by Corollary 2.21, conditional on
$\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, with probability at least
$1-C\exp(-cM^{2})$,

$S(x)<h_{1}+h_{2}-2\lambda|x-x_{*}|+C|x-x_{*}|L^{5/16}\log L+ML^{-1/4},$
(9.10)

for any $|x-x_{*}|\leq L^{-1/2}$, and

$S(x)<h_{1}+h_{2}-2\lambda|x-x_{*}|+C|x-x_{*}|L^{5/16}\log
L+M|x-x_{*}|^{1/2}(|\log(|x-x_{*}|LM^{-2})|+1),$ (9.11)

for any $x$ with $|x-x_{*}|\geq L^{-1/2}$, $|x|\leq W$.

To get (9.8), we note that by plugging (9.10) and (9.11) into (9.9), we have
that $\mathcal{L}^{\beta}[W]<h_{1}+h_{2}+\log(\lambda^{-1})+CM^{2}L^{-1/2}$
when $M>L^{1/4}$. (Note that the RHS of (9.11) is maximized when $|x-x_{*}|$
is of order $M^{2}/L$.) Relabeling $M$ completes the proof.

To get (9.6), we note that with probability $>1-C\exp(-cL^{3/2})$ conditional
on $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, the event
$|\pi^{*}(s_{1})-x_{*}|\leq L^{-1/2}$ implies that
$S(x)<h_{1}+h_{2}+ML^{-1/4}+C\exp(-cL)$ for all $|x|\leq W$. Plugging the
minimum of this bound and (9.11) into (9.9), we get that
$\mathcal{L}^{\beta}[W]<h_{1}+h_{2}+\log(\lambda^{-1})+CML^{-1/4}+CL^{-3/16}\log
L$. Using that $M>L^{1/16}\log L$ the estimate (9.6) follows.

Lower bound. As the comparison in Lemma 2.19 is for some interval with length
of order $L^{1/2}$, we need to do a truncation for $\pi^{*}(s_{1})$. Namely,
we claim

$\mathbb{P}(|\pi^{*}(s_{1})|>W\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h})<C\exp(-cL^{3/2}).$
(9.12)

Indeed, from Theorem 2.13 and Proposition 2.16, and Proposition 3.4, we can
deduce that

$\frac{\mathbb{P}\left(\sup_{|x|>W}\mathcal{L}^{\beta}(x_{-},0;x,s_{1})\geq\mathcal{L}^{\beta}(x_{-},0;x_{*},s_{1}),\mathcal{L}^{\beta}_{1}\approx
h_{1}\right)}{\mathbb{P}(\mathcal{L}^{\beta}_{1}\approx
h_{1})}<C\exp(-cL^{3/2}),$
$\frac{\mathbb{P}\left(\sup_{|x|>W}\mathcal{L}^{\beta}(x,s_{1};x_{+},s_{1}+s_{2})\geq\mathcal{L}^{\beta}(x_{*},s_{1};x_{+},s_{1}+s_{2}),\mathcal{L}^{\beta}_{2}\approx
h_{2}\right)}{\mathbb{P}(\mathcal{L}^{\beta}_{2}\approx
h_{2})}<C\exp(-cL^{3/2}).$

These together imply (9.12).

Now by definition

$\displaystyle\mathcal{L}^{\beta}[x_{*},L^{-1/2}(\log
L)^{2}]=\log\int_{-L^{-1/2}(\log L)^{2}}^{L^{-1/2}(\log
L)^{2}}\exp(S(x_{*}+x))\,\mathrm{d}x$
$\displaystyle=S(x_{*})+\log\int_{-L^{-1/2}(\log L)^{2}}^{L^{-1/2}(\log
L)^{2}}\exp\left(-2\lambda|x|+B(x)\right)\,\mathrm{d}x.$ (9.13)

Define events $A_{M}$ and $E$ as

$\begin{split}A_{M}&=\left\\{\min_{|x|\in[L^{-1/2}/4,L^{-1/2}/2]}B(x)\geq
ML^{-1/4}\right\\}\cap\left\\{\min_{|x|\leq
L^{-1/2}/4}B(x)\geq-L^{-1/4}\right\\}\\\
&\qquad\cap\left\\{B(x)\geq-|x|^{1/2}\log(|x|L^{1/2})\text{ for all
}|x|\in[L^{-1/2}/2,L^{-1/2}\log L]\right\\},\\\ E&=\left\\{\max_{|x|\geq
L^{-1/2},|x+x_{*}|\leq W}B(x)-2\lambda|x|<\max_{|x|\leq
L^{-1/2}}B(x)-2\lambda|x|-C\exp(-cL)\right\\}.\end{split}$

If $B$ were replaced by a two-sided Brownian motion with bounded drift, the
probability of $A_{M}\cap E$ would be $>c\exp(-CM^{2})$, by standard Brownian
motion computations. Then by Lemma 2.19,

$\mathbb{P}(A_{M}\cap
E\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h})>c\exp(-CM^{2})(1-C\exp(-cL))-C\exp(-cL^{3/2})\\\
>c\exp(-CM^{2})-C\exp(-cL^{3/2}).$

Assuming $A_{M}\cap E$, we can lower bound (9.1) by
$h_{1}+h_{2}+\log(\lambda^{-1})+cML^{-1/4}-CL^{-1/4}$; and
$|\pi^{*}(s_{1})|\leq L^{-1/2}$ outside an event with probability
$<C\exp(-cL^{3/2})$, by Proposition 3.4. Noting that $M$ is taken large
enough, together with (9.12), the lower bound (9.7) follows.

These complete the proof of the lemma in the $\beta=1$ case. We now turn to
the $\beta=\infty$ case.

_Zero temperature:_ We now need to bound the maximum of $S$ instead of its
integral. For the upper bound, since $h_{1},h_{2}=\Theta(L)$ and
$|x_{*}|,|x_{-}|,|x_{+}|\leq 2L^{5/16}\log L$, by Corollary 2.21, conditional
on $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, with probability at least
$1-C\exp(-cM^{2})$,

$S(x)<h_{1}+h_{2}-2\lambda|x-x_{*}|+C|x-x_{*}|L^{5/16}\log L+ML^{-1/2},$

for any $|x-x_{*}|\leq L^{-1}$, and

$S(x)<h_{1}+h_{2}-2\lambda|x-x_{*}|+C|x-x_{*}|L^{5/16}\log
L+M|x-x_{*}|^{1/2}(|\log(|x-x_{*}|LM^{-2})|+1),$

for any $x$ with $|x-x_{*}|\geq L^{-1}$, $|x|\leq W$. Then $S(x)$ in $[-W,W]$
is at most $h_{1}+h_{2}+CM^{2}L^{-1/2}$ (note that the RHS of the previous
display is maximized when $|x-x_{*}|$ is of order $M^{2}/L$), and we get
(9.8), by relabeling $M$.

Under the additional assumption that $|\pi^{*}(s_{1})-x_{*}|\leq L^{-1}$ (note
that $w_{\beta=\infty}=L^{-1}$), by Proposition 3.4, we have that $S(x)$ in
$[-W,W]$ is at most $h_{1}+h_{2}+ML^{-1/2}$ outside another event of
probability $<C\exp(-cL^{3/2})$. Thus we get (9.6).

For the lower bound, we note that (9.12) still holds with the same proof. Then
by Proposition 3.4, we just need to consider the probability of

$\max_{|x|\leq L^{-1}}B(x)\geq
ML^{-1/2},\quad\left|\operatorname*{argmax}_{|x+x_{*}|\leq
W}B(x)-2\lambda|x|\right|\leq L^{-1}.$

Again by Lemma 2.19, conditional on $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$,
the above happens with probability $>c\exp(-CM^{2})-C\exp(-cL^{3/2})$. This
completes the proof. ∎

Below we return to using the notations defined before this subsection, i.e.,
at the beginning of Section 9.

###  9.2. Proof of Proposition 9.1

Note that (with $y_{0}=y_{k}=0$)

$\displaystyle\mathcal{L}^{\beta}=\emax[(\beta)]_{y_{1},\ldots,y_{k-1}}\sum_{i=1}^{k}\mathcal{L}\left(y_{i-1},s_{i-1};y_{i},s_{i}\right).$

We want to simplify the RHS using coalescence (Proposition 3.4). Since we only
know that coalescence holds up to a distance of order $L^{1/2}$, we have to
first argue that we can restrict the $\max^{(\beta)}$ to be over an interval
centered at 0 of size much smaller than $L^{1/2}$, with high probability
conditionally on $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$.

For notational convenience let, for $R>0$ and $i=1,\ldots,k-1$, and
$x\in\mathbb{R}$,

$\displaystyle\mathcal{L}^{\beta}_{i}[x,R]=\emax[(\beta)]_{|y-x|\leq
R}\mathcal{L}^{\beta}(x_{i-1},s_{i-1};y,s_{i})+\mathcal{L}^{\beta}(y,s_{i};x_{i+1},s_{i+1}).$

Restricting the interval and coalescence: Let
$\delta=10^{-6}\min_{i=1,\ldots,k}(s_{i}-s_{i-1})$. Let $\mathcal{E}_{0}$
denote the event that

$\mathcal{L}^{\beta}\leq\mathcal{L}^{\beta}[\vec{0},\delta
L^{1/2}]+C\exp(-cL),$

where $\vec{0}\in\mathbb{R}^{k-1}$ is the vector with every entry being $0$.
Then we have

$\mathbb{P}(\mathcal{E}_{0}^{c}\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h})<\mathbb{P}(\mathcal{E}_{0}^{c},\mathcal{L}>H-C\log
L\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h})+\mathbb{P}(\mathcal{L}\leq
H-C\log L\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h}).$

The second term in the RHS is bounded by $C\exp(-cL^{2})$ by Lemma 6.1 and
Theorem 2.13; and the first term is bounded by

$\mathbb{P}(\mathcal{E}_{0}^{c}\mid\mathcal{L}>H-C\log
L)\frac{\mathbb{P}(\mathcal{L}>H-C\log
L)}{\mathbb{P}(\vec{\mathcal{L}}^{\beta}\approx\vec{h})}.$

By Lemma 5.4, we have
$\mathbb{P}(\mathcal{E}_{0}^{c}\mid\mathcal{L}>H)<C\exp(-cL^{3/2})$; and the
ratio is bounded by $C\exp(CL^{25/18})$ using Theorem 2.13 (and the assumption
that $\vec{x}\in[-L^{5/16}\log L,L^{5/16}\log L]^{k-1}$, and that each
$\left|h_{i}-(s_{i}-s_{i-1})H\right|<L^{8/9}$). Then we conclude that
$\mathbb{P}(\mathcal{E}_{0}^{c}\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h})<C\exp(-cL^{3/2})$.

By the convolution formula, this with Proposition 3.4 implies that
$\mathbb{P}(\mathrm{Coal}\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h})>1-C\exp(-cL^{3/2})$,
where the coalescence event $\mathrm{Coal}$ is defined by

$\mathrm{Coal}=\biggl{\\{}\mathcal{L}^{\beta}\in\sum_{i=1}^{k-1}\mathcal{L}^{\beta}_{i}[0,\delta
L^{1/2}]-\sum_{i=2}^{k-1}\mathcal{L}^{\beta}_{i}+[-C\exp(-cL),C\exp(-cL)]\biggr{\\}}.$

Upper bound: Under $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, the sum
$\sum_{i=2}^{k-1}\mathcal{L}^{\beta}_{i}\in\sum_{i=2}^{k-1}h_{i}+[0,(k-2)e^{-L}]$.
Therefore, assuming $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$ and
$\mathrm{Coal}$,

$\displaystyle\Bigl{\\{}\mathcal{L}^{\beta}>H-(k-1)\beta^{-1}\log(2H^{1/2})+Mr_{\beta}\Bigr{\\}}$
$\displaystyle\subseteq\Biggl{\\{}\sum_{i=1}^{k-1}\mathcal{L}^{\beta}_{i}[0,\delta
L^{1/2}]\geq\sum_{i=1}^{k-1}(h_{i}+h_{i+1})-(k-1)\beta^{-1}\log(2H^{1/2})+Mr_{\beta}-C\exp(-cL)\Biggr{\\}}.$

Let $P_{i}$ be the probability of

$\mathcal{L}^{\beta}_{i}[0,\delta L^{1/2}]\geq
h_{i}+h_{i+1}-\beta^{-1}\log(2H^{1/2})+k^{-1}M,$

conditional on $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$. Let $P_{i}^{\prime}$
be the probability of

$\mathcal{L}^{\beta}_{i}[0,\delta L^{1/2}]\geq
h_{i}+h_{i+1}-\beta^{-1}\log(2H^{1/2})+k^{-1}Mr_{\beta},$

and that $|\pi^{*}(s_{i})-x_{i}|\leq w_{\beta}$, conditional on
$\vec{\mathcal{L}}^{\beta}\approx\vec{h}$. By a union bound, it suffices to
upper bound $\sum_{i=1}^{k-1}P_{i}$ and $\sum_{i=1}^{k-1}P_{i}^{\prime}$.

For (9.4), we can then invoke (9.8) in Lemma 9.3 (using shear and translation
invariance properties) to obtain an upper bound on each $P_{i}$. Similarly,
for (9.2) we invoke (9.6) in Lemma 9.3 to obtain an upper bound on
$P_{i}^{\prime}$. We note that, since $|h_{i}-(s_{i}-s_{i-1})H|<L^{8/9}$, it
follows that
$\log((s_{i}-s_{i-1})^{-1/2}h_{i}^{1/2}+(s_{i+1}-s_{i})^{-1/2}h_{i+1}^{1/2})=\log(2H^{1/2})+O(L^{-1/9})$,
while $\beta^{-1}L^{-1/9}<r_{\beta}$. This completes the proof.

Lower bound: By the convolution formula and Proposition 3.4, conditional on
$\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, with probability at least
$1-C\exp(-cL^{3/2})$,

$\displaystyle\mathcal{L}^{\beta}[\vec{x},L^{-1/2}(\log L)^{2}]$
$\displaystyle\geq\sum_{i=1}^{k-1}\mathcal{L}^{\beta}_{i}[x_{i},L^{-1/2}(\log
L)^{2}]-\sum_{i=2}^{k-1}\mathcal{L}^{\beta}_{i}-C\exp(-cL)$
$\displaystyle\geq\sum_{i=1}^{k-1}\mathcal{L}^{\beta}_{i}[x_{i},L^{-1/2}(\log
L)^{2}]-\sum_{i=2}^{k-1}h_{i}-C\exp(-cL).$ (9.14)

Take $2\leq i\leq k-1$. Consider the processes
$\mathcal{L}^{\beta}(x_{i-1},s_{i-1};\cdot,s_{i})$ and
$\mathcal{L}^{\beta}(\cdot,s_{i-1};x_{i},s_{i})$ conditional on
$\mathcal{L}^{\beta}_{i}\approx h_{i}$, as a measure on
$\mathcal{C}([0,1],\mathbb{R})^{2}$. By Proposition 3.5, it can be replaced by
the product measure of its marginals (up to an error of $C\exp(-cL)$), on an
event with probability $>1-C\exp(-cL^{3/2})$, with Radon-Nikodym derivative
bounded between $1-C\exp(-cL)$ and $1+C\exp(-cL)$. By Lemma 9.3 (plus shear
and translation invariance properties), we can lower bound the probability of

$\mathcal{L}^{\beta}_{i}[x_{i},L^{-1/2}(\log L)^{2}]\geq
h_{i}+h_{i+1}-\beta^{-1}\log(2H^{1/2})+Mr_{\beta},|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta},$ (9.15)

conditional on $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, by
$c\exp(-CM^{2}L^{1/2}r_{\beta})-C\exp(-cL^{3/2})$. (Here we also use that

$\log((s_{i}-s_{i-1})^{-1/2}h_{i}^{1/2}+(s_{i+1}-s_{i})^{-1/2}h_{i+1}^{1/2})=\log(2H^{1/2})+O(L^{-1/9})$

and $\beta^{-1}L^{-1/9}<r_{\beta}$.) By the above coupling to independence,
the probability of (9.15) for each $i$, conditional on
$\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, is also
$>c\exp(-CM^{2}L^{1/2}r_{\beta})-C\exp(-cL^{3/2})$. Plugging these into (9.14)
finishes the proof of (9.3). ∎

##  10\. Global and segment maximizers

In Proposition 7.1 we have shown that (conditional on the upper tail) the
polymer $\Gamma_{0}$ at any $s\in(0,1)$ is close to the maximizer $\pi(s)$,
whose definition we recall is

$\pi(s)=\operatorname*{argmax}_{x\in\mathbb{R}}\mathcal{L}^{\beta}(0,0;x,s)+\mathcal{L}^{\beta}(x,s;0,1).$

In zero temperature, we also use $\pi$ to denote the geodesic $\pi_{0}$, which
is defined through the same expression.

As already alluded, in our computations to obtain finite dimensional
Gaussianity, it is more convenient to use another optimizer

$\pi^{*}(s)=\operatorname*{argmax}_{x\in\mathbb{R}}\mathcal{L}^{\beta}(0,s_{-};x,s)+\mathcal{L}^{\beta}(x,s;0,s_{+}).$

Here $s_{-}\in[0,s)$ and $s_{+}\in(s,1]$. In the rest of this section, all the
constants can depend on $s_{-},s,s_{+}$.

The main goal of this section is the following closeness between $\pi(s)$ and
$\pi^{*}(s)$. As in previous sections, we write
$\mathcal{L}^{\beta}=\mathcal{L}^{\beta}(0,0;0,1)$.

######  Proposition 10.1.

For any $L\geq 2$, we have

$\mathbb{P}\left(|\pi(s)-\pi^{*}(s)|>L^{-1/2}(\log L)^{2}\ \Big{|}\
\mathcal{L}^{\beta}>L\right)<C\exp(-c(\log L)^{2}).$

The case where $s_{-}=0$ and $s_{+}=1$ is obvious, since then
$\pi(s)=\pi^{*}(s)$. Below we write the proof for the case where $s_{-}>0$ and
$s_{+}<1$. The proof for the other two cases are similar and we omit them.

For the rest of this section, we shall take the shorthand
$\mathcal{L}^{\beta}_{i}=\mathcal{L}^{\beta}(x_{i-1},s_{i-1};x_{i},s_{i})$ for
some $\vec{x}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$, and $x_{0}=x_{4}=0$,
$s_{0}=0$, $s_{1}=s_{-}$, $s_{2}=s$, $s_{3}=s_{+}$, $s_{4}=1$. We also write
$\smash{\vec{\mathcal{L}}^{\beta}}$ for the vector
$\\{\mathcal{L}^{\beta}_{i}\\}_{i=1}^{4}$; in other words, we now have four
parts instead of two like in previous sections. For any vector
$\vec{h}\in\mathbb{R}^{k}$, $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$ denotes
the event where $\mathcal{L}^{\beta}_{i}\in h_{i}+[0,e^{-L}]$ for each $i$.

Our general idea is to replace the conditioning $\mathcal{L}^{\beta}>L$ by
$\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, for some reasonable
$\vec{x}\in\mathbb{R}^{3}$ and $\vec{h}$. Then we bound $|\pi(s)-x_{2}|$
conditional on $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, using coalescence
and the tent picture.

We first restrict $\pi^{*}$ to an interval of length $<L^{1/2}$.

######  Lemma 10.2.

For each $i=1,2,3$ and $L$ large,

$\mathbb{P}\left(|\pi^{*}(s_{i})|>L^{5/16}\log L\ \Big{|}\
\mathcal{L}^{\beta}>L\right)<C\exp(-c(\log L)^{2}).$

###### Proof.

We write the proof for $i=2$; the other two cases follow similarly.

Let $\mathcal{E}$ denote the event where $|\pi^{*}(s)|>L^{5/16}\log L$. By
Lemma 6.4 we have
$\mathbb{P}(\mathcal{E}_{1}\mid\mathcal{L}^{\beta}>L)>1-C\exp(-c(\log
L)^{2})$, where $\mathcal{E}_{1}$ is the event

$\inf_{|x_{1}|,|x_{2}|,|x_{3}|\leq L^{-1/4}\log
L}\sum_{i=1}^{4}\mathcal{L}^{\beta}_{i}\geq L-L^{5/8}\log L.$

We have that $\mathcal{E}\cap\mathcal{E}_{1}\subset\mathcal{E}_{2}$, where
$\mathcal{E}_{2}$ is the event

$\sup_{|x_{2}|>L^{5/16}\log
L,x_{1}=x_{3}=0}\sum_{i=1}^{4}\mathcal{L}^{\beta}_{i}\geq L-L^{5/8}\log L.$

By the shear invariance property and Corollary 6.2, and a union bound, we have
that $\mathbb{P}(\mathcal{E}_{2})<C\exp\left(-\frac{4}{3}L^{3/2}-cL^{9/8}(\log
L)^{2}\right)$. By combining this with the tail estimate Theorem 2.13 we get
that
$\mathbb{P}(\mathcal{E}_{2}\mid\mathcal{L}^{\beta}>L)<C\exp(-cL^{9/8}(\log
L)^{2})$. As we can upper bound
$\mathbb{P}(\mathcal{E}\mid\mathcal{L}^{\beta}>L)$ by
$\mathbb{P}(\mathcal{E}_{1}^{c}\mid\mathcal{L}^{\beta}>L)+\mathbb{P}(\mathcal{E}_{2}\mid\mathcal{L}^{\beta}>L)$,
the conclusion follows. ∎

In replacing the conditioning $\mathcal{L}^{\beta}>L$ into
$\vec{\mathcal{L}}^{\beta}\approx\vec{h}$, we will need to bound the ratio
$\mathbb{P}(\vec{\mathcal{L}}^{\beta}\approx\vec{h})\mathbb{P}(\mathcal{L}^{\beta}>L)^{-1}$,
for which we need that $\sum_{i=1}^{4}h_{i}>L-\log L$. For this we will invoke
Proposition 9.2, as well as a rough lower bound of $\mathcal{L}^{\beta}_{i}$
conditional on the upper tail $\mathcal{L}^{\beta}>L$.

######  Lemma 10.3.

For any $\vec{x}\in[-L^{5/16}\log L,L^{5/16}\log L]^{3}$, and
$i\in\\{1,2,3,4\\}$, we have

$\mathbb{P}\left(\mathcal{L}^{\beta}_{i}<(s_{i}-s_{i-1})L-L^{27/32}\log L)\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-c(\log L)^{2}).$

###### Proof.

We prove this for $i=2$, and the other cases would follow verbatim.

We use $\mathcal{L}^{\beta}_{j,0}$ to denote $\mathcal{L}^{\beta}_{j}$ with
each of $x_{1},x_{2},x_{3}$ replaced by $0$. Let $\mathcal{E}$ denote the
event $\mathcal{L}^{\beta}_{2}<(s_{2}-s_{1})L-L^{27/32}\log L$, and define
$\mathcal{E}_{0}$ by

$\mathcal{E}_{0}=\left\\{\mathcal{L}^{\beta}_{j,0}>(s_{j}-s_{j-1})L-L^{53/64}\right\\},$

for each $j=1,2,3,4$. By Lemma 6.4, we have

$\mathbb{P}\left(\sum_{j=1}^{4}\mathcal{L}^{\beta}_{j,0}<L-L^{5/8}\log L\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-c(\log L)^{2}).$ (10.1)

By Lemma 6.3, we have

$\mathbb{P}\left(\mathcal{E}_{0}^{c},\sum_{j=1}^{4}\mathcal{L}^{\beta}_{j,0}\geq
L-L^{5/8}\log L\right)\\\
<C\exp(-cL^{37/32})\cdot\mathbb{P}\left(\sum_{j=1}^{4}\mathcal{L}^{\beta}_{j,0}\geq
L-L^{5/8}\log L\right).$

By Theorem 2.13 and Corollary 6.2, we can bound the ratio of the last factor
over $\mathbb{P}(\mathcal{L}^{\beta}>L)$ by $C\exp(CL^{9/8}\log L)$. Therefore
we have

$\mathbb{P}\left(\mathcal{E}_{0}^{c},\sum_{j=1}^{4}\mathcal{L}^{\beta}_{j,0}\geq
L-L^{5/8}\log L\ \Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-cL^{37/32}).$

Combining this with (10.1) implies that
$\mathbb{P}(\mathcal{E}_{0}^{c}\mid\mathcal{L}^{\beta}>L)<C\exp(-c(\log
L)^{2})$.

We next upper bound $\mathbb{P}(\mathcal{E}\mid\mathcal{E}_{0})$. In the case
where $x_{1}x_{2}<0$, by the shift-invariance (Lemma 3.3)

$\mathbb{P}(\mathcal{E}\mid\mathcal{E}_{0})=\mathbb{P}\left(\mathcal{L}^{\beta}(0,s_{1};x_{2}-x_{1},s_{2})<(s_{2}-s_{1})L-L^{27/32}\log
L\ \Big{|}\ \mathcal{E}_{0}\right).$

By Lemma 2.20 and using that $|x_{1}|,|x_{2}|\leq L^{5/16}\log L$, this is
bounded by $C\exp(-cL^{11/8}\log L)$.

In the case where $x_{1}x_{2}\geq 0$, by Lemma 2.20 and using that
$|x_{1}|,|x_{2}|\leq L^{5/16}\log L$, we have

$\mathbb{P}\left(\mathcal{L}^{\beta}(0,s_{1};x_{2},s_{2})<(s_{2}-s_{1})L-L^{27/32}(\log
L)/2\ \Big{|}\ \mathcal{E}_{0}\right)<C\exp(-cL^{11/8}\log L),$
$\mathbb{P}\left(\mathcal{L}^{\beta}(x_{1},s_{1};0,s_{2})<(s_{2}-s_{1})L-L^{27/32}(\log
L)/2\ \Big{|}\ \mathcal{E}_{0}\right)<C\exp(-cL^{11/8}\log L).$

Then by the quadrangle inequalities (2.1) and (2.4), we still have
$\mathbb{P}(\mathcal{E}\mid\mathcal{E}_{0})<C\exp(-cL^{11/8}\log L)$. Then
since (by Theorem 2.13)
$\mathbb{P}(\mathcal{E}_{0})\mathbb{P}(\mathcal{L}^{\beta}>L)^{-1}<C\exp(CL^{85/64})$,
we have

$\begin{split}\mathbb{P}(\mathcal{E}\mid\mathcal{L}^{\beta}>L)&<C\exp(-c(\log
L)^{2})+\mathbb{P}(\mathcal{E}\cap\mathcal{E}_{0}\mid\mathcal{L}^{\beta}>L)\\\
&\leq C\exp(-c(\log
L)^{2})+\mathbb{P}(\mathcal{E}\cap\mathcal{E}_{0})\mathbb{P}(\mathcal{L}^{\beta}>L)^{-1}\\\
&\leq C\exp(-c(\log
L)^{2})+\mathbb{P}(\mathcal{E}\mid\mathcal{E}_{0})\mathbb{P}(\mathcal{E}_{0})\mathbb{P}(\mathcal{L}^{\beta}>L)^{-1}\\\
&<C\exp(-c(\log L)^{2}).\end{split}$

Thus the conclusion follows. ∎

We next prove that, given the tail $\vec{\mathcal{L}}^{\beta}\approx\vec{h}$,
$\pi(s)$ should be close to $x_{2}$. Define

$\mathrm{Val}^{4}=\Bigg{\\{}\vec{h}\in\mathbb{R}^{4}:h_{i}\geq(s_{i}-s_{i-1})L-L^{27/32}\log
L,\;\forall i=1,2,3,4;\\\ L-\log L\leq\sum_{i=1}^{4}h_{i}<L+(\log
L)^{2}\Bigg{\\}}.$

######  Lemma 10.4.

For any $\vec{x}\in[-L^{5/16}\log L,L^{5/16}\log L]^{3}$, and
$\vec{h}\in\mathrm{Val}^{4}$,

$\mathbb{P}\left(|\pi(s)-x_{2}|>L^{-1/2}(\log L)^{2}/2\ \Big{|}\
\vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)<C\exp(-cL^{1/2}(\log L)^{2}).$

###### Proof.

By Theorem 2.13 we have

$\mathbb{P}(\vec{\mathcal{L}}^{\beta}\approx\vec{h})>c\exp\left(-\frac{4}{3}L^{3/2}-CL^{9/8}(\log
L)^{2}\right).$ (10.2)

Therefore, using Lemma 6.1 we have

$\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;x_{2},s)<h_{1}+h_{2}-C\log L\
\Big{|}\ \vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)<C\exp(-cL^{2}),$
$\mathbb{P}\left(\mathcal{L}^{\beta}(x_{2},s;0,1)<h_{3}+h_{4}-C\log L\
\Big{|}\ \vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)<C\exp(-cL^{2}).$

Thus

$\mathbb{P}\left(\mathcal{L}^{\beta}(0,0;x_{2},s)+\mathcal{L}^{\beta}(x_{2},s;0,1)<L-C\log
L\ \Big{|}\ \vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)<C\exp(-cL^{2}).$
(10.3)

We next restrict the intervals. We define $\mathcal{L}^{\beta}_{\mathrm{out}}$
as follows. For $\beta=\infty$, we let

$\mathcal{L}^{\beta}_{\mathrm{out}}=\sup_{\vec{x}\in\mathbb{R}^{3}\setminus[-L^{3/8},L^{3/8}]^{3}}\sum_{i=1}^{4}\mathcal{L}^{\beta}_{i}.$

For $\beta=1$, we let

$\mathcal{L}^{\beta}_{\mathrm{out}}=\left(\sup_{|x_{2}|>L^{3/8}}\mathcal{L}^{\beta}(x_{2},s;0,1)+\mathcal{L}^{\beta}(0,0;x_{2},s)\right)\vee\\\
\left(\sup_{|x_{2}|\leq
L^{3/8}}(\mathcal{L}^{\beta}_{\mathrm{out},-}(x_{2})+\mathcal{L}^{\beta}(x_{2},s;0,1))\vee(\mathcal{L}^{\beta}(0,0;x_{2},s)+\mathcal{L}^{\beta}_{\mathrm{out},+}(x_{2}))\right),$

where

$\mathcal{L}^{\beta}_{\mathrm{out},-}(x_{2})=\log\int_{|x_{1}|\geq
L^{3/8}}\exp\left(\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}\right)\,\mathrm{d}x_{1},$
$\mathcal{L}^{\beta}_{\mathrm{out},+}(x_{2})=\log\int_{|x_{3}|\geq
L^{3/8}}\exp\left(\mathcal{L}^{\beta}_{3}+\mathcal{L}^{\beta}_{4}\right)\,\mathrm{d}x_{3}.$

Using Corollary 6.2 and shear invariance, and a union bound, we can upper
bound $\mathbb{P}(\mathcal{L}^{\beta}_{\mathrm{out}}>L-(\log L)^{3/2})$ by
$C\exp\left(-\frac{4}{3}L^{3/2}-cL^{5/4}(\log L)^{2}\right)$. Then by (10.2)
we have

$\mathbb{P}\left(\mathcal{L}^{\beta}_{\mathrm{out}}>L-(\log
L)^{3/2}\mid\vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)<C\exp(-cL^{5/4}(\log
L)^{2}).$ (10.4)

We next consider the restricted part.

$\mathcal{L}^{\beta}_{\mathrm{res}}(x_{2})=\max_{x_{1},x_{3}\in[-L^{3/8},L^{3/8}]}\sum_{i=1}^{4}\mathcal{L}^{\beta}_{i},$

for $\beta=\infty$, and

$\mathcal{L}^{\beta}_{\mathrm{res}}(x_{2})=\log\int_{|x_{1}|\leq
L^{3/8}}\exp\left(\mathcal{L}^{\beta}_{1}+\mathcal{L}^{\beta}_{2}\right)\,\mathrm{d}x_{1}+\log\int_{|x_{3}|\leq
L^{3/8}}\exp\left(\mathcal{L}^{\beta}_{3}+\mathcal{L}^{\beta}_{4}\right)\,\mathrm{d}x_{3},$

for $\beta=1$. Using Proposition 3.4 and Corollary 2.21, one can upper bound
$\mathcal{L}^{\beta}_{\mathrm{res}}$ assuming a coalescence event, as in the
proof of Proposition 9.1. Omitting the details, we can deduce that

$\mathbb{P}\left(\sup_{x\in I}\mathcal{L}^{\beta}_{\mathrm{res}}(x)>L-(\log
L)^{3/2}\ \Big{|}\
\vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)<C\exp(-cL^{1/2}(\log L)^{2}),$

where $I=[-L^{3/8},L^{3/8}]\setminus[x_{2}-L^{-1/2}(\log
L)^{2}/2,x_{2}+L^{-1/2}(\log L)^{2}/2]$. This with (10.4) implies that

$\mathbb{P}\left(\sup_{|x-x_{2}|>L^{-1/2}(\log
L)^{2}/2}\mathcal{L}^{\beta}(0,0;x_{2},s)+\mathcal{L}^{\beta}(x_{2},s;0,1)>L-(\log
L)^{3/2}\ \Big{|}\ \vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)\\\
<C\exp(-cL^{1/2}(\log L)^{2}).$

This and (10.3) imply the conclusion. ∎

We now finish proving Proposition 10.1, using all the above ingredients as
well as Proposition 9.2, which lower bounds the sum
$\sum_{i=1}^{4}\mathcal{L}^{\beta}_{i}$ by $L-\log L$, given that
$\mathcal{L}^{\beta}>L$.

###### Proof of Proposition 10.1.

We can assume that $L$ is large enough, since otherwise the conclusion follows
trivially.

Denote $w_{\beta=1}=L^{-1/2}$ and $w_{\beta=\infty}=L^{-1}$ as in the previous
section. By Lemmas 10.2 and 10.3, we can bound the LHS in the display by

$\sum_{\vec{x}\in([-L^{5/16}\log L,L^{5/16}\log L]\cap
w_{\beta}\mathbb{Z})^{3}}\mathbb{P}\Big{(}\max_{i=1,2,3}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta},|\pi(s)-x_{2}|>L^{-1/2}(\log L)^{2}/2\\\
\mathcal{L}^{\beta}_{i}\geq(s_{i}-s_{i-1})L-L^{27/32}\log L,\forall i=1,2,3,4\
\Big{|}\ \mathcal{L}^{\beta}>L\Big{)}+C\exp(-c(\log L)^{2}).$ (10.5)

By Lemma 6.1 and Theorem 4.1, we have

$\mathbb{P}\left(\sum_{i=1}^{4}\mathcal{L}^{\beta}_{i}\geq L+(\log L)^{2}\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-cL^{1/2}(\log L)^{2}).$

This together with Proposition 9.2 implies that, for any
$\vec{x}\in([-L^{5/16}\log L,L^{5/16}\log L]\cap w_{\beta}\mathbb{Z})^{3}$,

$\mathbb{P}\Big{(}\max_{i=1,2,3}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta},\vec{\mathcal{L}}^{\beta}\not\in\mathrm{Val}^{4},\\\
\mathcal{L}^{\beta}_{i}\geq(s_{i}-s_{i-1})L-L^{27/32}\log L,\forall i=1,2,3,4\
\Big{|}\ \mathcal{L}^{\beta}>L\Big{)}<C\exp(-c(\log L)^{2}).$

Then each summand in (10.5) can be bounded by

$\sum_{\vec{h}}\mathbb{P}\Big{(}\max_{i=1,2,3}|\pi^{*}(s_{i})-x_{i}|\leq
w_{\beta},|\pi(s)-x_{2}|>L^{-1/2}(\log
L)^{2}/2,\vec{\mathcal{L}}^{\beta}\approx\vec{h},\ \Big{|}\
\mathcal{L}^{\beta}>L\Big{)}\\\ +C\exp(-c(\log L)^{2}),$

where the sum is over $<\exp(CL)$ many $\vec{h}\in\mathrm{Val}^{4}$, such that
any element in $\mathrm{Val}^{4}$ is $\approx\vec{h}$ for one of them. We note
that the summand for each $\vec{h}$ is bounded by

$\mathbb{P}\left(|\pi(s)-x_{2}|>L^{-1/2}(\log L)^{2}/2\ \Big{|}\
\vec{\mathcal{L}}^{\beta}\approx\vec{h}\right)\mathbb{P}(\vec{\mathcal{L}}^{\beta}\approx\vec{h})\mathbb{P}(\mathcal{L}^{\beta}>L)^{-1}.$

By Lemma 10.4, the first factor is bounded by $C\exp(-cL^{1/2}(\log L)^{2})$.
Then the sum over $\vec{h}$ is bounded by

$C\exp(-cL^{1/2}(\log
L)^{2})\mathbb{P}(\vec{\mathcal{L}}^{\beta}\in\mathrm{Val}^{4})\mathbb{P}(\mathcal{L}^{\beta}>L)^{-1}.$

By Lemma 6.1 and Theorem 4.1, we can bound this by $C\exp(CL^{1/2}\log L)$,
since $\vec{\mathcal{L}}^{\beta}\in\mathrm{Val}^{4}$ implies that
$\sum_{i=1}^{4}\mathcal{L}^{\beta}_{i}\geq L-\log L$. Now since each summand
in (10.5) is bounded by $C\exp(-c(\log L)^{2})$, plugging this estimate back
leads to the conclusion. ∎

##  11\. Finite dimensional Brownian bridge limit

In this section, we prove finite dimensional convergence of $\pi_{0}$ and
$\Gamma_{0}$ to Brownian bridge (under upper tails). As in previous sections
we take the setup of
$(s_{1},\cdots,s_{k-1})\in\mathring{\Lambda}_{k-1}([0,1])$ for
$k\in\mathbb{N}$, and denote $s_{0}=0$ and $s_{k}=1$; and all the constants
within this section can depend on $k$ and $(s_{1},\cdots,s_{k-1})$. Also
recall that we define

$\pi^{*}(s_{i})=\operatorname*{argmax}_{x}\mathcal{L}^{\beta}(0,s_{i-1};x,s_{i})+\mathcal{L}^{\beta}(x,s_{i};0,s_{i+1}),$

for each $i=1,\ldots,k-1$. We adopt the shorthand
$\mathcal{L}^{\beta}=\mathcal{L}^{\beta}(0,0;0,1)$.

We shall prove that as $L\to\infty$, $L^{1/4}\\{\pi^{*}(s_{i})\\}_{i=1}^{k-1}$
conditional on $\mathcal{L}^{\beta}>L$ converges to a joint Gaussian, matching
that for a Brownian bridge. We note that even in zero temperature,
$\pi^{*}(s_{i})$ does not a priori coincide with $\pi_{0}(s_{i})=\pi(s_{i})$,
which is instead given as the maximizer of processes from height $0$ to
$s_{i}$ and $s_{i}$ to $1$ (instead of $s_{i-1}$ to $s_{i}$ and $s_{i}$ to
$s_{i+1}$ here), i.e.,
$\pi(s_{i})=\operatorname*{argmax}_{z}\mathcal{L}^{\beta}(0,0;z,s_{i})+\mathcal{L}^{\beta}(z,s_{i};0,1)$.
However, we have shown that, conditional on the upper tail, with high
probability $|\pi^{*}(s_{i})-\pi(s_{i})|$ is of order smaller than $L^{-1/4}$
(Proposition 10.1); and in positive temperature the polymer measure
concentrates at height $s_{i}$ in a window of order smaller than $L^{-1/4}$
around $\pi(s_{i})$ (Proposition 7.1). Therefore, the Gaussian limit of
$\\{\pi^{*}(s_{i})\\}_{i=1}^{k-1}$ as follows suffices for us to deduce our
main results.

We denote $w_{\beta=1}=L^{-1/2}$ and $w_{\beta=\infty}=L^{-1}$ as before. Fix
any compact set $\mathcal{K}\subseteq\mathbb{R}^{k-1}$, and all the constants
below can depend on $\mathcal{K}$.

######  Theorem 11.1.

As $L\to\infty$, uniformly over $\vec{x},\vec{y}\in\mathcal{K}$ (with
$x_{0}=x_{k}=y_{0}=y_{k}=0$ for the convenience of notations),

$\displaystyle\frac{\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|\leq
w_{\beta}\mid\mathcal{L}^{\beta}>L\right)}{\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|\leq
w_{\beta}\mid\mathcal{L}^{\beta}>L\right)}$
$\displaystyle\to\exp\left(-2\left[\sum_{i=1}^{k}\frac{(x_{i}-x_{i-1})^{2}-(y_{i}-y_{i-1})^{2}}{s_{i}-s_{i-1}}\right]\right).$

We note that the RHS is the ratio of the joint density of
$(\frac{1}{2}B(s_{1}),\ldots,\frac{1}{2}B(s_{k-1}))$ evaluated at $\vec{x}$
and $\vec{y}$, where $B$ is a standard Brownian bridge on $[0,1]$.

Note that this is a comparison of probabilities, while we wish to show weak
convergence. The following lemma allows the transition. Its proof is fairly
straightforward real analysis and we relegate it to Appendix A.

######  Lemma 11.2.

Let $d\geq 1$ and suppose $\\{\vec{X}_{\varepsilon}\\}_{\varepsilon>0}$ is a
family of $\mathbb{R}^{d}$-valued random vectors such that as $\varepsilon\to
0$, $\vec{X}_{\varepsilon}\to\vec{X}$ in distribution for some random vector
$\vec{X}$. Suppose also that there is a continuous strictly positive
integrable function $f:\mathbb{R}^{d}\to(0,\infty)$ such that, for every
compact set $\mathcal{K}\subseteq\mathbb{R}^{d}$, uniformly over
$\vec{x},\vec{y}\in\mathcal{K}$ as $\varepsilon\to 0$,

$\frac{\mathbb{P}\left(\vec{X}_{\varepsilon}\in\vec{x}+[-\varepsilon,\varepsilon)^{d}\right)}{\mathbb{P}\left(\vec{X}_{\varepsilon}\in\vec{y}+[-\varepsilon,\varepsilon)^{d}\right)}\cdot\frac{f(\vec{y})}{f(\vec{x})}\to
1.$ (11.1)

Then $\vec{X}$ is absolutely continuous with respect to the Lebesgue measure
of $\mathbb{R}^{d}$, and has density given by
$f(\vec{x})/\int_{\mathbb{R}^{d}}f(\vec{z})\,\mathrm{d}\vec{z}$.

With these results in hand, we may prove our main theorems.

###### Proofs of Theorems 1.1 and 1.2.

The tightness of $\\{L^{1/4}\pi_{0}\\}_{L\geq 2}$ and
$\\{L^{1/4}\Gamma_{0}\\}_{L\geq 2}$ conditional on $\mathcal{L}^{\beta}>L$ in
the space $\mathcal{C}([0,1],\mathbb{R})$ is given by Proposition 5.1, and it
remains to establish finite dimensional convergence.

For $B$ being a standard Brownian bridge on $[0,1]$, Theorem 11.1 and Lemma
11.2 imply that
$\\{2L^{1/4}\pi^{*}(s_{i})\\}_{i=1}^{k-1}\to\\{B(s_{i})\\}_{i=1}^{k-1}$ in
distribution. Proposition 10.1 along with the Borel-Cantelli lemma guarantees
that $L^{1/4}\pi(s_{i})-L^{1/4}\pi^{*}(s_{i})\to 0$ almost surely for every
$i$. This implies that, for $\beta=\infty$,
$\\{2L^{1/4}\pi_{0}(s_{i})\\}_{i=1}^{k-1}\to\\{B(s_{i})\\}_{i=1}^{k-1}$ in
distribution, completing the proof of Theorem 1.1.

For $\beta=1$, taking expectations in Proposition 7.1 further yields that
$\mathbb{P}(\max_{i=1,\ldots,k-1}|\Gamma_{0}(s_{i})-\pi^{*}(s_{i})|\leq
ML^{-1/2}\log L\mid\mathcal{L}^{\beta}>L)>1-L^{-cM}-C\exp(-c(\log L)^{2})$,
for $M$ being a large enough constant. Thus
$L^{1/4}\Gamma_{0}(s_{i})-L^{1/4}\pi^{*}(s_{i})\to 0$ almost surely for every
$i$. We then obtain that
$\\{2L^{1/4}\Gamma_{0}(s_{i})\\}_{i=1}^{k-1}\to\\{B(s_{i})\\}_{i=1}^{k-1}$ in
distribution, completing the proof of Theorem 1.2. ∎

The rest of this section and the next are devoted to proving Theorem 11.1.

###  11.1. Finite dimensional convergence

We introduce some useful shorthand to make the notation simpler:
$\vec{\mathcal{L}}^{\beta,\vec{x}}$ is a vector whose $i$th component is given
by

$\displaystyle\mathcal{L}^{\beta,\vec{x}}_{i}=\mathcal{L}^{\beta}(x_{i-1}L^{-1/4},s_{i-1};x_{i}L^{-1/4},s_{i})$

Below we also use $o(1)$ to denote any quantity that $\to 0$ as $L\to\infty$.

###### Proof of Theorem 11.1.

We start by noting that, by Lemma 6.4,

$\mathbb{P}\left(\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}<L-L^{5/8}\log
L\ \Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-c(\log L)^{2});$

and by Lemma 6.1, plus Theorem 2.13 and Theorem 4.1, for some $C_{0}>0$

$\mathbb{P}\left(\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}>L+C_{0}\log L\
\Big{|}\ \mathcal{L}^{\beta}>L\right)<C\exp(-cL^{1/2}\log L).$

So we can upper bound
$\mathbb{P}(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|\leq
w_{\beta},\mathcal{L}^{\beta}>L)$ by

$\mathbb{P}\left(\mathcal{E}_{\mathrm{sum}},\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}>L\right)+C\exp(-c(\log
L)^{2})\cdot\mathbb{P}(\mathcal{L}^{\beta}>L),$

where $\mathcal{E}_{\mathrm{sum}}$ is the event

$L-L^{5/8}\log L\leq\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}\leq
L+C_{0}\log L.$

Take $M_{*}$ to be a large constant, and let
$\mathcal{E}_{\mathrm{prop}}(M_{*})$ be defined by

$\displaystyle\mathcal{E}_{\mathrm{prop}}(M_{*})=\bigcap_{i=1}^{k}\left\\{\mathcal{L}^{\beta,\vec{x}}_{i}\geq(s_{i}-s_{i-1})L-M_{*}L^{7/8}\right\\}.$

Using Lemma 6.3 (proportionality statement), and Corollary 6.2 and Theorem
2.13 (tail bounds for the sum $\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}$
and $\mathcal{L}^{\beta}$), we have

$\displaystyle\mathbb{P}\left(\mathcal{E}_{\mathrm{prop}}^{c},\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}\geq
L-L^{5/8}\log L\right)$
$\displaystyle\leq\exp\left(-cM_{*}^{2}L^{5/4}\right)\cdot\mathbb{P}\left(\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}\geq
L-L^{5/8}\log L\right)$
$\displaystyle\leq\exp\left(-cM_{*}^{2}L^{5/4}-\frac{4}{3}(L-L^{5/8}\log
L)^{3/2}+CL^{3/4}\right)$
$\displaystyle\leq\exp\left(-cM_{*}^{2}L^{5/4}+CL^{9/8}\log
L\right)\cdot\mathbb{P}\left(\mathcal{L}^{\beta}>L\right).$

As $M_{*}$ is large enough, this is upper bounded by
$\exp(-cM_{*}^{2}L^{5/4})\cdot\mathbb{P}(\mathcal{L}^{\beta}>L)$.

Thus far, we have overall shown that

$\displaystyle\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}>L\right)$
$\displaystyle\leq\mathbb{P}\left(\mathcal{E}_{\mathrm{prop}}(M_{*}),\mathcal{E}_{\mathrm{sum}},\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}>L\right)+C\exp(-c(\log
L)^{2})\cdot\mathbb{P}(\mathcal{L}^{\beta}>L).$

Set (as before) $r_{\beta=1}=1$ and $r_{\beta=\infty}=L^{-1/2}$. Using
Proposition 9.2, we conclude that

$\displaystyle\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}>L\right)$
$\displaystyle\leq\bigl{(}1-e^{-cM_{*}^{2}L^{1/2}r_{\beta}}\bigr{)}^{-1}\mathbb{P}\left(\parbox{296.30743pt}{\centering$\mathcal{E}_{\mathrm{prop}}(M_{*}),\mathcal{E}_{\mathrm{sum}},\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}>L$\\\
$\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}>L+(k-1)\beta^{-1}\log(2L^{1/2})-M_{*}r_{\beta}$\@add@centering}\right)$
(11.2) $\displaystyle\qquad\quad+C\exp(-c(\log
L)^{2})\cdot\mathbb{P}\left(\mathcal{L}^{\beta}>L\right).$

We next do a restriction. Recall the notation $\emax[(\beta)]_{\vec{x}\in I}f$
for $f:\mathbb{R}^{k-1}\to\mathbb{R}$ and a set $I\subseteq\mathbb{R}^{k-1}$
from (9.1). Define

$\mathcal{L}^{\beta}[1]=\emax[(\beta)]_{\|\vec{z}\|_{\infty}\leq
1}\sum_{i=1}^{k}\mathcal{L}^{\beta}(s_{i-1},z_{i-1};s_{i},z_{i}),$ (11.3)

where $z_{0}=z_{k}=0$. Then, by the (quenched) one-point transversal
fluctuation estimate (Proposition 8.4 for $\beta=1$ and Lemma 5.2 for
$\beta=\infty$), (11.2) is bounded by

$\begin{split}&\bigl{(}1-e^{-cM_{*}^{2}L^{1/2}r_{\beta}}\bigr{)}^{-1}\mathbb{P}\left(\parbox{263.78505pt}{\centering$\mathcal{E}_{\mathrm{prop}}(M_{*}),\mathcal{E}_{\mathrm{sum}},\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta}$,\\\
$\mathcal{L}^{\beta}[1]>(1-\beta^{-1}e^{-cL^{1/4}})L$,\\\
$\sum_{i=1}^{k}\mathcal{L}^{\beta,\vec{x}}_{i}>L+(k-1)\beta^{-1}\log(2L^{1/2})-M_{*}r_{\beta}$\@add@centering}\right)\\\
&\qquad+C\exp(-c(\log
L)^{2})\cdot\mathbb{P}\left(\mathcal{L}^{\beta}>L\right).\end{split}$ (11.4)

To analyze the first term in the previous display, we define

$E_{\mathrm{val}}=\left\\{(h_{1},\ldots,h_{k})\in(e^{-L}\mathbb{Z})^{k}:\parbox{312.9803pt}{\centering$L+(k-1)\beta^{-1}\log(2L^{1/2})-(M_{*}+1)r_{\beta}\leq\sum_{i=1}^{k}h_{i}\leq
L+C_{0}\log L$, \\\ $h_{i}\in(s_{i}-s_{i-1})L+[-2M_{*}L^{7/8},kM_{*}L^{7/8}]$
for $i=1,\ldots,k$\<EMAIL_ADDRESS>(11.5)

Here we take the fine mesh $(e^{-L}\mathbb{Z})^{k}$ instead of
$\mathbb{R}^{k}$ because (as mentioned before) we will need to apply
Proposition 3.4 and Proposition 3.5. We also recall the notation
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$ for the event that each
$\mathcal{L}^{\beta,\vec{x}}_{i}\in h_{i}+[0,e^{-L}]$.

Now we see that the probability in the first term of (11.4) is bounded by

$\sum_{\vec{h}\in
E_{\mathrm{val}}}\mathbb{P}\Bigl{(}\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}[1]>(1-\beta^{-1}e^{-cL^{1/4}})L\
\Big{|}\
\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\Bigr{)}\cdot\mathbb{P}\Bigl{(}\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\Bigr{)}.$
(11.6)

Our goal is to relate each of these terms to the corresponding one with
$\vec{x}$ replaced by $\vec{y}$, which we state precisely in the next two
lemmas to be proved later. The first factor is essentially unchanged:

######  Lemma 11.3.

For $\vec{h}=(h_{1},\ldots,h_{k})\in E_{\mathrm{val}}$,

$\displaystyle\mathbb{P}\Bigl{(}\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}[1]>(1-\beta^{-1}e^{-cL^{1/4}})L\
\Big{|}\ \vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\Bigr{)}$
$\displaystyle\leq(1+o(1))\mathbb{P}\Bigl{(}\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}[1]>L\
\Big{|}\ \mathcal{L}^{\beta,\vec{y}}\approx\vec{h}\Bigr{)}.$

The proof of this lemma is somewhat involved and will be given in Section 12.
For the second factor in (11.6), we have the following statement, which is the
source of the Brownian bridge density in our result. Its proof is a
straightforward consequence of Theorem 4.1, and we give it after completing
the proof of Theorem 11.1.

######  Lemma 11.4.

For $(h_{1},\ldots,h_{k})\in E_{\mathrm{val}}$,

$\displaystyle\mathbb{P}\left(\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\right)$
$\displaystyle=(1+o(1))\exp\left(-2\sum_{i=1}^{k}\frac{(x_{i}-x_{i-1})^{2}-(y_{i}-y_{i-1})^{2}}{s_{i}-s_{i-1}}\right)\cdot\mathbb{P}\left(\mathcal{L}^{\beta,\vec{y}}\approx\vec{h}\right).$

Inputting the information from the previous two lemmas into (11.6) and (11.4)
yields that

$\displaystyle\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|\leq
w_{\beta},\mathcal{L}^{\beta}>L\right)$ $\displaystyle<C\exp(-c(\log
L)^{2})\cdot\mathbb{P}\left(\mathcal{L}^{\beta}>L\right)$
$\displaystyle\quad+(1+o(1))\left(1+\exp(-cM_{*}^{2}L^{1/2}r_{\beta})\right)\exp\left(-2\sum_{i=1}^{k}\frac{(x_{i}-x_{i-1})^{2}-(y_{i}-y_{i-1})^{2})}{s_{i}-s_{i-1}}\right)$
$\displaystyle\qquad\times\sum_{\vec{h}\in
E_{\mathrm{val}}}\mathbb{P}\Bigl{(}\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}[1]>L\
\Big{|}\
\mathcal{L}^{\beta,\vec{y}}\approx\vec{h}\Bigr{)}\cdot\mathbb{P}(\mathcal{L}^{\beta,\vec{y}}\approx\vec{h}).$

Therefore we have, since $\\{\mathcal{L}^{\beta,\vec{y}}\approx\vec{h}\\}$ are
disjoint events for distinct $\vec{h}\in E_{\mathrm{val}}$,

$\begin{split}&\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|\leq
w_{\beta}\ \Big{|}\ \mathcal{L}^{\beta}>L\right)\\\
&\leq(1+o(1))\left(1+e^{-cM_{*}^{2}L^{1/2}r_{\beta}}\right)\exp\left(-2\sum_{i=1}^{k}\frac{(x_{i}-x_{i-1})^{2}-(y_{i}-y_{i-1})^{2})}{s_{i}-s_{i-1}}\right)\\\
&\qquad\times\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|\leq
w_{\beta}\ \Big{|}\ \mathcal{L}^{\beta}>L\right)+C\exp(-c(\log
L)^{2}).\end{split}$ (11.7)

We need to lower bound
$\mathbb{P}(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|\leq
w_{\beta}\mid\mathcal{L}^{\beta}>L)$ in order to ensure that the error term
$C\exp(-c(\log L)^{2})$ is not dominating. For this we sum the previous
display over $\vec{x}\in[-KL^{-1/4},KL^{-1/4}]^{k}\cap(w_{\beta}Z)^{k}$, where
$K$ is a large constant such that

$\sum_{\vec{x}\in[-KL^{-1/4},KL^{-1/4}]^{k}\cap(w_{\beta}Z)^{k}}\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|\leq
w_{\beta}\ \Big{|}\ \mathcal{L}^{\beta}>L\right)\geq\frac{1}{2};$

that this is possible follows from combining transversal fluctuation bounds
for $\pi(s_{i})$ (Proposition 7.1 and Proposition 7.3 for $\beta=1$ or Lemma
5.2 for $\beta=\infty$) and the bound on the closeness of $\pi(s_{i})$ and
$\pi^{*}(s_{i})$ (Proposition 10.1). Then from (11.7) we have that

$\displaystyle\left(\tfrac{1}{2}-C\exp(-c(\log L)^{2})\right)$
$\displaystyle\leq
C\cdot\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|\leq
w_{\beta}\ \Big{|}\ \mathcal{L}^{\beta}>L\right)$
$\displaystyle\quad\times\sum_{\vec{x}\in[-KL^{-1/4},KL^{-1/4}]^{k}\cap(w_{\beta}Z)^{k}}\exp\left(-2\sum_{i=1}^{k}\frac{(x_{i}-x_{i-1})^{2}-(y_{i}-y_{i-1})^{2})}{s_{i}-s_{i-1}}\right).$

This yields that
$\mathbb{P}(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|\leq
w_{\beta}\mid\mathcal{L}^{\beta}>L)$ is lower bounded by a polynomial in
$L^{-1}$. Thus we conclude that

$\displaystyle\frac{\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|\leq
w_{\beta}\mid\mathcal{L}^{\beta}>L\right)}{\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-y_{i}L^{-1/4}|\leq
w_{\beta}\mid\mathcal{L}^{\beta}>L\right)}$
$\displaystyle\leq(1+o(1))\left(1+e^{-cM_{*}^{2}L^{1/2}r_{\beta}}\right)\exp\left(-2\sum_{i=1}^{k}\frac{(x_{i}-x_{i-1})^{2}-(y_{i}-y_{i-1})^{2})}{s_{i}-s_{i-1}}\right)$
$\displaystyle+C\exp(-c(\log L)^{2}).$

We can then get a lower bound of the same ratio by swapping $\vec{x}$ and
$\vec{y}$. Taking $L\to\infty$ followed by $M_{*}\to\infty$ completes the
proof. ∎

###### Proof of Lemma 11.4.

It suffices to prove the case $\vec{y}=\vec{0}$ and apply the resulting
statement twice. By independence,
$\mathbb{P}\left(\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\right)=\prod_{i=1}^{k}\mathbb{P}\left(\mathcal{L}^{\beta,\vec{x}}_{i}\in
h_{i}+[0,e^{-L}]\right)$. So we have to show that

$\displaystyle\mathbb{P}\left(\mathcal{L}^{\beta,\vec{x}}_{i}\in
h_{i}+[0,e^{-L}]\right)=(1+o(1))\exp\left(-\frac{2x_{i}^{2}}{s_{i}-s_{i-1}}\right)\cdot\mathbb{P}\left(\mathcal{L}^{\beta,\vec{0}}_{i}\in
h_{i}+[0,e^{-L}]\right).$

Now, by shear invariance and using Theorem 4.1 in the third line,

$\displaystyle\mathbb{P}\left(\mathcal{L}^{\beta,\vec{x}}_{i}\in
h_{i}+[0,e^{-L}]\right)$
$\displaystyle=\mathbb{P}\left(\mathcal{L}^{\beta,\vec{x}}_{i}\geq
h_{i}\right)-\mathbb{P}\left(\mathcal{L}^{\beta,\vec{x}}_{i}\geq
h_{i}+e^{-L}\right)$
$\displaystyle=\mathbb{P}\left(\mathcal{L}^{\beta,\vec{0}}_{i}\geq
h_{i}+\frac{(x_{i}-x_{i-1})^{2}L^{-1/2}}{s_{i}-s_{i-1}}\right)-\mathbb{P}\left(\mathcal{L}^{\beta,\vec{0}}_{i}\geq
h_{i}+e^{-L}+\frac{(x_{i}-x_{i-1})^{2}L^{-1/2}}{s_{i}-s_{i-1}}\right)$
$\displaystyle=(1+o(1))\exp\left(-2(s_{i}-s_{i-1})^{-3/2}h_{i}^{1/2}(x_{i}-x_{i-1})^{2}L^{-1/2}\right)\cdot\mathbb{P}\left(\mathcal{L}^{\beta,\vec{0}}_{i}\geq
h_{i}\right)$
$\displaystyle\quad-(1+o(1))\exp\left(-2(s_{i}-s_{i-1})^{-3/2}(h_{i}+e^{-L})^{1/2}(x_{i}-x_{i-1})^{2}L^{-1/2}\right)\cdot\mathbb{P}\left(\mathcal{L}^{\beta,\vec{0}}_{i}\geq
h_{i}+e^{-L}\right).$

Since $(h_{i}+e^{-L})^{1/2}=h_{i}^{1/2}+O(e^{-L})$, and
$h_{i}^{1/2}=(s_{i}-s_{i-1})^{1/2}L^{1/2}+o(L^{1/2})$ due to that
$h_{i}=(s_{i}-s_{i-1})L+o(L)$, the previous display equals

$\displaystyle(1+o(1))\exp\left(-\frac{2(x_{i}-x_{i-1})^{2}}{s_{i}-s_{i-1}}\right)\left[\mathbb{P}\left(\mathcal{L}^{\beta,\vec{0}}_{i}\geq
h_{i}\right)-\mathbb{P}\left(\mathcal{L}^{\beta,\vec{0}}_{i}\geq
h_{i}+e^{-L}\right)\right],$

which is what we wanted to show. ∎

##  12\. Joint comparison of maximizer location and free energy across peaks

In this section, we give the proof of Lemma 11.3. We use the setup there: in
particular, $C_{0},M_{*}$ are large constants, $L$ is taken to be large
(depending on $C_{0},M_{*}$), $\vec{h}\in E_{\mathrm{val}}$ for
$E_{\mathrm{val}}$ defined in (11.5), $\vec{x},\vec{y}\in\mathcal{K}$ for a
compact set $\mathcal{K}\subseteq\mathbb{R}^{k-1}$, and
$(s_{1},\cdots,s_{k-1})\in\mathring{\Lambda}_{k-1}([0,1])$ for
$k\in\mathbb{N}$, with $s_{0}=0$ and $s_{k}=1$. All the constants within this
section can depend on $C_{0}$, $M_{*}$, $\mathcal{K}$, $k$, and
$(s_{1},\cdots,s_{k-1})$.

The proof strategy is to do a resampling on a small interval $I$ (with size of
order $L^{-1/2}\log L$) in a way that a certain conditional probability of the
event in question (the location of $\pi^{*}(s_{i})$ for $i=1,\ldots,k$ and a
lower bound on $\mathcal{L}^{\beta}[1]$) is a function of the endpoint values
at the boundary of $I$. The proof then comes down to showing that the density
of these endpoint values under the conditioning
$\smash{\mathcal{L}^{\vec{x}}\approx\vec{h}}$ (which recall is shorthand for
$\mathcal{L}^{\beta,\vec{x}}_{i}\in h_{i}+[0,e^{-L}]$ for $i=1,\ldots,k$) is
$1+o(1)$ of the density of the same under the conditioning
$\mathcal{L}^{\vec{y}}\approx\vec{h}$.

However, note that neither of the events
$\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta}$ and
$\mathcal{L}^{\beta}[1]>(1-\beta^{-1}\exp(-cL^{1/4}))L$ are functions of the
profile on an interval of size of order $L^{-1/2}\log L$. Thus we will need to
first argue that we can consider different events which do have this localized
property. For the first event, we simply consider the maximizer on $I$ as a
proxy, instead of on $\mathbb{R}$; clearly, if the first event holds, then it
also holds that the restricted maximizer is $w_{\beta}$-close to
$x_{i}L^{-1/4}$. We will then show (Lemma 12.7) that with high probability the
restricted maximizer and true maximizer coincide. In the $\beta=\infty$ case,
on this event we also have that $\mathcal{L}^{\beta}[1]$ is a function of the
profile on an interval of size of order $L^{-1/2}\log L$.

Modifying the second event, $\mathcal{L}^{\beta}[1]>(1-\exp(-cL^{1/4}))L$, to
be a local function of the profile in the $\beta=1$ case is more difficult. A
naive argument (invoking Proposition 7.1 and Proposition 10.1) shows that the
free energy when restricted to an interval of size $ML^{-1/2}\log L$ (for a
large number $M$) would capture a $(1-L^{-M})$ fraction of the total free
energy. But at the end of the comparison we need to be able to return to the
event of being larger than $\mathcal{L}^{\beta}[1]>L$, and so we need to argue
that, conditional on $\mathcal{L}^{\vec{x}}\approx\vec{h}$ and
$\mathcal{L}^{\beta}[1]>(1-\exp(-cL^{1/4}))L$ (as well as the location of the
restricted maximizer), with high probability $\mathcal{L}^{\beta}$ will
actually be larger than $L+L^{-E_{*}}$ for some $E_{*}$, i.e., the free energy
overshoots by a polynomial amount. Then we know that restricting to a
$ML^{-1/2}\log L$ window still results in the free energy being larger than
$L$ (by picking $M$ large enough), and we can do our localized comparison
argument with this event.

In the following Section 12.1 we give the argument for saying that,
conditional on $\mathcal{L}^{\vec{x}}\approx\vec{h}$ and
$\mathcal{L}^{\beta}[1]>(1-\exp(-cL^{1/4}))L$, it holds with high probability
that $\mathcal{L}^{\beta}[1]>L+L^{-E_{*}}$ (Lemma 12.1). In Section 12.2 we
argue that the free energy (in a form suitable for analysis in the upcoming
proof of Lemma 11.3) from the smaller interval of size $ML^{-1/2}\log L$ is
also $L+L^{-E_{*}}$ (Lemma 12.5). Then in Section 12.3 we give the proof of
Lemma 11.3.

Denote $\mathcal{E}_{\mathrm{cond}}$ to be the event where
$\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})|<w_{\beta},\mathcal{L}^{\beta}[1]>(1-\beta^{-1}e^{-cL^{1/4}})L,\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$.
The following will be frequently used, to relate the conditioning on
$\mathcal{E}_{\mathrm{cond}}$ and
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$ for $\vec{h}\in
E_{\mathrm{val}}$ with $E_{\mathrm{val}}$ as defined in (11.5):

$\mathbb{P}\Bigg{(}\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})|<w_{\beta},\mathcal{L}^{\beta}[1]>(1-\beta^{-1}e^{-cL^{1/4}})L\
\Big{|}\
\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\Bigg{)}>c\exp(-CL^{1/2}).$
(12.1)

It follows from the lower bound in Proposition 9.1 and $\vec{h}\in
E_{\mathrm{val}}$.

###  12.1. Conditional overshoot at positive temperature

We work in the case of $\beta=1$ only in this subsection, in which case we
recall that $w_{\beta}=L^{-1/2}$.

######  Lemma 12.1.

There exists a constant $\rho>0$ such that

$\mathbb{P}\Big{(}\mathcal{L}^{\beta}[1]<L+\rho L^{-3}\ \Big{|}\
\mathcal{E}_{\mathrm{cond}}\Big{)}<CL^{-1}.$

This lemma is proved by a resampling argument, and we explain the setup next.
For a function $f:\mathbb{R}\to\mathbb{R}$ and an interval $[a,b]$, we define
the _bridge of $f$ on $[a,b]$_, denoted $f^{[a,b]}$, by

$\displaystyle f^{[a,b]}(x)=f(x)-\frac{x-a}{b-a}f(b)-\frac{b-x}{b-a}f(a);$

in words, it is the function obtained by affinely shifting $f$ to equal $0$ at
$a$ and $b$.

For any $x\in\mathbb{R}$, we define

$\mathcal{L}^{\beta}_{+}[1](x)=\emax[(\beta)]_{|z_{2}|,\ldots,|z_{k-1}|\leq
1}\sum_{i=2}^{k}\mathcal{L}^{\beta}(z_{i-1},s_{i-1};z_{i},s_{i}),$

with $z_{1}=x$, $z_{k}=0$; note that this is a function of
$\mathcal{L}^{\beta}$ on the temporal strip $[s_{1},1]$.

Let $\mathfrak{h}^{\beta}_{s_{1},1}$ and $\mathfrak{h}^{\beta}_{s_{1},2}$ be
the top two lines in the line ensemble associated with
$\smash{\mathcal{L}^{\beta}(0,0;\cdot,s_{1})}$; in particular
$\smash{\mathfrak{h}^{\beta}_{s_{1},1}=\mathcal{L}^{\beta}(0,0;\cdot,s_{1})}$.
Let $\mathcal{F}$ be the $\sigma$-algebra generated by

  * $\bullet$

$\mathfrak{h}^{\beta}_{s_{1},1}(z)$ for
$z\in(-\infty,x_{1}L^{-1/4}]\cup[x_{1}L^{-1/4}+w_{\beta},\infty)$,

  * $\bullet$

$(\mathfrak{h}^{\beta}_{s_{1},1})^{x_{1}L^{-1/4}+[0,\frac{1}{2}w_{\beta}]}$,
$(\mathfrak{h}^{\beta}_{s_{1},1})^{x_{1}L^{-1/4}+[\frac{1}{2}w_{\beta},w_{\beta}]}$,

  * $\bullet$

$\mathfrak{h}^{\beta}_{s_{1},2}$ and $\mathcal{L}^{\beta}_{+}[1]$.

In particular, conditional on $\mathcal{F}$, the remaining randomness (to
determine $\mathfrak{h}^{\beta}_{s_{1},1}$) is the value of
$U:=\smash{\mathcal{L}^{\beta}(0,0;x_{1}L^{-1/4}+\frac{1}{2}w_{\beta},s_{1})}=\smash{\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4}+\frac{1}{2}w_{\beta})}$,
by linear interpolation. Therefore, for
$z\in[x_{1}L^{-1/4},x_{1}L^{-1/4}+\frac{1}{2}w_{\beta}]$ and $u\in\mathbb{R}$,
we denote

$\displaystyle\mathfrak{h}^{\beta,u}_{s_{1},1}(z):=\frac{z-x_{1}L^{-1/4}}{\frac{1}{2}w_{\beta}}\bigl{(}u-\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4})\bigr{)}+\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4})+(\mathfrak{h}^{\beta}_{s_{1},1})^{x_{1}L^{-1/4}+[0,\frac{1}{2}w_{\beta}]}(z)$
(12.2)

and a similar expression for
$z\in\smash{[x_{1}L^{-1/4}+\frac{1}{2}w_{\beta},x_{1}L^{-1/4}+w_{\beta}]}$.
Further, given $\mathcal{F}$ and $U$, we also determine the value of
$\mathcal{L}^{\beta}[1]$ via the formula for $\mathfrak{h}^{\beta}_{s_{1},1}$
as well as the convolution formula, since $\mathcal{L}^{\beta}_{+}[1]$ is
$\mathcal{F}$-measurable. We also observe that the dependence of
$\smash{\mathcal{L}^{\beta}[1]}$ on $U$ is increasing since the convolution
formula has an increasing dependence on
$\smash{\mathfrak{h}^{\beta}_{s_{1},1}}$, which itself depends on $U$ in an
increasing manner.

Now we apply the Brownian Gibbs property. It implies that the distribution of
$U$ is a normal random variable of $\mathcal{F}$-measurable mean $\mu$ and
variance $\sigma^{2}$ given by

$\begin{split}\mu&=\tfrac{1}{2}\left(\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4}+w_{\beta})+\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4})\right)\\\
\sigma^{2}&=2\cdot\frac{\frac{1}{2}w_{\beta}\cdot\frac{1}{2}w_{\beta}}{w_{\beta}}=\tfrac{1}{2}w_{\beta},\end{split}$
(12.3)

tilted by the Radon-Nikodym derivative $W^{\mathrm{pt}}(U)/Z^{\mathrm{pt}}$,
where $W^{\mathrm{pt}}$ and $Z^{\mathrm{pt}}$ are given by

$W^{\mathrm{pt}}(u)=W(\mathfrak{h}^{\beta,u}_{s_{1},1},\mathfrak{h}^{\beta}_{s_{1},2}),\quad
Z^{\mathrm{pt}}=\mathbb{E}_{\mathcal{F}}\left[W^{\mathrm{pt}}(U)\right].$

where $W(\mathfrak{h}^{\beta,u}_{s_{1},1},\mathfrak{h}^{\beta}_{s_{1},2})$ is
from (2.6) for the interval $[x_{1}L^{-1/4},x_{1}L^{-1/4}+w_{\beta}]$.

Now, if we require that $\mathcal{L}^{\beta}[1]>(1-e^{-cL^{1/4}})L$, this is
equivalent to $U$ being larger than some $\mathcal{F}$-determined value, due
to the increasing dependence of $\mathcal{L}^{\beta}[1]$ on $U$ already noted.
Further, if we also require that $|\pi^{*}(s_{1})|\leq w_{\beta}$, it is not
hard to see that this also is equivalent to a lower bound on $U$ (which may be
$-\infty$).

These in turn imply that, conditional on $\mathcal{E}_{\mathrm{cond}}$, and
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, the distribution of $U$ is
given by a Gaussian random variable of mean $\mu$ and variance $\sigma^{2}$ as
given in (12.3) tilted by $W^{\mathrm{pt}}(U)/Z^{\mathrm{pt}}$, and is further
conditioned to be larger than an $\mathcal{F}$-measurable random variable,
which we denote by $\mathrm{Cor}$ (short for “corner”, which is terminology
introduced for an analogous object in [Ham16]). Now, since $W^{\mathrm{pt}}$
is an increasing function, this Gaussian random variable stochastically
dominates the Gaussian random variable $X$ with the same mean and variance
which is conditioned only to be larger than $\mathrm{Cor}$.

If we know that $\mathrm{Cor}$ is not too high (say less than order $L$) and
$\mu$ is not too low (say $-\mu$ less than order $L$), then the Gaussian
random variable will overshoot by an amount at least polynomial in $L^{-1}$.
This upper bound on $\mathrm{Cor}$ and lower bound on $\mu$ are recorded in
the next two lemmas and will be proved shortly.

######  Lemma 12.2.

There exists $K_{0}$ such that for $K>K_{0}$,

$\displaystyle\mathbb{P}\left(\mathrm{Cor}>h_{1}+K\ \Big{|}\
\mathcal{E}_{\mathrm{cond}}\right)\leq\exp(-cKL^{1/2}).$

######  Lemma 12.3.

There exists $K_{0}$ such that,

$\displaystyle\mathbb{P}\left(\mu<-K_{0}L\ \Big{|}\
\mathcal{E}_{\mathrm{cond}}\right)\leq\exp(-cL^{3/2}).$

However, knowing that the Gaussian overshoots by a polynomial-in-$L^{-1}$
amount does not immediately imply that the free energy overshoots $L$ by a
comparable amount. For this we need to additionally know that the contribution
to the free energy from the interval that gets perturbed by resampling $U$ is
non-trivial (in fact, it contributes a positive proportion of the free energy,
as we will show). This is recorded in the next lemma.

######  Lemma 12.4.

There exists $\rho>0$ such that, conditional on $\mathcal{E}_{\mathrm{cond}}$,
with probability at least $1-\exp(-cL^{1/2})$,

$\displaystyle\int_{x_{1}L^{-1/4}+\frac{1}{4}w_{\beta}}^{x_{1}L^{-1/4}+\frac{3}{4}w_{\beta}}\exp\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)+\mathcal{L}^{\beta}_{+}[1](x)\right)\,\mathrm{d}x\geq\rho\mathcal{L}^{\beta}[1].$

With the previous three lemmas in hand, we now give the proof of Lemma 12.1.
Afterward, we will give the proofs of those lemmas.

###### Proof of Lemma 12.1.

For every $u\in\mathbb{R}$, we define $\mathcal{L}^{\beta,u}[1]$ the same way
as $\mathcal{L}^{\beta}[1]$ through (11.3), except for replacing
$\mathcal{L}^{\beta}(0,0;s_{1},\cdot)=\mathfrak{h}^{\beta}_{s_{1},1}$ by
$\mathfrak{h}^{\beta,u}_{s_{1},1}$. We observe that, for any $\delta>0$,

$\displaystyle\exp\left(\mathcal{L}^{\beta,U-\delta}[1]\right)$
$\displaystyle=\int_{[-1,1]\setminus[x_{1}L^{-1/4}+\frac{1}{4}w_{\beta},x_{1}L^{-1/4}+\frac{3}{4}w_{\beta}]}\exp\left(\mathfrak{h}^{\beta,U-\delta}_{1}(x)+\mathcal{L}^{\beta}_{+}[1](x)\right)\,\mathrm{d}x$
$\displaystyle\qquad+\int_{x_{1}L^{-1/4}+\frac{1}{4}w_{\beta}}^{x_{1}L^{-1/4}+\frac{3}{4}w_{\beta}}\exp\left(\mathfrak{h}^{\beta,U-\delta}_{1}(x)+\mathcal{L}^{\beta}_{+}[1](x)\right)\,\mathrm{d}x.$

Notice that
$\mathfrak{h}_{s_{1},1}^{\beta,U-\delta}(x)\leq\mathfrak{h}^{\beta}_{s_{1},1}(x)-\frac{1}{2}\delta$
on $x_{1}L^{-1/4}+[\frac{1}{4}w_{\beta},\frac{3}{4}w_{\beta}]$ (by the formula
(12.2)). So the second term in the previous display is upper bounded by

$\displaystyle\int_{x_{1}L^{-1/4}+\frac{1}{4}w_{\beta}}^{x_{1}L^{-1/4}+\frac{3}{4}w_{\beta}}\exp\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)+\mathcal{L}^{\beta}_{+}[1](x)-\tfrac{1}{2}\delta\right)\,\mathrm{d}x.$

Note also that
$\mathfrak{h}_{s_{1},1}^{\beta,U-\delta}(x)\leq\mathfrak{h}_{s_{1},1}^{\beta,U}(x)=\mathfrak{h}_{s_{1},1}^{\beta}(x)$,
which gives an upper bound on the first term two displays above. Then by Lemma
12.4, there exists $\rho>0$ such that, with conditional probability at least
$1-\exp(-cL^{1/2})$, $\exp(\mathcal{L}^{\beta,U-\delta}[1])$ is upper bounded
by

$\displaystyle\left[1-\rho+e^{-\frac{1}{2}\delta}\rho\right]\int_{[-1,1]}\exp\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)+\mathcal{L}^{\beta}_{+}[1](x)\right)\,\mathrm{d}x=\left[1-\rho+e^{-\frac{1}{2}\delta}\rho\right]\exp(\mathcal{L}^{\beta}[1]).$

Since $e^{-x}\leq 1-x/2$ for $x\in[0,1]$, we see that the square bracket
factor is at most $1-\rho\delta/4$. In summary, with conditional probability
at least $1-\exp(-cL^{1/2})$, for any $\delta>0$ sufficiently small,

$\displaystyle\exp\left(\mathcal{L}^{\beta,U-\delta}[1]\right)\leq(1-\rho\delta/4)\exp\left(\mathcal{L}^{\beta}[1]\right).$
(12.4)

Recall from the discussion preceding Lemma 12.2 that, conditionally on
$\mathcal{F}$, $U$ stochastically dominates a random variable $X$ which is
Gaussian with mean $\mu$ and variance $\sigma^{2}$ as given in (12.3) and
conditioned to stay above $\mathrm{Cor}$ (i.e., $X$ is not tilted by
$W^{\mathrm{pt}}(U)/Z^{\mathrm{pt}}$). Now, since (by Lemma 12.2 and Lemma
12.3) with conditional probability at least $1-\exp(-cL^{1/2})$ it holds that
$\mathrm{Cor}\leq h_{1}+K_{0}$ (so that in particular $\mathrm{Cor}=O(L)$) and
$\mu\geq-K_{0}L$, it follows that, with conditional probability at least
$1-L^{-1}$ that

$\displaystyle X\geq\mathrm{Cor}+L^{-3},$

using standard estimates for the normal tail bound (Lemma 2.22).

In particular, since $U$ stochastically dominates $X$, taking $\delta=L^{-3}$
implies that $U-\delta\geq\mathrm{Cor}$. Since $\mathrm{Cor}$ is such that for
any $u\geq\mathrm{Cor}$ it holds that
$\mathcal{L}^{\beta,u}[1]\geq(1-e^{-cL^{1/4}})L$, we obtain from combining
these observations with (12.4) that

$\displaystyle\exp(\mathcal{L}^{\beta}[1])\geq(1+\tfrac{1}{5}\rho
L^{-3})\exp(L).$

Taking logarithms and relabeling $\rho$ completes the proof. ∎

We next prove the three lemmas that have been assumed.

###### Proof of Lemma 12.2.

We observe that by the definition of $\mathrm{Cor}$, it holds almost surely
that
$\mathrm{Cor}\leq\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4}+\frac{1}{2}w_{\beta})$.
So we obtain

$\displaystyle\mathbb{P}\left(\mathrm{Cor}>h_{1}+K\ \Big{|}\
\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\right)\leq\mathbb{P}\left(\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4}+\tfrac{1}{2}w_{\beta})>h_{1}+K\
\Big{|}\ \vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\right).$

By Theorem 2.17 this is upper bounded by $\exp(-cKL^{1/2})$. Combining this
with (12.1) and taking $K$ large enough, we obtain the claimed bound. ∎

###### Proof of Lemma 12.3.

Since
$\mu=\frac{1}{2}(\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4}+w_{\beta})+\mathfrak{h}^{\beta}_{s_{1},1}(x_{1}L^{-1/4}))$,
it holds by Theorem 2.17 and the independence of
$\mathcal{L}^{\beta,\vec{x}}_{i}$ across $i$ that
$\mathbb{P}\left(\mu<-K_{0}L\mid\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\right)<\exp(-cL^{3/2})$.
Combining this with (12.1) completes the proof. ∎

###### Proof of Lemma 12.4.

Take $C_{*}$ to be a large constant. By using Proposition 9.1 for a mesh of
$L^{2}$ many $x\in[-1,1]$, taking a union bound, and applying the
unconditional local fluctuations estimates Proposition 2.16, we have

$\mathbb{P}\Bigg{(}\max_{|x|\leq
1}\mathcal{L}^{\beta}(x,s_{1};0,1)>\sum_{i=2}^{k}h_{i}-(k-2)\log(2L^{1/2})+\frac{1}{2}C_{*}\
\Big{|}\
\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\Bigg{)}<\exp(-cC_{*}L^{1/2}).$
(12.5)

Further by Corollary 2.21,

$\mathbb{P}\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)\leq
h_{1}-C_{*}-L^{1/2}|x-x_{1}L^{-1/4}|/2,\;\forall x:|x-x_{1}L^{-1/4}|\geq
10C_{*}w_{\beta},|x|\leq 1\ \Big{|}\
\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\right)\\\
>1-C\exp(-cC_{*}L^{1/2}).$

We can replace the conditioning
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$ in the above two estimates
by $\mathcal{E}_{\mathrm{cond}}$, using (12.1). Then from these, we have that
with probability $>1-C\exp(-cL^{1/2})$ conditioning on
$\mathcal{E}_{\mathrm{cond}}$,

$\displaystyle\int_{|x-x_{1}L^{-1/4}|\geq 10C_{*}w_{\beta},|x|\leq
1}\exp\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)+\mathcal{L}^{\beta}(x,s_{1};0,1)\right)\,\mathrm{d}x$
$\displaystyle\leq\exp\left(\sum_{i=1}^{k}h_{i}-(k-2)\log(2L^{1/2})-\log(L^{1/2})-\tfrac{1}{2}C_{*}\right)$
$\displaystyle\leq\exp(L-\tfrac{1}{4}C_{*})\leq\exp(-\tfrac{1}{4}C_{*})\mathcal{L}^{\beta}[1],$

using that $\vec{h}\in E_{\mathrm{val}}$ and $C_{*}$ is large for the second
inequality, and that we have conditioned on $\mathcal{L}^{\beta}[1]>L$ for the
last inequality. This implies that

$\displaystyle\mathcal{L}^{\beta}[1]\leq(1-\exp(-\tfrac{1}{4}C_{*}))^{-1}\int_{|x-x_{1}L^{-1/4}|\leq
10C_{*}w_{\beta}}\exp\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)+\mathcal{L}^{\beta}_{+}[1](x)\right)\,\mathrm{d}x.$
(12.6)

It also follows from Corollary 2.21 that, with probability at least
$1-C\exp(-cC_{*}L^{1/2})$ conditioning on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$,

$h_{1}-20C_{*}\leq\inf_{|x-x_{1}L^{-1/4}|\leq
10C_{*}w_{\beta}}\mathfrak{h}^{\beta}_{s_{1},1}(x)\leq\sup_{|x-x_{1}L^{-1/4}|\leq
10C_{*}w_{\beta}}\mathfrak{h}^{\beta}_{s_{1},1}(x)\leq h_{1}+C_{*}.$

And by Corollary 2.21 and Proposition 3.4, with probability at least
$1-C\exp(-cC_{*}L^{1/2})$ conditioning on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, there is

$\mathcal{L}^{\beta}(x,s_{i-1};y,s_{i})>h_{i}-30C_{*},$

for any $i=2,\ldots,k$, and $|x-x_{i-1}L^{-1/4}|,|y-x_{i}L^{-1/4}|\leq
10C_{*}w_{\beta}$; thus

$\inf_{|x-x_{1}L^{-1/4}|\leq
10C_{*}w_{\beta}}\mathcal{L}^{\beta}_{+}[1](x)\geq\sum_{i=2}^{k}h_{i}-(k-2)\log(2L^{1/2})-30kC_{*}.$

Combining these two estimates with (12.5), we get that, with probability at
least $1-C\exp(-cC_{*}L^{1/2})$ conditioning on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$,

$\displaystyle\frac{\int_{x_{1}L^{-1/4}+\frac{1}{4}w_{\beta}}^{x_{1}L^{-1/4}+\frac{3}{4}w_{\beta}}\exp\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)+\mathcal{L}^{\beta}(x,s_{1};0,1)\right)\,\mathrm{d}x}{\int_{x_{1}L^{-1/4}-10C_{*}w_{\beta}}^{x_{1}L^{-1/4}+10C_{*}w_{\beta}}\exp\left(\mathfrak{h}^{\beta}_{s_{1},1}(x)+\mathcal{L}^{\beta}(x,s_{1};0,1)\right)\,\mathrm{d}x}\geq(40C_{*})^{-1}\exp(-40kC_{*}).$

We can further replace the conditioning by $\mathcal{E}_{\mathrm{cond}}$,
using (12.1). Then together with (12.6) the proof completes. ∎

###  12.2. Free energy of restricted interval

In this subsection, we prove the following statement.

We introduce some notation to describe the free energy profiles in disjoint
temporal strips defined by $[s_{i-1},s_{i}]$. We define

$\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}=\mathcal{L}^{\beta}(x_{i-1}L^{-1/4},s_{i-1};x_{i}L^{-1/4}+\cdot,s_{i}),\quad\mathfrak{h}^{\mathrm{bot},i,\vec{x}}_{1}=\mathcal{L}^{\beta}(x_{i-1}L^{-1/4}+\cdot,s_{i-1};x_{i}L^{-1/4},s_{i}),$
(12.7)

for each $i=1,\ldots,k$. For any vector $\vec{z}\in\mathbb{R}^{k-1}$, we
always write $z_{0}=z_{k}=0$ for the convenience of notations. We denote

$\mathfrak{h}^{\mathrm{sum}}(\vec{z})=\sum_{i=1}^{k-1}\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}(z_{i})+\sum_{i=2}^{k}\mathfrak{h}^{\mathrm{bot},i,\vec{x}}_{1}(z_{i-1}),$
(12.8)

We also write $I_{M,L}=[-ML^{-1/2}\log L,ML^{-1/2}\log L]$, and
$I_{\vec{x}}=L^{-1/4}\vec{x}+I_{M,L}^{k-1}\subset\mathbb{R}^{k-1}$. Recall the
$\emax[(\beta)]$ notation from (9.1).

######  Lemma 12.5.

When $\beta=1$, there exists $\rho>0$ such that, conditional on
$\mathcal{E}_{\mathrm{cond}}$, it holds with probability at least $1-CL^{-1}$
that

$\displaystyle\emax[(\beta)]_{\vec{z}\in
I^{k-1}_{M,L}}\mathfrak{h}^{\mathrm{sum}}(\vec{z})\geq L+\rho
L^{-3}+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i}.$

When $\beta=\infty$, conditional on $\mathcal{E}_{\mathrm{cond}}$, it holds
with probability at least $1-\exp(-cL^{1/2})$ that

$\displaystyle\max_{\vec{z}\in
I^{k-1}_{M,L}}\mathfrak{h}^{\mathrm{sum}}(\vec{z})\geq
L+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i}.$

To prove this in the $\beta=1$ case, the main thing we need is that the
contribution to the free energy $\mathcal{L}^{\beta}[1]$ from outside
$I_{\vec{x}}$ is small, which we isolate in the following statement.

######  Lemma 12.6.

When $\beta=1$, conditional on $\mathcal{E}_{\mathrm{cond}}$, with probability
at least $1-\exp(-cML^{1/2})$,

$\displaystyle\emax[(\beta)]_{\vec{z}\in[-1,1]^{k-1}\setminus
I_{\vec{x}}}\sum_{i=1}^{k}\mathcal{L}^{\beta}(z_{i-1},s_{i-1};z_{i},s_{i})\leq
L-\frac{1}{2}M\log L.$

To handle the $\beta=\infty$ case, we will need the following statement, which
is that the maximizer restricted to $I_{\vec{x}}$ is with high probability the
same as the global maximize. We note that it also holds when $\beta=1$.

We define the restricted (in terms of the interval over which the maximization
is performed) version of $\pi^{*}(s_{i})$: for $i=1,\ldots,k-1$ we let

$\displaystyle\pi^{*,\mathrm{res},\vec{x}}(s_{i})=\operatorname*{argmax}_{|z|\leq
ML^{-1/2}\log
L}\left(\mathcal{L}^{\beta}(0,s_{i-1};x_{i}L^{-1/4}+z,s_{i})+\mathcal{L}^{\beta}(x_{i}L^{-1/4}+z,s_{i};0,s_{i+1})\right),$
(12.9)

for $M$ being a large number.

######  Lemma 12.7.

With probability $1-\exp(-cML^{1/2}\log L)$ conditional on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, for each $i=1,\ldots,k$,

$\displaystyle\pi^{*}(s_{i})=x_{i}L^{-1/4}+\pi^{*,\mathrm{res},\vec{x}}(s_{i}).$

We give the proof of Lemma 12.5 before turning to proving Lemmas 12.6 and
12.7.

###### Proof of Lemma 12.5.

In this proof, by ‘conditional probability’, we refer to the conditioning
$\mathcal{E}_{\mathrm{cond}}$.

_Case of $\beta=\infty$_: We let $\mathcal{E}_{\mathrm{eq}}$ denote the event
where $\pi^{*}(s_{i})=x_{i}L^{-1/4}+\pi^{*,\mathrm{res},\vec{x}}(s_{i})$ for
each $i=1,\ldots,k-1$. We first note that by Lemma 12.7 and (12.1), for all
large enough $M$, $\mathcal{E}_{\mathrm{eq}}$ holds with conditional
probability $>1-\exp(-cML^{1/2})$. Now on $\mathcal{E}_{\mathrm{eq}}$ it
follows by definition that $\mathcal{L}^{\beta}[1]=\max_{\vec{z}\in
I_{\vec{x}}}\sum_{i=1}^{k}\mathcal{L}^{\beta}(z_{i-1},s_{i-1};z_{i},s_{i})$.
Further by coalescence, which holds with conditional probability
$>1-C\exp(-cL^{3/2})$ by Proposition 3.4 and (12.1), it follows that

$\max_{\vec{z}\in
I^{k}_{M,L}}\mathfrak{h}^{\mathrm{sum}}(\vec{z})=\max_{\vec{z}\in
I_{\vec{x}}}\sum_{i=1}^{k}\mathcal{L}^{\beta}(z_{i-1},s_{i-1};z_{i},s_{i})+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i}\geq
L+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i},$

completing the proof.

_Case of $\beta=1$_: By Lemma 12.1 there exists $\rho>0$ such that, with
conditional probability at least $1-CL^{-1}$, we have
$\mathcal{L}^{\beta}[1]\geq L+\rho L^{-3}$. By Lemma 12.6, with conditional
probability at least $1-\exp(-cML^{1/2})$,

$\displaystyle\exp(\mathcal{L}^{\beta}[1])-\exp\left(\emax[(\beta)]_{\vec{z}\in
I_{\vec{x}}}\sum_{i=1}^{k}\mathcal{L}^{\beta}(z_{i-1},s_{i-1};z_{i},s_{i})\right)\leq
L^{-\frac{1}{2}M}\exp(L).$

The previous two inequalities, along with $\exp(x)\geq 1+x$, imply that

$\displaystyle\exp\left(\emax[(\beta)]_{\vec{z}\in
I_{\vec{x}}}\sum_{i=1}^{k}\mathcal{L}^{\beta}(z_{i-1},s_{i-1};z_{i},s_{i})\right)$
$\displaystyle\geq\exp(L+\rho L^{-3})-L^{-\frac{1}{2}M}\exp(L)$
$\displaystyle\geq\exp(L)\left[1+\rho L^{-3}-L^{-\frac{1}{2}M}\right]$
$\displaystyle\geq\exp(L)\left[1+\tfrac{1}{2}\rho L^{-3}\right].$

By coalescence (Proposition 3.4), with conditional probability at least
$1-C\exp(-cL^{3/2})$,

$\emax[(\beta)]_{\vec{z}\in
I^{k}_{M,L}}\mathfrak{h}^{\mathrm{sum}}(\vec{z})\geq\emax[(\beta)]_{\vec{z}\in
I_{\vec{x}}}\sum_{i=1}^{k}\mathcal{L}^{\beta}(z_{i-1},s_{i-1};z_{i},s_{i})+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i}-kCe^{-cL}\\\
>L+\tfrac{1}{4}\rho
L^{-3}-kCe^{-cL}+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i},$

Since $kCe^{-cL}\ll\rho L^{-1}$, the proof is complete by relabeling $\rho$. ∎

The main step in proving Lemma 12.6 is to handle the case of $k=2$, which is
given in the following statement; then we can obtain the statement for general
$k$ by making use of coalescence (Proposition 3.4).

######  Lemma 12.8.

Conditional on $\mathcal{E}_{\mathrm{cond}}$, it holds with probability at
least $1-\exp(-cML^{1/2})$ that, for each $i=1,\ldots,k-1$,

$\displaystyle\int_{(-x_{i}L^{-1/4}+[-1,1])\setminus
I_{M,L}}\exp\bigl{(}\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}(z)+\mathfrak{h}^{\mathrm{bot},i+1,\vec{x}}_{1}(z)\bigr{)}\,\mathrm{d}z\leq
L^{-M}\exp\left(h_{i}+h_{i+1}\right).$

and

$\displaystyle\int_{-x_{i}L^{-1/4}+[-1,1]}\exp\bigl{(}\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}(z)+\mathfrak{h}^{\mathrm{bot},i+1,\vec{x}}_{1}(z)\bigr{)}\,\mathrm{d}z\leq\exp\left(h_{i}+h_{i+1}-\log(2L^{1/2})+M\right).$

###### Proof.

The second inequality follows from (9.6) in Lemma 9.3 and (12.1).

For the first inequality, by (12.1), it suffices to show that the displayed
inequality holds with probability at least $1-\exp(-cML^{1/2})$ given only
that $\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$. Indeed, conditional on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, from Corollary 2.21, it
holds with probability at least $1-C\exp(-cML^{1/2}\log L)$ that, for all
$z\not\in I_{M,L}$, $z+x_{i}L^{-1/4}\in[-1,1]$,

$\displaystyle\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}(z)$ $\displaystyle\leq
h_{i}-\tfrac{1}{2}M\log L-\tfrac{1}{4}L^{1/2}|z|,$
$\displaystyle\mathfrak{h}^{\mathrm{bot},i+1,\vec{x}}_{1}(z)$
$\displaystyle\leq h_{i+1}-\tfrac{1}{2}M\log L-\tfrac{1}{4}L^{1/2}|z|.$

By plugging these into the integral, the conclusion follows. ∎

###### Proof of Lemma 12.6.

By coalescence Proposition 3.4 and (12.1), with probability at least
$1-C\exp(-cL^{3/2})$ conditional on $\mathcal{E}_{\mathrm{cond}}$, the LHS is
at most

$\emax[(\beta)]_{\vec{z}+L^{-1/4}\vec{x}\in[-1,1]^{k-1}\setminus
I_{\vec{x}}}\sum_{i=1}^{k-1}\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}(z_{i})+\mathfrak{h}^{\mathrm{bot},i+1,\vec{x}}_{1}(z_{i})+Ce^{-cL}-\sum_{i=2}^{k-1}h_{i}.$

Recall that the first term is the logarithm of an integral over
$[-1,1]^{k-1}\setminus I_{\vec{x}}$. We can upper bound the integral by
replacing $[-1,1]^{k-1}\setminus I_{\vec{x}}$ with the $k-1$ dimension product
where all but the $i$th one is $[-1,1]$ and the $i$th one is
$[-1,1]\setminus(x_{i}L^{-1/4}+I_{M,L})$, and summing over $i$ (i.e., the sum
is inside the logarithm). Using Lemma 12.8, we get an upper bound of

$\displaystyle\sum_{i=1}^{k}h_{i}+Ce^{-cL}-M\log L-(k-2)\log(2L^{1/2})+M+C\log
k.$

Using that $\vec{h}\in E_{\mathrm{val}}$, by taking $M$ large enough the
conclusion follows. ∎

###### Proof of Lemma 12.7.

We need to show that with high probability, conditional on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, it holds that
$\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|\leq ML^{-1/2}\log L$ so
that $\pi^{*}(s_{i})=x_{i}L^{-1/4}+\pi^{*,\mathrm{res},\vec{x}}(s_{i})$.

Let $J_{i}$ denote the interval $\\{z:|z+x_{i}L^{-1/4}|\leq
10^{-6}(s_{i}-s_{i-1})^{1/2}\wedge(s_{i+1}-s_{i})^{1/2}\\}$. By Proposition
3.4, conditional on $\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, with
probability $>1-C\exp(-cL^{3/2})$ the event $\pi^{*}(s_{i})\neq
x_{i}L^{-1/4}+\pi^{*,\mathrm{res},\vec{x}}(s_{i})$ implies that

$\sup_{z\in J_{i}\setminus
I_{M,L}}\mathfrak{h}_{1}^{\mathrm{top},i,\vec{x}}(z)+\mathfrak{h}_{1}^{\mathrm{bot},i+1,\vec{x}}(z)>h_{i}+h_{i+1}-C\exp(-cL),$
(12.10)

or

$\sup_{z\not\in
J_{i}}\mathcal{L}^{\beta}(0,s_{i-1};x_{i}L^{-1/4}+z,s_{i})+\mathcal{L}^{\beta}(x_{i}L^{-1/4}+z,s_{i};0,s_{i+1})\\\
\geq\mathcal{L}^{\beta}(0,s_{i-1};x_{i}L^{-1/4},s_{i})+\mathcal{L}^{\beta}(x_{i}L^{-1/4},s_{i};0,s_{i+1}).$
(12.11)

We have that (12.10) further implies that either

$\sup_{z\in J_{i}\setminus
I_{M,L}}\mathfrak{h}_{1}^{\mathrm{top},i,\vec{x}}(z)>h_{i+1}-C\exp(-cL),$

or

$\sup_{z\in J_{i}\setminus
I_{M,L}}\mathfrak{h}_{1}^{\mathrm{bot},i+1,\vec{x}}(z)>h_{i}-C\exp(-cL),$

happens; and by Corollary 2.21, each happens, conditional on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, with probability
$<\exp(-cML^{1/2}\log L)$.

As for (12.11), by Corollary 2.21, with probability $>1-C\exp(-cL^{3/4})$
conditional on $\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$, we have that
the RHS is $>h_{i}+h_{i+1}-L^{1/4}\log L$. For the LHS, for any fixed
$z\not\in J_{i}$ we have

$\mathbb{P}\left(\mathcal{L}^{\beta}(0,s_{i-1};x_{i}L^{-1/4}+z,s_{i})+\mathcal{L}^{\beta}(x_{i}L^{-1/4}+z,s_{i};0,s_{i+1})>h_{i}+h_{i+1}-L^{1/4}\log
L\right)\\\
<C\exp\left(-\tfrac{4}{3}(s_{i+1}-s_{i-1})L^{3/2}-cz^{2}L^{1/2}\right)$

by Theorem 2.13, and that $\vec{h}\in E_{\mathrm{val}}$. Then by a union bound
over all $z\not\in J_{i}$, $z\in L^{-2}L$, and using the continuity estimate
Proposition 2.16, we get that the LHS is $\leq h_{i}+h_{i+1}-L^{1/4}\log L$
with probability
$>1-C\exp\left(-\tfrac{4}{3}(s_{i+1}-s_{i-1})L^{3/2}-cL^{3/2}\right)$. Then
since

$\mathbb{P}\left(\mathcal{L}^{\beta,\vec{x}}_{i}\in
h_{i}+[0,e^{-L}]\right)>c\exp\left(-\tfrac{4}{3}(s_{i}-s_{i-1})L^{3/2}-CL^{3/4}\right)$

and

$\mathbb{P}\left(\mathcal{L}^{\beta,\vec{x}}_{i+1}\in
h_{i+1}+[0,e^{-L}]\right)>c\exp\left(-\tfrac{4}{3}(s_{i+1}-s_{i})L^{3/2}-CL^{3/4}\right),$

by Theorem 2.13, we conclude that (12.11) happens with probability
$<C\exp(-cL^{3/4})$ conditional on
$\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}$. This completes the proof. ∎

###  12.3. The density comparison

We now finish the proof of Lemma 11.3.

Recall the setup at the beginning of this section. Also recall the definition
of $\pi^{*,\mathrm{res},\vec{x}}(s_{i})$ from (12.9) as a version of
$\pi^{*}(s_{i})$ defined for $i=1,\ldots,k-1$ but with a restricted interval
over which the maximization is performed, and the definition of
$\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}$ and
$\mathfrak{h}^{\mathrm{bot},i,\vec{x}}_{1}$ from (12.7), and
$\mathfrak{h}^{\mathrm{sum}}$ from (12.8).

As before, we let $I_{M,L}=[-ML^{-1/2}\log L,ML^{-1/2}\log L]$, and
$w_{\beta=1}=L^{-1/2}$, $w_{\beta=\infty}=L^{-1}$.

###### Proof of Lemma 11.3.

We want to estimate the conditional probabilities on both the LHS and the RHS
and show that they are within $1+o(1)$ of each other. Towards this, for either
$\iota=0$ or $1$ (we allow both as the RHS in Lemma 11.3 does not have the
$\smash{\beta^{-1}e^{-cL^{1/4}}}$ term), we claim that

$\mathbb{P}\Bigl{(}\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\mathcal{L}^{\beta}[1]>(1-\iota\beta^{-1}e^{-cL^{1/4}})L\
\Big{|}\
\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\Bigr{)}=(1+O(L^{-1}))\\\
\times\mathbb{P}\left(\max_{i=1,\ldots,k-1}|\pi^{*}(s_{i})-x_{i}L^{-1/4}|<w_{\beta},\emax[(\beta)]_{\vec{z}\in
I_{M,L}^{k-1}}\mathfrak{h}^{\mathrm{sum}}(\vec{z})\geq L+\beta^{-1}\rho
L^{-3}+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i}\ \Big{|}\
\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h}\right).$ (12.12)

Indeed, by Lemma 12.5, the LHS is upper bounded by the RHS; while by
Proposition 3.4,

$\mathcal{L}^{\beta}[1]+\sum_{i=2}^{k-1}\mathcal{L}^{\beta,\vec{x}}_{i}\geq\emax[(\beta)]_{\vec{z}\in
I_{M,L}^{k-1}}\mathfrak{h}^{\mathrm{sum}}(\vec{z}),$

with probability $>1-C\exp(-cL^{3/2})$, conditional on
$\mathcal{L}^{\beta,\vec{x}}\approx\vec{h}$. Then since the LHS is lower
bounded by $c\exp(-CL^{1/2})$ ($\beta=1$) or $c$ ($\beta=\infty$) by
Proposition 9.1 and $\vec{h}\in E_{\mathrm{val}}$, the LHS is lower bounded by
the RHS.

We now let $\mathcal{E}_{\pm}$ denote the events where for each
$i=1,\ldots,k-1$,

$\sup_{|z|\leq
w_{\beta}}\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}(z)+\mathfrak{h}^{\mathrm{bot},i+1,\vec{x}}_{1}(z)>\sup_{|z|\geq
w_{\beta},z\in
I_{M,L}}\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}(z)+\mathfrak{h}^{\mathrm{bot},i+1,\vec{x}}_{1}(z)\mp
C\exp(-cL),$

and

$\emax[(\beta)]_{\vec{z}\in
I_{M,L}^{k-1}}\mathfrak{h}^{\mathrm{sum}}(\vec{z})\geq L+\beta^{-1}\rho
L^{-3}+\sum_{i=2}^{k-1}h_{i}\mp C\exp(-cL).$

By Lemma 12.7 and Proposition 9.1, the probability in the RHS of (12.12) is in

$\left[\mathbb{P}(\mathcal{E}_{-}\mid\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h})-\exp(-cML^{1/2}),\;\mathbb{P}(\mathcal{E}_{+}\mid\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h})+\exp(-cML^{1/2})\right].$
(12.13)

Now we wish to make use of Proposition 3.5 to replace the processes
$\mathfrak{h}^{\mathrm{top},i,\vec{x}}_{1}$ and
$\mathfrak{h}^{\mathrm{bot},i,\vec{x}}_{1}$ by a collection of processes given
by independent Brownian bridges, up to an exponentially small $L^{\infty}$
error and on an event with probability $1-\exp(-cL)$.

Denote

$G=L+\beta^{-1}\rho L^{-3}-\sum_{i=1}^{k}h_{i}$

Since $\vec{h}\in E_{\mathrm{val}}$ from (11.5), we have
$G<-(k-1)\beta^{-1}\log(2L^{1/2})+Cr_{\beta}$ (recall that $r_{\beta=1}=1$ and
$r_{\beta=\infty}=L^{-1/2}$).

Reduce to Brownian bridges. We denote
$\theta=10^{-7}\min_{i=1,\ldots,k}(s_{i}-s_{i-1})L^{1/2}$. We let $\Theta$
denote the collection of all

$\vec{b}=(\vec{b}^{\mathrm{bot},\mathrm{L}},\vec{b}^{\mathrm{bot},\mathrm{R}},\vec{b}^{\mathrm{top},\mathrm{L}},\vec{b}^{\mathrm{top},\mathrm{R}})=\bigl{(}(b^{\mathrm{bot},\mathrm{L}}_{i})_{i=2}^{k},(b^{\mathrm{bot},\mathrm{R}}_{i})_{i=2}^{k},(b^{\mathrm{top},\mathrm{L}}_{i})_{i=1}^{k-1},(b^{\mathrm{top},\mathrm{R}}_{i})_{i=1}^{k-1}\bigr{)}\in\mathbb{R}^{4(k-1)}.$

For any $\vec{b}\in\Theta$, we denote by $B^{\dagger,i,\vec{b}}$ a rate $2$
Brownian bridge from $(-\theta,-2\theta
L^{1/2}+b^{\mathrm{\dagger},\mathrm{L}}_{i})$, to $(0,0)$, then to
$(\theta,-2\theta L^{1/2}+b^{\mathrm{\dagger},\mathrm{R}}_{i})$, for
$\dagger=\mathrm{top}$, $i=1,\ldots,k-1$, and $\dagger=\mathrm{bot}$,
$i=2,\ldots,k$. All these $B^{\dagger,i}$ are independent of each other. One
should think of $B^{\dagger,i}$ as being approximately
$\mathfrak{h}_{1}^{\dagger,i,\vec{x}}-h_{i}$, which we will use in a more
precise sense via Proposition 3.5.

We then denote by $\mathcal{E}_{\vec{b}}$ the event where for each
$i=1,\ldots,k-1$,

$\sup_{|z|\leq
w_{\beta}}B^{\mathrm{top},i,\vec{b}}(z)+B^{\mathrm{bot},i+1,\vec{b}}(z)>\sup_{|z|\geq
w_{\beta},z\in
I_{M,L}}B^{\mathrm{top},i,\vec{b}}(z)+B^{\mathrm{bot},i+1,\vec{b}}(z),$

and

$\emax[(\beta)]_{\vec{z}\in
I_{M,L}^{k-1}}\sum_{i=1}^{k-1}B^{\mathrm{top},i,\vec{b}}(z_{i})+\sum_{i=2}^{k}B^{\mathrm{bot},i,\vec{b}}(z_{i-1})\geq
G.$

We also denote by $\mathcal{E}_{\vec{b},\pm}$ the event where for each
$i=1,\ldots,k-1$,

$\sup_{|z|\leq
w_{\beta}}B^{\mathrm{top},i,\vec{b}}(z)+B^{\mathrm{bot},i+1,\vec{b}}(z)>\sup_{|z|\geq
w_{\beta},z\in
I_{M,L}}B^{\mathrm{top},i,\vec{b}}(z)+B^{\mathrm{bot},i+1,\vec{b}}(z)\mp
C\exp(-cL),$

and

$\emax[(\beta)]_{\vec{z}\in
I_{M,L}^{k-1}}\sum_{i=1}^{k-1}B^{\mathrm{top},i,\vec{b}}(z_{i})+\sum_{i=2}^{k}B^{\mathrm{bot},i,\vec{b}}(z_{i-1})\geq
G\mp C\exp(-cL).$

It is straightforward to check that

$|\mathbb{P}(\mathcal{E}_{\vec{b}})-\mathbb{P}(\mathcal{E}_{\vec{b},\pm})|<C\exp(-cL).$
(12.14)

For each $a>0$ we let $\Theta_{a}$ denote the collection of all
$\vec{b}\in\Theta$, where each coordinate is in $[-a,a]$. We write $D=\log L$.

Since $\vec{x}\in\mathcal{K}$ which is a compact set, applying Proposition
3.5, and using Lemma 2.20 to bound each $\mathfrak{h}^{\mathrm{top},i}_{1}$
and $\mathfrak{h}^{\mathrm{bot},i}_{1}$ at $\pm\theta$, we have that

$\mathbb{P}(\mathcal{E}_{-}\mid\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h})\geq(1-C\exp(-cD^{2}))\min_{\vec{b}\in\Theta_{DL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b},-})-C\exp(-cL),$
(12.15)

and

$\mathbb{P}(\mathcal{E}_{+}\mid\vec{\mathcal{L}}^{\beta,\vec{x}}\approx\vec{h})\leq\max_{\vec{b}\in\Theta_{DL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b},+})+\sum_{i=2}^{\lfloor
L^{1/4}\rfloor}C\exp(-ci^{2}D^{2})\max_{\vec{b}\in\Theta_{iDL^{1/4}}}\mathbb{P}(\mathcal{E}_{\vec{b},+})\\\
+C\exp(-cL)+C\exp(-cD^{2}L^{1/2}).$ (12.16)

By (12.14), we can replace each $\mathbb{P}(\mathcal{E}_{\vec{b},\pm})$ by
$\mathbb{P}(\mathcal{E}_{\vec{b}})$. We also note that these bounds are
independent of $\vec{x}$.

We next state a comparison lemma for different $\vec{b}$.

######  Lemma 12.9.

For any large enough $K>0$, and $\vec{b},\vec{g}\in\Theta$ such that