Hence taking $t=1/\sqrt{\omega}$, if we let $\mathcal{E}$ be the event that

$|\Delta_{k}^{(j)}-\Delta_{k_{r-1}}^{(j)}|\leq|\Delta_{k_{r-1}}^{(j)}|+(1/\sqrt{\omega}),\quad\text{for
all $r\geq 1$ such that $k_{r-1}\leq n$},$

then this event holds with probability $1-e^{-\Omega(n^{\delta})}$. We turn to
controlling $\Delta_{k_{r}}^{(j)}$ on the event $\mathcal{E}$ using
Proposition 5.6. From Lemma 5.3,
$A_{r+\lfloor{n_{1}^{+}}/\varkappa\rfloor}=\Delta_{k_{r-1}}^{(j)}$ satisfies
(5.13) with $V=C/\beta$, $W=C/(\beta\omega)$, $\epsilon=1$ and
$E=1/\sqrt{\omega}$.

$\mathbb{P}\left[\bigl{\\{}\max_{r:k_{r-1}\leq
n}|A_{r+\lfloor{n_{1}^{+}}/\varkappa\rfloor}|\geq
x\sqrt{1/\omega}\bigr{\\}}\cap\mathcal{E}\right]\leq
2\exp\left(-\frac{1}{C}\frac{\beta(\log x)^{2}}{\log(n/{n_{1}^{+}})}\right).$

Hence taking $x=n^{\delta}$ completes the proof for the imaginary part.

Once more, for the real part the proof is simpler: having controlled the
difference of imaginary parts, the difference of real parts admits a block
martingale structure. We suppress the details. ∎

###  5.3 Coupling to Brownian motions

We augment the probability space by creating a family of complex Brownian
motions $\\{\widehat{(\mathfrak{W}}_{t}^{j}:t\geq
0):j\in\mathcal{D}_{n/k_{1}}\\}$ having

$\sqrt{\tfrac{2}{k+1}}Z_{k}^{(j)}=\widehat{\mathfrak{W}}^{j}_{H_{k+1}}-\widehat{\mathfrak{W}}^{j}_{H_{k}}$

for all $k\geq n_{1}^{+}$. They may be constructed so that conditionally on
all $\\{Z_{k}^{(j)}:k,j\\}$, the bridges

$\biggl{\\{}(\widehat{\mathfrak{W}}^{j}_{t}-\widehat{\mathfrak{W}}^{j}_{H_{k}}:t\in[H_{k},H_{k+1}]),j\in\mathcal{D}_{n/k_{1}},k\geq
n_{1}^{+}\biggr{\\}}$

are independent. By construction we may extend $\lambda^{(j)}$ to a continuous
function of time by setting (c.f. (5.6))

$\lambda_{t}^{(j)}(\theta)=\lambda_{k}^{(j)}(\theta)+i\theta(t-k)+\sqrt{\tfrac{4}{\beta}}(\widehat{\mathfrak{W}}^{j}_{H_{t}}-\widehat{\mathfrak{W}}^{j}_{H_{k}})e^{i\Im\lambda^{*,(j)}_{k}(\theta)}\quad\text{for
}t\in[k,k+1],$

and where $H_{t}=H_{k}+\tfrac{t-k}{k+1}.$ We make a time change by setting

$k_{n}(t)=n_{1}^{+}\exp\biggl{(}\frac{\log(n/n_{1}^{+})}{T+-T_{-}}\bigl{(}t-T_{-}\bigr{)}\biggr{)}\quad
t\in[T_{-},T_{+}].$

In terms of this time change, we set

$\widehat{\mathfrak{L}}^{j}_{t}(\theta)=\lambda_{k_{n}(t)}^{(j)}(\theta_{j}+\tfrac{\theta}{n})-i(k_{n}(t)+1)\theta_{j}.$

Finally, we define the Brownian motion $\mathfrak{W}_{t}^{j}$ by the identity

$d\mathfrak{W}_{t}^{j}=\frac{e^{i(k_{n}(t)+1)\theta_{j}}d\widehat{\mathfrak{W}}_{H_{k_{n}(t)}}^{j}}{\sqrt{\tfrac{d}{dt}(H_{k_{n}(t)})}}\quad\text{on}\quad
t\in[T_{-},T_{+}].$ (5.9)

Finally, we recall $\mathfrak{L}_{t}^{j}$ which solves . The function
$\widehat{\mathfrak{L}}^{j}_{t}(\theta)$ is an approximate solution of the
same equation, and we can compare the two solutions.

######  Lemma 5.5.

For any $C>0$ and any $\delta>0$ sufficiently small, for all $n$ sufficiently
large

$\sup_{|\theta|\leq
n^{1-\delta}}\mathbb{P}_{M}\bigl{[}\sup_{t\in[T_{-},T_{+}]}|\widehat{\mathfrak{L}}^{j}_{t}(\theta)-\mathfrak{L}^{j}_{t}(\theta)|>n^{-\delta/2}\bigr{]}\leq
n^{-C}\quad\operatorname{a.s.},$

###### Proof.

We begin by posing a stochastic differential equation for
$\widehat{\mathfrak{L}}^{j}_{t}(\theta)$. We have that it is a strong solution
of the differential equation

$d\widehat{\mathfrak{L}}^{j}_{t}(\theta)=i\frac{\theta
k_{n}^{\prime}(t)}{n}+\sqrt{\tfrac{4}{\beta}}d\bigl{(}\widehat{\mathfrak{W}}_{H_{k_{n}(t)}}^{j}\bigr{)}e^{i\Im\lambda^{*,(j)}_{k_{n}(t)}(\theta)}.$
(5.10)

We note the derivative $k_{n}^{\prime}(t)$ satisfies

$\tfrac{1}{n}k_{n}^{\prime}(t)=e^{t}k_{1}^{-1}\bigl{(}1+O_{k_{1}}(1/n)\bigr{)}.$

Similarly, almost everywhere,

$\tfrac{d}{dt}\bigl{(}H_{k_{n}(t)}\bigr{)}=\frac{k_{n}^{\prime}(t)}{\lfloor
k_{n}(t)\rfloor+1}=t+O_{k_{1}}(1/n).$

Thus we can express the SDE as

$d\widehat{\mathfrak{L}}^{j}_{t}(\theta)=i\theta(e^{t}k_{1}^{-1}+E_{1}(t))dt+\sqrt{\tfrac{4}{\beta}}d\mathfrak{W}^{j}_{t}e^{i\Im\widehat{\mathfrak{L}}^{j}_{t}(\theta)+iE_{3}(t)}(1+E_{2}(t)),$
(5.11)

for deterministic errors $E_{1},E_{2}$ which are $O_{k_{1}}(1/n)$ and a random
error $E_{3}(t)$ which is controlled by Lemma 5.1.

Hence if we form the difference
$D_{t}\coloneqq\widehat{\mathfrak{L}}^{j}_{t}(\theta)-{\mathfrak{L}}^{j}_{t}(\theta)$,
we have that

$dD_{t}=i\theta
E_{1}(t)dt+\sqrt{\tfrac{4}{\beta}}d\mathfrak{W}^{j}_{t}e^{i\Im{\mathfrak{L}}^{j}_{t}(\theta)}(e^{i\Im
D_{t}}-1+(e^{i\Im D_{t}+iE_{3}(t)}-e^{i\Im D_{t}})+E_{2}(t)).$ (5.12)

We can furthermore check that $f_{t}=\log(1+n^{2\delta}|D_{t}|^{2})$ has both
drift and diffusion coefficient which are bounded above by $O_{k_{1}}(1)$
uniformly in $\theta$ with probability at least $1-n^{-C}$ by Lemma 5.1. It
follows that we have a Gaussian tail bound for the difference with a variance
which is $O_{k_{1}}(1)$, which implies the claim. ∎

The lemmas assembled give a proof of Proposition 2.12.

###### Proof of Proposition 2.12.

We briefly survey the role of each lemma. We apply these lemmas with
$\varkappa=n^{1-\epsilon}$ and $\omega=2n^{\epsilon}$. Note we may pick the
$\delta$’s that appear in the Lemmas as small as desired with respect to
$\epsilon$ (say $\epsilon/1000$). Lemma 5.5 connects the SDE
$\mathfrak{L}_{t}^{j}(\theta)$ to an approximate solution
$\widehat{\mathfrak{L}}^{j}_{t}(\theta)$ This approximate solution is a
time–changed and spatially scaled version of the process
$\lambda_{k}^{(j)}(\theta)$. Lemma 5.4 bounds the difference between
$\lambda_{k}^{(j)}(\theta)$ and $\lambda_{k}(\theta)$. This shows that we can
replace the driving Gaussian noise by band–resampled Gaussians, for which
Fourier modes that are far from those $\theta_{j}$ are resampled. Finally
Lemma 5.2 shows that $\lambda_{k}(\theta)$, which is a locally linearized (in
time) version of $2\log\Phi^{*}(k(\theta)$ is indeed close to it. ∎

###  5.4 Logarithmic ladder

We suppose that $\left\\{A_{k}\right\\}$ is a sequence of random variables
which roughly has the type of multiplicative recurrence structure of the
Prüfer phases. This is to say, we let
$\mathscr{F}_{k}=\sigma(A_{1},A_{2},\dots,A_{k})$ and we suppose there are
constants $V$ and $W$ so that for all $\lambda\in{\mathbb{R}}$ and all
$k\geq{n_{1}^{+}}$ for some ${n_{1}^{+}}\in{\mathbb{N}},$

$\mathbb{E}\left[e^{\lambda A_{k+1}}\middle|\mathscr{F}_{k}\right]\leq
e^{\lambda A_{k}+\frac{\lambda^{2}}{k}(VA_{k}^{2}+W)}.$ (5.13)

######  Proposition 5.6.

Let $\epsilon,E>0$ be given and suppose that $E\leq\sqrt{W/V}.$ Let
$\mathcal{E}$ be the event such that

$A_{k+1}\leq(1+\epsilon)A_{k}+E\quad\text{for all }{n_{1}^{+}}\leq k\leq
n\quad\text{ and }A_{{n_{1}^{+}}}\leq E.$

There is an absolute constant $C>0$ so that for all
$x\geq\max\\{(2+\epsilon)^{2},CV(2+\epsilon)^{3}\log(2+\epsilon)\log(n/{n_{1}^{+}})\\},$

$\mathbb{P}\bigl{[}\bigl{\\{}\max_{{n_{1}^{+}}\leq k\leq n}A_{k}\geq
x\sqrt{W/V}\bigr{\\}}\cap\mathcal{E}\bigr{]}\leq\exp\left(-\tfrac{1}{C}\tfrac{(\log
x)^{2}}{V(2+\epsilon)^{3}\log(2+\epsilon)^{2}\log(n/{n_{1}^{+}})}\right).$

###### Proof.

Set $\eta=1+\epsilon$. We define stopping times $\\{\tau_{p}\\}$, for
$p\in{\mathbb{N}}$, as the first times $k\geq{n_{1}^{+}}$ that $A_{k}$ exceeds
$\eta^{p}E$ or that $k=n$. Then it follows that on $\mathcal{E}$

$A_{\tau_{p}}\leq(1+\epsilon)\eta^{p}E+E.$

Define, for any $p\in{\mathbb{N}}$ the process

$M_{j}=e^{\lambda
A_{j}(\theta)-\lambda^{2}(V\eta^{2p+2}E^{2}+W)H_{j}},\quad\quad\text{where
}H_{j}=\sum_{k=1}^{j}\frac{1}{k}.$

When stopped at $\tau_{p+1},$ $\left\\{M_{j}\right\\}$ is a supermartingale,
and so

$\mathbb{E}\left[M_{\tau_{p+1}}\middle|\mathscr{F}_{\tau_{p}}\right]\leq
M_{\tau_{p}}.$

It follows that

$\mathbb{E}\left[e^{\lambda(\eta^{p}E(\eta-1-\epsilon)-E)-(H_{\tau_{p+1}}-H_{\tau_{p}})\lambda^{2}(V\eta^{2p+2}E^{2}+W)}\mathbf{1}\left\\{{\mathcal{E}}\right\\}\middle|\mathscr{F}_{\tau_{p}}\right]\leq
1.$

Now $H_{\tau_{p+1}}-H_{\tau_{p}}\leq\log(\tau_{p+1}/\tau_{p}).$ On the event
that $\tau_{p+1}/\tau_{p}\leq t_{p}$ for some $t_{p}\geq 1,$ we conclude that

$\mathbb{P}\left[\\{\tau_{p+1}/\tau_{p}\leq
t_{p}\\}\cap\mathcal{E}\middle|\mathscr{F}_{\tau_{p}}\right]\leq
e^{-\lambda(\eta^{p}(\eta-1-\epsilon)-1)E+\log(t_{p})\lambda^{2}(V\eta^{2p+2}E^{2}+W)}.$

Finally, optimizing in $\lambda,$ it follows that

$\displaystyle\mathbb{P}\left[\\{\tau_{p+1}/\tau_{p}\leq
t_{p}\\}\cap\mathcal{E}\middle|\mathscr{F}_{\tau_{p}}\right]$
$\displaystyle\leq\exp\left(-\frac{(\eta^{p}(\eta-1-\epsilon)-1)^{2}E^{2}}{4\log(t_{p})(V\eta^{2p+2}E^{2}+W)}\right)$
$\displaystyle\leq\exp\left(-\frac{\eta^{2p-1}E^{2}}{4\log(t_{p})(V\eta^{2p+2}E^{2}+W)}\right),$

where in the final equality we have used that $\eta=2+\epsilon$ and $p\geq 1.$

Let $r_{0}\geq{n_{1}^{+}}$ be the smallest integer such that
$V\eta^{2r_{0}}E^{2}\geq W.$ Then by iterating the previous conditional
expectation,

$\displaystyle\mathbb{P}\left[\cap_{p=r_{0}}^{r}\\{\tau_{p+1}/\tau_{p}\leq
t_{p}\\}\cap\mathcal{E}\middle|\mathscr{F}_{\tau_{r_{0}}}\right]$
$\displaystyle\leq\exp\left(-\sum_{p=r_{0}}^{r}\frac{\eta^{2p-1}E^{2}}{4\log(t_{p})(V\eta^{2p+2}E^{2}+W)}\right)$
$\displaystyle\leq\exp\left(-\sum_{p=r_{0}}^{r}\frac{1}{8V\eta^{3}\log(t_{p})}\right)$
$\displaystyle\leq\mathbb{P}\left[\cap_{p=r_{0}}^{r}\biggl{\\{}\tfrac{1}{8V\eta^{3}X_{p}}\leq\log(t_{p})\biggr{\\}}\right],$

where $\left\\{X_{p}\right\\}$ are a family of independent
$\operatorname{Exp}(1)$ random variables. Hence, we may couple
$\left\\{\tau_{p}\\!:\\!p\geq r_{0}\right\\}$ with $\left\\{X_{p}\right\\}$ in
such a way that on $\mathcal{E}$

$\tau_{p+1}/\tau_{p}\geq e^{\tfrac{1}{8V\eta^{3}X_{p}}},\quad\text{ for all
}p\geq r_{0}.$

Moreover, we conclude that for any $t\geq 0,$

$\displaystyle\mathbb{P}\left[\\{\tau_{r}/\tau_{r_{0}}\leq
t\\}\cap\mathcal{E}\right]\leq\mathbb{P}\left[\exp\left({\textstyle\sum_{p=r_{0}}^{r-1}}\tfrac{1}{8V\eta^{3}X_{p}}\right)\leq
t\right]$
$\displaystyle=\mathbb{P}\left[{\textstyle\sum_{p=r_{0}}^{r-1}}\tfrac{1}{X_{p}}\leq
8V\eta^{3}\log(t)\right].$

Using the harmonic–mean–arithmetic–mean inequality,

$\frac{r-r_{0}}{\sum_{p=r_{0}}^{r-1}X_{p}}\leq\frac{1}{r-r_{0}}\sum_{p=r_{0}}^{r-1}\tfrac{1}{X_{p}}.$

Hence we arrive at, under the assumption that
$\tfrac{(r-r_{0})^{2}}{8V\eta^{3}\log(t)}\geq 2(r-r_{0}),$

$\mathbb{P}\left[\\{\tau_{r}/\tau_{r_{0}}\leq
t\\}\cap\mathcal{E}\right]\leq\mathbb{P}\left[{\textstyle\sum_{p=r_{0}}^{r-1}X_{p}}\geq\tfrac{(r-r_{0})^{2}}{8V\eta^{3}\log(t)}\right]\leq\exp\left(-\frac{1}{C}\min\left\\{\tfrac{(r-r_{0})^{2}}{V\eta^{3}\log(t)},\tfrac{(r-r_{0})^{3}}{V^{2}\eta^{6}\log(t)^{2}}\right\\}\right),$
(5.14)

using Bernstein’s inequality for subexponential random variables [Ver18,
Theorem 2.8.1]. Observe that under the assumption the minimum is always
attained by the first term.

Finally, we observe that for $r$ such that $V\eta^{2r}E^{2}\geq W,$

$\mathbb{P}\bigl{[}\bigl{\\{}\max_{{n_{1}^{+}}\leq k\leq
n}A_{k}\geq\eta^{r}E\bigr{\\}}\cap\mathcal{E}\bigr{]}\leq\mathbb{P}\bigl{[}\\{\tau_{r}/\tau_{r_{0}}\leq
n/{n_{1}^{+}}\\}\cap\mathcal{E}\bigr{]}\leq\exp\bigl{(}-\tfrac{1}{C}\tfrac{(r-r_{0})^{2}}{V\eta^{3}\log(n/{n_{1}^{+}})}\bigr{)},$

provided $r-r_{0}\geq 16V\eta^{3}\log(n/{n_{1}^{+}}).$ Moreover, for $x\geq
1,$ if we take $r=r_{0}+\lfloor\log(x)/\log(\eta)\rfloor-1$

$\displaystyle\mathbb{P}\bigl{[}\bigl{\\{}\max_{{n_{1}^{+}}\leq k\leq
n}A_{k}\geq x\sqrt{W/V}\bigr{\\}}\cap\mathcal{E}\bigr{]}$
$\displaystyle\leq\mathbb{P}\bigl{[}\bigl{\\{}\max_{{n_{1}^{+}}\leq k\leq
n}A_{k}\geq\eta^{r-r_{0}+1}\sqrt{W/V}\bigr{\\}}\cap\mathcal{E}\bigr{]}$
$\displaystyle\leq\mathbb{P}\bigl{[}\bigl{\\{}\max_{{n_{1}^{+}}\leq k\leq
n}A_{k}\geq\eta^{r}E\bigr{\\}}\cap\mathcal{E}\bigr{]}.$

Hence for $x\geq\eta^{2},$ we conclude there is an absolute constant $C>0$
such that

$\mathbb{P}\bigl{[}\bigl{\\{}\max_{{n_{1}^{+}}\leq k\leq n}A_{k}\geq
x\sqrt{W/V}\bigr{\\}}\cap\mathcal{E}\bigr{]}\leq\exp\bigl{(}-\tfrac{1}{C}\tfrac{(\log
x)^{2}}{V\eta^{3}\log(\eta)^{2}\log(n/{n_{1}^{+}})}\bigr{)},$

provided $\log(x)\geq\log(\eta)+16V\eta^{3}\log(\eta)\log(n/{n_{1}^{+}}).$
This completes the proof. ∎

##  6 Changing the initial condition

In our application, we will want to consider changing the initial conditions
of $\mathfrak{L}_{T_{-}}.$ The real part of $(\mathfrak{L}_{t}:t)$ does not
influence the evolution of the diffusion, and therefore any initial condition
specified for $\Re\mathfrak{L}_{T_{-}}$ will simply appear as an additive
perturbation to solution of $(\mathfrak{L}_{t}:t)$ with
$\Re\mathfrak{L}_{T_{-}}(\theta)=0.$ On the other hand, we wish to show that
for the imaginary part, the probability that a small perturbation of initial
condition grows in magnitude _and_ the random walk performs an unusual growth
(as is needed to be relevant for the maximum) is small. In fact, it will be
important in the real case $\sigma=1$ that having a large real part tends to
compress the relative Prüfer phase.

######  Proposition 6.1.

Fix some $j\in\mathcal{D}_{n/k_{1}}$ and some $\theta\in[-2\pi k_{1},0].$ Let
$(\mathfrak{L}_{t}:t\in[T_{-},T_{+}])$ solve (2.35) and let
$(\mathfrak{L}_{t}^{o}:t\in[T_{-},T_{+}])$ solve (2.44). Suppose
$|\Im(\mathfrak{L}_{T_{-}}-\mathfrak{L}_{t}^{o})(\theta)|\leq\frac{k_{1}(\log
k_{1})^{50}}{k_{1}^{+}}.$ Set
$\Delta_{t}\coloneqq\Im\mathfrak{L}_{t}(\theta)-\Im\mathfrak{L}_{t}^{o}(\theta)-\Im\mathfrak{L}_{T_{-}}(\theta_{j}).$
On the event

$\sqrt{\tfrac{8}{\beta}}\mathcal{A}_{T_{-}}^{-}\leq\mathfrak{U}_{T_{-}}(\theta)\leq\sqrt{\tfrac{8}{\beta}}\mathcal{A}_{T_{-}}^{+},$

there is a $\delta>0$ so that for all $k_{1}$ sufficiently large

$\mathbb{P}(|\Delta_{T_{+}}|>\sqrt{\tfrac{k_{1}^{+}}{k_{1}}},\left\\{\mathfrak{U}_{T_{+}}(\theta)\in[-(\log
k_{1})^{1/100},3k_{6}]\right\\}~{}|~{}\mathscr{F}_{n_{1}^{+}})\leq\frac{1}{k_{1}^{+}}e^{-\delta(\log
k_{1})^{19/20}}.$

The same holds if we replace $\mathfrak{U}$ by $\mathfrak{U}^{o}$ in the above
two equations. _(Addendum for $\sigma=1$): _ Having controlled the difference
of imaginary parts, we can then control the difference of real parts. Setting
$\Delta^{r}_{t}\coloneqq\Re\mathfrak{L}_{t}(\theta)-\Re\mathfrak{L}_{t}^{o}(\theta),$
we draw the same conclusion in the previous display with $\Delta_{t}$ replaced
by $\Delta^{r}_{t}-\Delta^{t}_{T_{-}}.$

###### Proof.

We show the case $\sigma=1$ first. We shall show how to modify the argument
for $\sigma=i$ after completing the $\sigma=1$ case.

##### Step 1: change of measure.

In this case, we have
$d\mathfrak{U}_{t}=-\sqrt{\tfrac{4}{\beta}}\Re(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j}).$
Let
$d\mathfrak{B}_{t}=\sqrt{\tfrac{4}{\beta}}\Im(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j}),$
which (for fixed $\theta$) is an independent Brownian motion of
$\mathfrak{U}$. Define a change of measure

$\frac{d{\mathbb{Q}}}{d\mathbb{P}}=\exp\left(\sqrt{\frac{\beta}{2}}(\mathfrak{U}_{T_{+}}(\theta)-\mathfrak{U}_{T_{-}}(\theta))-(T_{+}-T_{-})\right)$

then under ${\mathbb{Q}},$
$d\mathfrak{U}_{t}=d\mathfrak{X}_{t}+\sqrt{\tfrac{8}{\beta}}dt$ on
$[T_{-},T_{+}]$ for a ${\mathbb{Q}}$–Brownian motion
$(\mathfrak{X}_{t}:t\in[T_{-},T_{+}])$ (with quadratic variation
$\tfrac{4}{\beta}(t-T_{-})$). Under ${\mathbb{Q}},$ $\mathfrak{B}$ remains an
independent Brownian motion with the same quadratic variation as
$\mathfrak{X}.$

On the event given for $\mathfrak{U}_{T_{+}}(\theta)$ in the statement of the
Lemma, this Radon–Nikodym derivative is also in control and is given by
$e^{(\log k_{1}^{+})+O(\log k_{1})^{9/10}}.$ Hence it suffices to show that

$\mathbb{Q}(|\Delta_{T_{+}}|>\sqrt{\tfrac{k_{1}^{+}}{k_{1}}}~{}|~{}\mathscr{F}_{n_{1}^{+}})\leq
e^{-\delta(\log k_{1})^{19/20}}.$

To prove the statements with $\mathfrak{U}^{o}$, we instead need to use the
change of measure

$\frac{d{\mathbb{Q}}^{o}}{d\mathbb{P}}=\exp\left(\sqrt{\frac{\beta}{2}}(\mathfrak{U}^{o}_{T_{+}}(\theta)-\mathfrak{U}^{o}_{T_{-}}(\theta))-(T_{+}-T_{-})\right),$

but everything proceeds with obvious changes. We continue with the case
$\mathfrak{U}.$

The difference $\Delta_{t}$ satisfies the SDE

$\displaystyle d\Delta_{t}$ $\displaystyle=\theta
e^{t}k_{1}^{-1}\mathbf{1}\left\\{{t\leq
T_{\dagger}}\right\\}dt+\sqrt{\tfrac{4}{\beta}}\Im\biggl{(}(e^{i\Im\mathfrak{L}_{t}(\theta)}-e^{i\Im(\mathfrak{L}_{t}^{o}(\theta)+\mathfrak{L}_{T_{-}}^{j}(\theta_{j}))})d\mathfrak{W}_{t}^{j}\biggr{)}$
(6.1) $\displaystyle=\theta e^{t}k_{1}^{-1}\mathbf{1}\left\\{{t\leq
T_{\dagger}}\right\\}dt+\sqrt{\tfrac{4}{\beta}}\biggl{(}\Im(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j})(1-\cos\Delta_{t})+\Re(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j})\sin\Delta_{t}\biggr{)}$
$\displaystyle=\theta e^{t}k_{1}^{-1}\mathbf{1}\left\\{{t\leq
T_{\dagger}}\right\\}dt+d\mathfrak{B}_{t}(1-\cos\Delta_{t})-d\mathfrak{U}_{t}\sin\Delta_{t}.$

In the case that $\Delta_{T-}>0$, the process remains nonnegative for all
time.

##### Step 2: no movement before $T_{\dagger}$.

Set $\mathfrak{e}(t)=\exp(\sqrt{\tfrac{8}{\beta}}(t-T_{-})).$ Note that

$\displaystyle d(\Delta_{t}\mathfrak{e}(t))$
$\displaystyle=\mathfrak{e}(t)\theta e^{t}k_{1}^{-1}\mathbf{1}\left\\{{t\leq
T_{\dagger}}\right\\}dt+\mathfrak{e}(t)d\mathfrak{B}_{t}(1-\cos\Delta_{t})-\mathfrak{e}(t)d\mathfrak{X}_{t}\sin\Delta_{t}-\sqrt{\tfrac{8}{\beta}}\mathfrak{e}(t)dt(\sin\Delta_{t}-\Delta_{t}).$

Let $\vartheta$ be the first time in $[T_{-},T_{\dagger}]$ that
$|\Delta(t)|>4\exp(T_{\dagger})$. Then

$\Delta_{t}=\frac{\Delta_{T_{-}}}{\mathfrak{e}(t)}+\frac{\theta}{k_{1}\mathfrak{e}(t)}\int_{T_{-}}^{t}\mathfrak{e}(s)e^{s}\,ds+\frac{1}{\mathfrak{e}(t)}\bigl{(}M(t)+\mathcal{Y}(t)\bigr{)}$

for a martingale $M$ and a finite variation term $\mathcal{Y}$. The first two
terms, prior to $T_{\dagger}$ are bounded by $3\exp(T_{\dagger})$. The final
term is bounded, before $T_{\dagger}\wedge\vartheta$, by $Ce^{3T_{\dagger}}$.
The martingale, up to time $T_{\dagger}\wedge\vartheta$, has quadratic
variation bounded above by $\mathfrak{e}^{2}(t)e^{4T_{\dagger}}$. Thus,
summing over integer times between $[T_{-},T_{\dagger}]$, the probability that
it reaches height $\mathfrak{e}(t)e^{1.9T_{\dagger}}$ is at most
$\exp(-c\exp(-cT_{\dagger}))$ for some $c>0$. As $\Delta_{t}$ is continuous we
conclude that with probability $1-\exp(-c\exp(-cT_{\dagger}))$,
$\vartheta=\infty,$ which is to say $|\Delta(T_{\dagger})|\leq
4\exp(T_{\dagger}).$

##### Step 3: self-stabilizing after $T_{\dagger}$.

From time $t>T_{\dagger}$ the SDE (6.1) becomes

$d\Delta_{t}=d\mathfrak{B}_{t}(1-\cos\Delta_{t})-d\mathfrak{U}_{t}\sin\Delta_{t}.$

From the vanishing of the drift and diffusion terms, this equation cannot
cross any multiple of $2\pi{\mathbb{Z}}$. By passing to its negative if
necessary, we may assume $\Delta_{T_{\dagger}}\in(0,2\pi)$ We let $\tau$ be
the first hitting time of $\Delta_{t}$ to $\delta\pi.$ Then by comparison,
before $\tau,$ $\Delta_{t}$ for any $\epsilon>0$ there is $\delta>0$
sufficiently small that $\Delta_{t}$ is dominated by the solution to

$d\Delta_{t}^{\prime}=d\mathfrak{B}_{t}(1-\cos\Delta_{t}^{\prime})-d\mathfrak{X}_{t}\sin\Delta_{t}^{\prime}-(\sqrt{\tfrac{8}{\beta}}-\epsilon)\Delta_{t}^{\prime}dt,$

at for all $t\leq\tau.$ Taking logarithms, we have from Itô’s Lemma,

$d\log(\Delta_{t}^{\prime})=d\mathfrak{B}_{t}\frac{(1-\cos\Delta_{t}^{\prime})}{\Delta^{\prime}_{t}}-d\mathfrak{X}_{t}\frac{\sin\Delta_{t}^{\prime}}{\Delta^{\prime}_{t}}-\biggl{(}\sqrt{\frac{8}{\beta}}-\epsilon\biggr{)}dt-\frac{4}{\beta}\frac{(1-\cos\Delta_{t}^{\prime})}{(\Delta^{\prime}_{t})^{2}}dt.$

The stopped martingale part has uniformly bounded quadratic variation. The
drift is bounded as well, using $\cos(x)\leq 1-x^{2}(\tfrac{1}{2}-\epsilon)$
before the stopping time $\tau^{\prime},$ the first time $\Delta_{t}^{\prime}$
reaches $\pi\delta^{\prime}$ for $\delta^{\prime}$ sufficiently small. In
particular for $t\leq\tau^{\prime}$,

$d\log(\Delta_{t}^{\prime})\leq
dM_{t}+\left(\sqrt{\frac{8}{\beta}}-2\epsilon+\frac{2}{\beta}\right)dt,$

for a martingale $M_{t}$ with $d\langle M\rangle_{t}\leq\frac{C}{\beta}dt$ and
$M_{T_{\dagger}}=0.$ Hence to bound the ${\mathbb{Q}}$-probability that
$\log(\Delta_{T_{+}}^{\prime})\geq\log(\Delta_{T_{\dagger}}^{\prime})+(\log
k_{1})^{19/20},$ we can instead bound the probability that there is a
$t\leq\tau^{\prime}$ such that

$M_{t}\geq(\log
k_{1})^{19/20}+\left(\sqrt{\frac{8}{\beta}}-2\epsilon+\frac{2}{\beta}\right)(t-T_{\dagger}),$

noting that on the complement of this event
$\tau^{\prime}>(T_{+}-T_{\dagger})$ and so also $\tau>T_{+}-T_{\dagger}.$ From
a time-change, this is probability is dominated above by the probabillity that
a $\frac{\beta}{C}t$–quadratic variation Brownian motion crosses the same
linear barrier, and so

$\displaystyle{\mathbb{Q}}(\log(\Delta_{T_{+}\wedge\tau^{\prime}}^{\prime})\geq\log(\Delta_{T_{\dagger}}^{\prime})+(\log
k_{1})^{19/20})$ $\displaystyle\leq\exp\left(-(\log
k_{1})^{19/20}\frac{\sqrt{\beta}}{\sqrt{C}}\left(\sqrt{\frac{8}{\beta}}-2\epsilon+\frac{2}{\beta}\right)\right)$
$\displaystyle=e^{-c(\log k_{1})^{19/20}}.$

for some $c(\beta)>0$ and for all $k_{1}$ sufficiently large.

##### Step 4: Control of the real part.

Having controlled the difference of imaginary parts, we can then control the
difference of real parts $\Delta^{r}.$ We derive the diffusion for
$\Delta^{r},$ whose behavior is determined entirely by that of $\Delta:$

$\displaystyle d\Delta_{t}^{r}$
$\displaystyle=\sqrt{\tfrac{4}{\beta}}\Re\biggl{(}(e^{i\Im\mathfrak{L}_{t}(\theta)}-e^{i\Im(\mathfrak{L}_{t}^{o}(\theta)+\mathfrak{L}_{T_{-}}^{j}(\theta_{j}))})d\mathfrak{W}_{t}^{j}\biggr{)}$
$\displaystyle=\sqrt{\tfrac{4}{\beta}}\biggl{(}\Re(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j})(1-\cos\Delta_{t})-\Im(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j})\sin\Delta_{t}\biggr{)}$
$\displaystyle=d\mathfrak{U}_{t}(1-\cos\Delta_{t})-d\mathfrak{B}_{t}\sin\Delta_{t}$
$\displaystyle=dt\sqrt{\tfrac{8}{\beta}}(1-\cos\Delta_{t})+d\mathfrak{X}_{t}(1-\cos\Delta_{t})-d\mathfrak{B}_{t}\sin\Delta_{t}.$

The ${\mathbb{Q}}$–probability of the event that $|\Delta_{t}|\leq
e^{-0.99(\log k_{1})^{19/20}-\epsilon(t-T_{-})}$ for all $t\in[T_{-},T_{+}]$
is $1-e^{-\delta(\log k_{1})^{19/20}}$ for some $\delta,\epsilon>0$ and all
$k_{1}$ sufficiently large. On that event, both the drift and the the
martingale part of $\Delta_{t}^{r}$ have quadratic variation bounded by
$O(e^{-1.97(\log k_{1})^{19/20}})$ for all $k_{1}$ sufficiently large. Thus
the probability that this reaches $e^{-0.5(\log k_{1})^{19/20}}$ has the
claimed probability.

##### Step 5: The imaginary case.

We now have

$d\mathfrak{U}_{t}=-\sqrt{\tfrac{4}{\beta}}\Re(\sigma
e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j})=\sqrt{\tfrac{4}{\beta}}\Im(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j}),$

and we let

$d\mathfrak{B}_{t}=\sqrt{\tfrac{4}{\beta}}\Im(\sigma
e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j})=\sqrt{\tfrac{4}{\beta}}\Re(e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j})$

which again is an independent Brownian motion. The difference $\Delta_{t}$
satisfies the SDE (see the first two lines of (6.1))

$\displaystyle d\Delta_{t}$ $\displaystyle=\theta
e^{t}k_{1}^{-1}\mathbf{1}\left\\{{t\leq
T_{\dagger}}\right\\}dt+d\mathfrak{U}_{t}(1-\cos\Delta_{t})+d\mathfrak{B}_{t}\sin\Delta_{t}.$

Step 1 and 2 proceed in exactly the same way. For step 3, after the change of
measure we have the SDE for $t\geq T_{\dagger}$,

$\displaystyle
d\Delta_{t}=\sqrt{\tfrac{8}{\beta}}(1-\cos\Delta_{t})dt+d\mathfrak{X}_{t}(1-\cos\Delta_{t})+d\mathfrak{B}_{t}\sin\Delta_{t}.$

In the case $\sigma=i,$ the drift is positive, but weak (as it is quadratic in
$\Delta_{t}$). In particular before $\tau,$ we can dominate the solution by

$\displaystyle
d\Delta_{t}^{\prime}=\sqrt{\tfrac{8}{\beta}}\delta\Delta_{t}^{\prime}dt+d\mathfrak{X}_{t}(1-\cos\Delta_{t}^{\prime})+d\mathfrak{B}_{t}\sin\Delta_{t}^{\prime}.$

The proof now continues the same way as in the real case. ∎

##  7 Calculus estimates for the Prüfer phases

######  Lemma 7.1.

Let $\lambda_{1},\lambda\in\mathbb{C}$ and let $\alpha,\Gamma\in{\mathbb{R}}$
with $\Gamma>1$ be fixed real numbers and define, for $z\in\mathbb{C},$

$\displaystyle
F(z)=\Re\left\\{\lambda_{1}\log(1-u(z)e^{i\alpha})-\lambda_{1}\log(1-u(z))+\lambda_{2}\log(1-u(z))\right\\},$
$\displaystyle\text{where}\quad u(z)=\frac{z}{\sqrt{|z|^{2}+\Gamma}}.$

Then there is an absolute constant $C>0$ so that for
$x^{2}+y^{2}=r^{2}\leq\Gamma/2$

$\displaystyle|\partial_{x}F(x+iy)+\Re\\{\lambda_{1}(e^{i\alpha}-1)+\lambda_{2}\\}\Gamma^{-1/2}|\leq\frac{C(|\lambda_{1}(e^{i\alpha}-1)|+|\lambda_{2}|)r}{\Gamma},\text{
and }$ (7.1)
$\displaystyle|\partial_{y}F(x+iy)-\Im\\{\lambda_{1}(e^{i\alpha}-1)+\lambda_{2}\\}\Gamma^{-1/2}|\leq\frac{C(|\lambda_{1}(e^{i\alpha}-1)|+|\lambda_{2}|)r}{\Gamma},$

###### Proof.

We begin by computing the partial of $F$ with respect to $u,$ giving

$\frac{d}{du}F=-\Re\left\\{\lambda_{1}\frac{e^{i\alpha}-1}{(1-ue^{i\alpha})(1-u)}+\lambda_{2}\frac{1}{1-u(z)}\right\\}$
(7.2)

Further

$\partial_{x}(u(x+iy))=\frac{y^{2}+\Gamma-
ixy}{(x^{2}+y^{2}+\Gamma)^{3/2}}\quad\text{and}\quad\partial_{y}(u(x+iy))=\frac{i(x^{2}+\Gamma)-xy}{(x^{2}+y^{2}+\Gamma)^{3/2}}.$

We have that $\partial_{x}(u(x+iy))$ is nearly real, as is
$\partial_{y}(u(x+iy))$ nearly imaginary, when $x$ and $y$ are much smaller
than $\Gamma:$

$\max\\{|\Im\partial_{x}(u(x+iy))|,|\Re\partial_{y}(u(x+iy))|\\}\leq\frac{|xy|}{\Gamma^{3/2}}\leq\frac{r^{2}}{\Gamma^{3/2}}.$
(7.3)

As for the principal terms, using $r^{2}\leq\Gamma,$

$|\Re\partial_{x}(u(x+iy))-\Gamma^{-1/2}|\leq\frac{(x^{2}+y^{2}+\Gamma)^{3/2}-\Gamma^{1/2}(y^{2}+\Gamma)}{(x^{2}+y^{2}+\Gamma)^{3/2}\Gamma^{1/2}}\leq\frac{2r^{2}}{\Gamma^{3/2}}.$

Likewise,

$|\Im\partial_{y}(u(x+iy))-i\Gamma^{-1/2}|\leq\frac{2r^{2}}{\Gamma^{3/2}}.$

Hence using $|u(x+iy)|\leq r\Gamma^{-1/2}\leq\frac{1}{\sqrt{2}}$, we conclude
(7.1) for an absolute constant $C>0$ by combining the previous displays. ∎

As an immediate corollary, using that $F(0)=0$ and integrating $F$ from $0$ to
$x+iy,$ we obtain:

######  Corollary 7.2.

Let $\lambda_{1},\lambda\in\mathbb{C}$ and let $\alpha,\Gamma\in{\mathbb{R}}$
with $\Gamma>1$ and let $F$ be as in Lemma 7.1. Then there is an absolute
constant $C>0$ so that for $x^{2}+y^{2}=r^{2}\leq\Gamma/2$

$|F(x+iy)+\Re\\{(\lambda_{1}(e^{i\alpha}-1)+\lambda_{2})(x+iy)\\}\Gamma^{-1/2}|\leq\frac{C(|\lambda_{1}(e^{i\alpha}-1)|+|\lambda_{2}|)r^{2}}{\Gamma}.$

##  8 Polynomial interpolation

We will use some results for the _a priori_ stability of polynomials. The
first of these is a classical inequality due to Bernstein:

######  Theorem 8.1.

For any polynomial $Q$ of degree $k\geq 1,$

$\max_{|z|=1}|Q^{\prime}(z)|\leq k\cdot\max_{|z|=1}|Q(z)|.$

See [RS02, Chapter 14].

We also need a quantitative interpolation result for polynomials of a given
degree. The following is in some sense a generalization of [CMN18, Lemma 4.3].
Related inequalities have been published before, see especially [RS06] and
[FRR85, Theorem 8].

######  Theorem 8.2.

For any polynomial $Q$ of degree $k\geq 1,$ and any natural number $m\geq 2,$

$\max_{|z|=1}|Q(z)|^{2}\leq\frac{m}{m-1}\cdot\max_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}.$

Furthermore, if for any $b>0$ we partition the $(2mk)$-th roots of unity into
$\mathcal{N}$ and $\mathcal{F}$ so that $\mathcal{N}$ are all those roots of
unity $\omega$ so that $|\omega-1|\leq\frac{2b}{k},$ then there is an absolute
constant $C>0$ so that

$\displaystyle\max_{\begin{subarray}{c}|z-1|\leq\frac{b}{k},\\\
|z|=1\end{subarray}}|Q(z)|^{2}\leq\frac{m}{m-1}\cdot\max_{\omega\in\mathcal{N}}|Q(\omega)|^{2}+\frac{C}{b(m-1)}\cdot\max_{\omega\in\mathcal{F}}|Q(\omega)|^{2}\quad\text{and}$
$\displaystyle\min_{\begin{subarray}{c}|z-1|\leq\frac{b}{k},\\\
|z|=1\end{subarray}}|Q(z)|^{2}\geq\frac{m}{m-1}\cdot\min_{\omega\in\mathcal{N}}|Q(\omega)|^{2}-\biggl{(}1+\frac{C}{b}\biggr{)}\frac{1}{(m-1)}\cdot\max_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}.$

We give a proof of this fact. For any $m\in{\mathbb{N}}$ let $F_{m}$ be the
Fejér kernel, which for $|z|=1$ has the representation

$F_{m}(z)=\frac{1}{m}\sum_{r=0}^{m-1}\sum_{s=-r}^{r}z^{s}=\frac{1}{m}\biggl{[}\sum_{s=0}^{m-1}z^{s}\biggr{]}\biggl{[}\sum_{s=0}^{m-1}z^{-s}\biggr{]}=\frac{1}{m}\frac{|1-z^{m}|^{2}}{|1-z|^{2}}.$
(8.1)

We will need the following identity:

######  Lemma 8.3.

For all $m,r\in{\mathbb{N}}$ and all $t\in{\mathbb{R}},$

$\sum_{j=1}^{rm}F_{m}(e(t+j/(rm))=rm.$

See [Hof02] for a discussion of this. We give a proof below:

###### Proof.

Observe that using (8.1) we can write

$F_{m}(e(t+j/(rm))=\frac{1}{m}\frac{\sin({\pi tm}+\tfrac{\pi
j}{r})^{2}}{\sin({\pi t}+\tfrac{\pi j}{rm})^{2}}.$

We use the well-known identity that for any
$x\in\mathbb{C}\setminus\pi{\mathbb{Z}}$

$\frac{1}{\sin(x)^{2}}=\sum_{k\in{\mathbb{Z}}}\frac{1}{(x+k\pi)^{2}}.$ (8.2)

By continuity, it suffices to establish the identity for irrational $t.$ We
have, by grouping the terms in the sum over $j$ according to their residue
class $\ell$ modulo $r$

$\displaystyle\sum_{j=1}^{rm}F_{m}(e(t+j/(rm))$
$\displaystyle=\sum_{\ell=0}^{r-1}\sum_{p=1}^{m}\frac{1}{m}\frac{\sin({\pi
tm}+\tfrac{\pi\ell}{r})^{2}}{\sin({\pi
t}+\tfrac{\pi(\ell+rp)}{rm})^{2}}=\sum_{\ell=0}^{r-1}\sum_{p=1}^{m}\sum_{k\in{\mathbb{Z}}}\frac{1}{m}\frac{\sin({\pi
tm}+\tfrac{\pi\ell}{r})^{2}}{({\pi t}+\tfrac{\pi(\ell+rp+rmk)}{rm})^{2}}$
$\displaystyle=m\sum_{\ell=0}^{r-1}\frac{\sin({\pi
tm}+\tfrac{\pi\ell}{r})^{2}}{\sin(\pi tm+\tfrac{\pi\ell}{r})^{2}}=rm.$

In the penultimate display, we have extracted a factor of $m$ and again
applied (8.2). ∎

We can now give a proof of Theorem 8.2.

###### Proof.

Define for any $m\in{\mathbb{N}}$ with $m>1$

$R(z)=\frac{kmF_{km}(z)-kF_{k}(z)}{k(m-1)(2mk)}.$

Then we can write

$R(z)=\frac{1}{2mk}\sum_{s=-km}^{km}\lambda_{s}z^{s},$

where $\lambda_{s}=1$ for $-k\leq s\leq k.$ In particular, for any $0\leq
r<km$

$\sum_{\omega:\omega^{2mk}=1}\omega^{r}R(z\bar{\omega})=\frac{1}{2mk}\sum_{s=-km}^{km}\lambda_{s}z^{s}\cdot\biggl{[}\sum_{\omega:\omega^{2mk}=1}\omega^{r}\bar{\omega}^{s}\biggr{]}=\lambda_{r}z^{r},$

and so it follows that for any polynomial $Q(z)$ of degree $k$ and $z$ on the
unit circle,

$|Q(z)|^{2}=\sum_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}R(z\bar{\omega})\leq\sum_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}\frac{F_{km}(z\bar{\omega})}{2k(m-1)}\eqqcolon
X(z),$ (8.3)

where the first equality follows as $|Q(z)|^{2}$ for $|z|=1$ can be
represented as a Laurent polynomial with Fourier support contained in $[-k,k]$
and the inequality follows from the positivity of the Fejér kernel.

Using Lemma 8.3, we have

$\sum_{\omega:\omega^{2mk}=1}\frac{F_{km}(\omega)}{2k(m-1)}=\frac{m}{m-1},$

and hence for all $|z|=1,$

$X(z)\leq\frac{m}{m-1}\cdot\max_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}.$

If we further partition the roots of unity into $\mathcal{N}$ and
$\mathcal{F}$ as in the statement of the theorem, we can bound $X(z)$ using
(8.1) by

$X(z)\leq\frac{m}{m-1}\cdot\max_{\omega\in\mathcal{N}}|Q(\omega)|^{2}+\max_{\omega\in\mathcal{F}}|Q(\omega)|^{2}\cdot\sum_{\omega\in\mathcal{F}}\frac{2}{km(m-1)|1-z\bar{\omega}|^{2}}.$

If $z$ satisfies that $|z-1|\leq\frac{b}{k},$ then $|\omega-z|\geq\frac{b}{k}$
and there is therefore an absolute constant $C>0$ so that

$\sum_{\omega\in\mathcal{F}}\frac{2}{km(m-1)|1-z\bar{\omega}|^{2}}\leq\frac{C}{b(m-1)},$
(8.4)

which completes the proof of the upper bound.

For the lower bound, starting from (8.3),

$\displaystyle|Q(z)|^{2}$
$\displaystyle=\sum_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}R(z\bar{\omega})$
$\displaystyle\geq\sum_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}\frac{F_{km}(z\bar{\omega})}{2k(m-1)}-\max_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}\cdot\sum_{\omega:\omega^{2mk}=1}\frac{F_{k}(z\bar{\omega})}{2mk(m-1)}.$

Using Lemma 8.3, we conclude

$|Q(z)|^{2}\geq
X(z)-\frac{1}{m-1}\cdot\max_{\omega:\omega^{2mk}=1}|Q(\omega)|^{2}.$

Furthermore

$X(z)\geq\min_{\omega\in\mathcal{N}}|Q(\omega)|^{2}\cdot\sum_{\omega\in\mathcal{N}}\frac{F_{km}(z\bar{\omega})}{2k(m-1)}\geq\min_{\omega\in\mathcal{N}}|Q(\omega)|^{2}\biggl{(}\frac{m}{m-1}-\frac{C}{b(m-1)}\biggr{)},$

using (8.4). ∎

##  9 Convergence of the derivative martingale

We recall from (1.10)

$\varphi_{k+1}(\theta)=\varphi_{k}(\theta)+2\Re\\{\sigma\left(\log(1-\gamma_{k}e^{i\Psi_{k}(\theta)})\right)\\},\quad\varphi_{0}(\theta)=0,$

for $\sigma\in\left\\{1,i\right\\}.$

We also recall (from (2.1)) that $\gamma_{j}$ are independent, rotationally
invariant in law, and have $|\gamma_{j}|^{2}$ distributed as
$\operatorname{Beta}(1,\beta_{j})$ where
$\beta_{j}^{2}=\tfrac{\beta}{2}(j+1)$. Hence we have an explicit expression
for the moment generating functions of $\varphi,$ given by (see [CMN18,
Proposition 2.5] or [BHNY08, Lemma 2.3]).

######  Lemma 9.1.

For any $s,t\in{\mathbb{C}}$ with $\Re s\geq-1$

$\mathbb{E}[e^{s\Re(\log(1-\gamma_{j}))+t\Im(\log(1-\gamma_{j}))}]=\frac{\Gamma(1+\beta_{j}^{2})\Gamma(1+s+\beta_{j}^{2})}{\Gamma(1+\beta_{j}^{2}+(s+it)/2)\Gamma(1+\beta_{j}^{2}+(s-it)/2)}.$

For $s\geq 0$ and $t\in{\mathbb{R}},$

$\mathbb{E}[e^{s\Re(\log(1-\gamma_{j}))+t\Im(\log(1-\gamma_{j}))}]\leq\exp\left(\frac{s^{2}+t^{2}}{2}\frac{1}{1+\beta(j+1)}\right).$

We define for any $j\in{\mathbb{N}}$ and any $|\sigma|=1,$
$H^{(\sigma)}_{j}(s)=H_{j}(s)\coloneqq\log\mathbb{E}[e^{2s\Re(\sigma\log(1-\gamma_{j}))}].$
Define

$\mathscr{M}_{j}(\theta,s)\coloneqq
e^{s\varphi_{j}(\theta)-\sum_{k=1}^{j}H_{k}(s)},$

which is a martingale. Set $s_{\beta}\coloneqq\sqrt{\tfrac{\beta}{2}}$ and
define

$\widehat{\mathscr{D}}_{j}(\theta)\coloneqq-\partial_{s}\mathscr{M}_{j}(\theta,s)|_{s=s_{\beta}}=e^{s_{\beta}\varphi_{j}(\theta)-\sum_{k=1}^{j}H_{k}(s_{\beta})}\biggl{(}\sum_{k=1}^{j}H_{k}^{\prime}(s_{\beta})-\varphi_{j}(\theta)\biggr{)},$

which is also a martingale. Define

$\widehat{\mathscr{B}}_{j}\coloneqq\frac{1}{2\pi}\int_{0}^{2\pi}\widehat{\mathscr{D}}_{j}(\theta)d\theta.$
(9.1)

We need an elementary computation of the asymptotic behavior of the sums of
$H_{k}.$

######  Lemma 9.2.

The following limits exist and are finite

$\lim_{j\to\infty}\sum_{k=1}^{j}H_{k}(s_{\beta})-\log
j=\mathfrak{g}_{\beta},\quad\text{and}\quad\lim_{j\to\infty}\sum_{k=1}^{j}H_{k}^{\prime}(s_{\beta})-\sqrt{\frac{8}{\beta}}\log
j=\mathfrak{h}_{\beta}.$

###### Proof.

We recall the ratio asymptotic for the $\Gamma$ function (see [OOL+, 5.11])

$\frac{\Gamma(\beta_{k}^{2}+x)}{\Gamma(\beta_{k}^{2}+y)}=\beta_{k}^{2(x-y)}\left(1+\frac{\tfrac{1}{2}(x-y)(x+y-1)}{\beta_{k}^{2}}+O(\beta_{k}^{-4})\right).$

Then for any $s,t\in{\mathbb{C}},$

$\mathbb{E}[e^{s\Re(\log(1-\gamma_{k}))+t\Im(\log(1-\gamma_{k}))}]=1+\frac{s^{2}+t^{2}}{4\beta_{k}^{2}}+O(\beta_{k}^{-4}),$

and the asymptotic can be differentiated on both sides with respect to $s$ and
$t$ as well. Thus

$H_{k}(s)=\frac{s^{2}}{\beta_{k}^{2}}+O(\beta_{k}^{-4}).$

For $H_{k}(s_{\beta})$ we therefore have

$H_{k}(s_{\beta})=\frac{1}{k+1}+O(k^{-2}),$

which leads directly to the claimed asymptotic. For the derivative, we have
that

$H_{k}^{\prime}(s_{\beta})=\frac{2s_{\beta}}{\beta_{k}^{2}}+O(\beta_{k}^{-4})=\sqrt{\frac{8}{\beta}}\frac{1}{k+1}+O(\beta_{k}^{-4}).$

∎

We observe that the field $\\{\sqrt{\frac{8}{\beta}}\log
j-\varphi_{j}(\theta)\\}$ is rarely very negative, and is in fact almost
surely positive for all $j$ sufficiently large (but random). This can be done
by comparison to the marginally Gaussian process $G_{j}(\theta)$.

######  Lemma 9.3.

Define for any $\kappa>0$ the event $\mathcal{E}_{\kappa}$

$\mathcal{E}_{\kappa}\coloneqq\biggl{\\{}\sup\biggl{\\{}\max\\{|G_{j}(\theta)-\varphi_{j}(\theta)|,\varphi_{j}(\theta)-\sqrt{\tfrac{8}{\beta}}(\log
j-\tfrac{1}{8}\log\log j)\\}:j\in
2^{{\mathbb{N}}},\theta\in[0,2\pi]\biggr{\\}}\leq\sqrt{\tfrac{8}{\beta}}\kappa\biggr{\\}}.$

Then
$\mathbf{1}\left\\{{\mathcal{E}_{\kappa}}\right\\}\xrightarrow[\kappa\to\infty]{\mathbb{P}}1$.

###### Proof.

We use Proposition [CMN18, Propositions 3.1, 4.5], due to which

$\sup\biggl{\\{}\sup_{\theta\in[0,2\pi]}|\varphi_{j}(\theta)-G_{j}(\theta)|:j\in{\mathbb{N}}\biggr{\\}}+\sup\biggl{\\{}\sup_{\theta\in[0,2\pi]}\varphi_{j}(\theta)-\sup_{0\leq
k\leq 2j}G_{j}(\tfrac{\pi
k}{j}):j\in{\mathbb{N}}\biggr{\\}}<\infty\operatorname{a.s.}$ (9.2)

We start by giving an initial barrier estimate $\max_{k}G_{j}(\tfrac{\pi
k}{j})$. Let $\mathcal{E}^{\prime}$ be the event

$\mathcal{E}^{\prime}\coloneqq\biggl{\\{}\limsup_{j\in
2^{{\mathbb{N}}}}\biggl{(}\sup_{0\leq k\leq 2j}G_{j}(\tfrac{\pi
k}{j})-\sqrt{\tfrac{8}{\beta}}(\log j+\log\log
j)\biggr{)}=-\infty\biggr{\\}}.$

By a direct application of Borel-Cantelli and the standard Gaussian tail bound
$\mathbb{P}(Z>x)\leq Ce^{-x^{2}/2}/\sqrt{x}$, this has probability $1$. Thus,
the events

$\mathcal{E}^{\prime\prime}_{2^{n}}\coloneqq\biggl{\\{}\sup_{\begin{subarray}{c}j=2^{\ell},\\\
0\leq\ell\leq n\end{subarray}}\biggl{(}\sup_{0\leq k\leq 2j}G_{j}(\tfrac{\pi
k}{j})-\sqrt{\tfrac{8}{\beta}}(\log
j)\biggr{)}\leq\sqrt{\tfrac{8}{\beta}}(\log\log 2^{n})\biggr{\\}}$

for $n\in{\mathbb{N}}$ exhaust the probability space, up to a null set. Using
the Ballot theorem, with $j\in 2^{\mathbb{N}}$, there is a $C_{\beta}$ so that
for all $t\geq 0$,

$\mathbb{P}[\mathcal{E}_{j}^{\prime\prime}\cap\\{\sup_{0\leq k\leq
2j}G_{j}(\tfrac{\pi k}{j})>\sqrt{\tfrac{8}{\beta}}(\log j-\tfrac{3}{4}\log\log
j+t)\\}]\leq C_{\beta}(\log\log j)^{2}e^{-2t}$

for some constant $C_{\beta}>0.$ By Borel–Cantelli applied to the sequence
$j\in 2^{\mathbb{N}},$ we have for any $\epsilon>0$ there is almost surely a
$j_{0}<\infty$ so that $j\in 2^{\mathbb{N}}$ with $j>j_{0}$,

$\sup_{0\leq k\leq 2j}G_{j}(\tfrac{\pi k}{j})\leq\sqrt{\tfrac{8}{\beta}}(\log
j-(\tfrac{1}{4}-\epsilon)\log\log j).$

∎

On the event $\mathcal{E}_{\kappa}$ we can use Girsanov and the ballot theorem
for the Gaussian random walk $j\mapsto G_{j}(\theta)$ to conclude for any
$\delta>0$ any $\theta\in[0,2\pi],$

$\mathbb{P}[\mathcal{E}_{\kappa},\log
j+\kappa-\sqrt{\tfrac{\beta}{8}}\varphi_{j}\in[{k-1},k]]\leq\begin{cases}\frac{C_{\beta,\kappa,\delta}k}{(\log
j)^{3/2}}{\exp\left(-\frac{(\log j-k)^{2}}{\log
j}\right)},&k\leq(1-\delta)\log j,\\\ \frac{C_{\beta,\kappa,\delta}}{(\log
j)^{1/2}}{\exp\left(-\frac{(\log j-k)^{2}}{\log j}\right)},&k>(1-\delta)\log
j.\end{cases}$ (9.3)

We begin with the observation that the improperly normalized mass tends almost
surely to $0$ at the critical $s=s_{\beta}.$

######  Lemma 9.4.

For any $\sigma\in\left\\{1,i\right\\}$ and any $\beta>0,$

$Z_{j}\coloneqq\frac{1}{2\pi}\int_{0}^{2\pi}\mathscr{M}_{j}(\theta,s_{\beta})d\theta\xrightarrow[j\to\infty]{\operatorname{a.s.}}0.$

Furthermore $\\{Z_{j}\sqrt{\log j}:j\in{\mathbb{N}}\\}$ is tight, and for any
$\epsilon>0,$ there is a compact $K$ so that if

$\chi(\theta)=\mathbf{1}\left\\{{(\sqrt{\tfrac{8}{\beta}}\log
j-\varphi_{j}(\theta))/\sqrt{\log j}\not\in K}\right\\}$

then for any $j\in{\mathbb{N}}$

$\mathbb{P}\biggl{(}\int_{0}^{2\pi}\mathscr{M}_{j}(\theta,s_{\beta})\bigl{|}\sqrt{\tfrac{8}{\beta}}\log
j-\varphi_{j}(\theta)\bigr{|}\chi(\theta)d\theta>\epsilon\biggr{)}<\epsilon.$

###### Proof.

The process $Z_{j}$ is a positive martingale and so it converges almost
surely. After establishing the claimed tightness, it follows that
$Z_{j}\xrightarrow[j\to\infty]{\mathbb{P}}0.$ Hence along some subsequence, it
converges almost surely to $0.$ The almost sure convergence of $Z_{j}$ then
completes the proof of the first point.

The remaining statements now follow by taking expectations of $Z_{j}$ on the
event $\mathcal{E}_{\kappa}.$ In particular using (9.3) we have (with $\log
j-\sqrt{\tfrac{8}{\beta}}\varphi_{j}={u}{\sqrt{\log j}}$) and
$t=(1-\delta)\sqrt{\log j}$

$\begin{aligned}
\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathcal{E}_{\kappa}}\right\\}\mathscr{M}_{j}(\theta,s_{\beta})\biggr{)}&\leq
C_{\beta,\kappa}\int\limits_{0}^{t}\frac{u}{\sqrt{\log j}}\exp(-2u\sqrt{\log
j}+\log j)\exp\left(-\frac{(\log j-u\sqrt{\log j})^{2}}{\log j}\right)du\\\
&+C_{\beta,\kappa}\int\limits_{t}^{\infty}\exp(-2u\sqrt{\log j}+\log
j)\exp\left(-\frac{(\log j-u\sqrt{\log j})^{2}}{\log
j}\right)du\end{aligned}.$

This simplifies to

$\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathcal{E}_{\kappa}}\right\\}\mathscr{M}_{j}(\theta,s_{\beta})\biggr{)}\leq
C_{\beta,\kappa}\int\limits_{0}^{t}\frac{u}{\sqrt{\log
j}}\exp(-u^{2})du+O((\log j)^{(1-\delta)^{2}}).$

In particular, we conclude that

$\biggl{\\{}\mathbf{1}\\{\mathcal{E}_{\kappa}\\}\sqrt{\log
j}\int_{0}^{2\pi}\mathscr{M}_{j}(\theta,s_{\beta})\,d\theta:j\in{\mathbb{N}}\biggr{\\}}$

is tight, and as $\cup_{\kappa\in{\mathbb{N}}}\mathcal{E}_{\kappa}$ has
probability $1$, the tightness without the indicator holds.

Essentially the same computation shows the claimed estimates for the final
display of the lemma. With $K=[\eta/2,2\eta^{-1}],$ for all $j$ sufficiently
large

$\mathbb{E}\biggl{(}\mathbf{1}\left\\{{\mathcal{E}_{\kappa}}\right\\}\mathscr{M}_{j}(\theta,s_{\beta})|\log
j-\sqrt{\tfrac{8}{\beta}}\varphi_{j}(\theta)|\chi(\theta)\biggr{)}\leq
C_{\beta,\kappa}\biggl{(}\int\limits_{0}^{\eta}+\int\limits_{\eta^{-1}}^{\infty}\biggr{)}u^{2}\exp(-u^{2})du.$

This may be made as small as desired by picking $\eta$ small. ∎

We will use this convergence to compare $\widehat{\mathscr{B}}_{j}$ and
${\mathscr{B}_{j}},$ and $\widehat{\mathscr{D}}_{j}$ and ${\mathscr{D}_{j}}$.
Before doing this comparison, we will show the convergence of
$\widehat{\mathscr{B}}_{j}$ and of
$\int\widehat{\mathscr{D}}_{j}(\theta)f(\theta)d\theta$ for positive bounded
test functions $f$. Similar ideas appear in [DRSV14].

######  Lemma 9.5.

There is an almost surely nonnegative finite random variable
$\widehat{\mathscr{B}}_{\infty}$ and nonnegative measure
$\widehat{\mathscr{D}}_{\infty}$ so that for any bounded positive
deterministic test function $f$,

$\widehat{\mathscr{B}}_{2^{\ell}}\xrightarrow[\ell\to\infty]{\operatorname{a.s.}}\widehat{\mathscr{B}}_{\infty},\qquad\int_{0}^{2\pi}\widehat{\mathscr{D}}_{2^{\ell}}(\theta)f(\theta)d\theta\xrightarrow[\ell\to\infty]{\operatorname{a.s.}}\int_{0}^{2\pi}\widehat{\mathscr{D}}_{\infty}(\theta)f(\theta)d\theta.$

###### Proof.

Define the positive and negative parts

$\mathscr{D}^{\pm}_{j}(\theta)\coloneqq
e^{s_{\beta}\varphi_{j}(\theta)-\sum_{k=1}^{j}H_{k}(s_{\beta})}\biggl{(}\sum_{k=1}^{j}H_{k}^{\prime}(s_{\beta})-\varphi_{j}(\theta)\biggr{)}_{\pm},$

and define

$\widehat{\mathscr{B}}_{2^{\ell}}^{\pm}\coloneqq\frac{1}{2\pi}\int_{0}^{2\pi}\mathscr{D}^{\pm}_{2^{\ell}}(\theta)d\theta.$

Define

$Y_{2^{\ell}}\coloneqq\mathbb{E}[\widehat{\mathscr{B}}_{2^{\ell+1}}^{-}~{}|~{}\mathscr{F}_{2^{\ell}}].$

We will show $\sum_{\ell=1}^{\infty}Y_{2^{\ell}}<\infty$ almost surely. Having
done so, the lemma will follow, as we now explain. Observe

$\mathbb{E}[\widehat{\mathscr{B}}_{{2^{\ell+1}}}^{+}~{}|~{}\mathscr{F}_{2^{\ell}}]=\mathbb{E}[\widehat{\mathscr{B}}_{{2^{\ell+1}}}~{}|~{}\mathscr{F}_{2^{\ell}}]+Y_{2^{\ell}}=\widehat{\mathscr{B}}_{{2^{\ell}}}+Y_{2^{\ell}}\leq\widehat{\mathscr{B}}_{{2^{\ell}}}^{+}+Y_{2^{\ell}},$
(9.4)

it follows that both
$T_{2^{\ell}}^{\pm}\coloneqq\widehat{\mathscr{B}}_{2^{\ell}}^{\pm}-\sum_{k=1}^{{\ell-1}}Y_{2^{k}}$
are supermartingales with respect to the filtration
$(\mathscr{F}_{2^{\ell}}:\ell\in{\mathbb{N}})$. Letting $\tau$ be the stopping
time that $\ell\mapsto\sum_{k=1}^{\ell}Y_{2^{k}}$ exceeds $R,$ then
$T_{2^{j\wedge\tau}}^{\pm}$ is a supermartingale bounded below by $-R$ (by
predictability) which converges almost surely. As this holds for any
$R\in{\mathbb{N}}$ it follows that $T_{2^{\ell}}^{\pm}$ converges almost
surely. As the sum of $Y_{2^{k}}$ converges almost surely as well,
$\widehat{\mathscr{B}}_{2^{\ell}}^{\pm}$ converges almost surely. Hence so
does their difference.

######  Remark 9.6.

From Lemma 9.3, it follows that in fact $\widehat{\mathscr{B}}_{j}^{-}$ is
eventually $0$ for all $j$ sufficiently large, almost surely.

The statement concerning
$\int_{0}^{2\pi}\widehat{\mathscr{D}_{j}}(\theta)f(\theta)d\theta$ follows
similarly: instead of (9.4), use that

$\displaystyle\mathbb{E}[\int_{0}^{2\pi}\widehat{\mathscr{D}}_{{2^{\ell+1}}}^{+}(\theta)f(\theta)d\theta~{}|~{}\mathscr{F}_{2^{\ell}}]$
$\displaystyle\leq\mathbb{E}[\int_{0}^{2\pi}\widehat{\mathscr{D}}_{{2^{\ell+1}}}f(\theta)d\theta~{}|~{}\mathscr{F}_{2^{\ell}}]+2\pi\|f\|_{\infty}Y_{2^{\ell}}$
(9.5)
$\displaystyle=\int_{0}^{2\pi}\widehat{\mathscr{D}}_{{2^{\ell}}}(\theta)f(\theta)d\theta+Y_{2^{\ell}}\leq\int_{0}^{2\pi}\widehat{\mathscr{D}}_{{2^{\ell}}}^{+}(\theta)f(\theta)d\theta+2\pi\|f\|_{\infty}Y_{2^{\ell}},$

and the rest follows as in the treatment of
$\widehat{\mathscr{B}}_{2^{\ell}}$.

So, it remains to show $\sum_{j=1}^{\infty}Y_{2^{j}}<\infty$ almost surely.
There is a $C_{\beta}$ so that for all $j$ sufficiently large

$Y_{2^{j+1}}\leq
C_{\beta}\int_{0}^{2\pi}\mathscr{M}_{2^{j}}(\theta,s_{\beta})\mathbb{E}\left[e^{\sqrt{\beta/2}\left(\varphi_{2^{j+1}}(\theta)-\varphi_{2^{j}}(\theta)\right)-\log
2}\bigl{(}\varphi_{2^{j+1}}-\sqrt{\tfrac{8}{\beta}}\log
2^{j+1}\bigr{)}_{+}~{}|~{}\mathscr{F}_{2^{j}}\right].$

The increment
$\sqrt{\beta/2}\bigl{(}\varphi_{2^{j+1}}(\theta)-\varphi_{2^{j}}(\theta)\bigr{)}$
is uniformly subgaussian over all $j$ (from Lemma 9.1). Hence on the event
$\mathcal{E}_{\kappa},$ the restriction that

$\varphi_{2^{j+1}}-\varphi_{2^{j}}>\sqrt{\tfrac{8}{\beta}}\log
2^{j+1}-\varphi_{2^{j}}>\sqrt{\tfrac{8}{\beta}}(\kappa+\tfrac{1}{8}\log\log
2^{j})$

implies that there is a constant $c_{\beta,\kappa}>0$ so that for all
$j\in{\mathbb{N}}$

$\mathbb{E}\left[e^{\sqrt{\beta/2}\left(\varphi_{2^{j+1}}(\theta)-\varphi_{2^{j}}(\theta)\right)-\log
2}\bigl{(}\varphi_{2^{j+1}}-\sqrt{\tfrac{8}{\beta}}\log
2^{j+1}\bigr{)}_{+}~{}|~{}\mathscr{F}_{2^{j}}\right]\leq
e^{-c_{\beta,\kappa}(\log j)^{2}}.$

Hence we arrive at the conclusion that almost surely

$\sum_{j=1}^{\infty}\bigl{(}\mathbf{1}\\{\mathcal{E}_{\kappa}\\}Y_{2^{j}}\bigr{)}<\sum_{j=1}^{\infty}\biggl{(}e^{-c_{\beta,\kappa}(\log
j)^{2}}\int_{0}^{2\pi}\mathscr{M}_{2^{j}}(\theta,s_{\beta})d\theta\biggr{)}\quad\operatorname{a.s.}$

From Lemma 9.4 this is finite almost surely, which completes the proof as the
$(\mathcal{E}_{\kappa}:\kappa\in{\mathbb{N}})$ exhaust the probability space.
∎

We conclude with the proof of Theorem 1.6.

###### Proof of Theorem 1.6.

We begin by noting that $\widehat{\mathscr{D}}_{j}$ and $\mathscr{D}_{j}$ are
exactly related by the identity

$\sqrt{\tfrac{4}{\beta}}\mathscr{D}_{j}(\theta)e^{\log
j-\sum_{k=1}^{j}H_{k}(s_{\beta})}-\widehat{\mathscr{D}_{j}}(\theta)=\mathscr{M}_{j}(\theta,s_{\beta})\biggl{(}\sum_{k=1}^{j}H_{k}^{\prime}(s_{\beta})-\sqrt{\tfrac{8}{\beta}}\log
j\biggr{)}$

Hence integrating against a positive bounded test function $f$, from Lemma
9.2, Lemma 9.4, and Lemma 9.5,

$\int_{0}^{2\pi}f(\theta)\mathscr{D}_{2^{\ell}}(\theta)d\theta\xrightarrow[\ell\to\infty]{\operatorname{a.s.}}\sqrt{\tfrac{\beta}{4}}e^{\mathfrak{g}_{\beta}}\int_{0}^{2\pi}f(\theta)\widehat{\mathscr{D}}_{\infty}(\theta)d\theta\quad\text{and}\quad\mathscr{B}_{2^{\ell}}\xrightarrow[\ell\to\infty]{\operatorname{a.s.}}\sqrt{\tfrac{\beta}{4}}e^{\mathfrak{g}_{\beta}}\widehat{\mathscr{B}}_{\infty}.$
(9.6)

It remains to prove the non-atomicity of $\widehat{\mathscr{D}}_{\infty}$.
Mimicking Lemma 9.4, with $\epsilon\in(1/3,1/2)$, introduce the interval
$K_{k}=[k^{-\epsilon},k^{\epsilon}]$ and, for $j_{0}$ fixed, the function

$\widehat{\chi}_{j_{0},j}(\theta)=\prod_{k=j_{0}}^{j}\mathbf{1}\left\\{{(\sqrt{\tfrac{8}{\beta}}k\log
2-\varphi_{2^{k}}(\theta)/\sqrt{k}\in K_{k}}\right\\}.$

The same argument as in Lemma 9.4 shows that

$\int\mathscr{M}_{2^{j}}(\theta,s_{\beta})\biggl{|}\biggl{(}\sum_{k=1}^{2^{j}}H_{k}^{\prime}(s_{\beta})-\sqrt{\tfrac{8}{\beta}}j\log
2\biggr{)}\biggr{|}(1-\widehat{\chi}_{j_{0},j}(\theta))d\theta\leq
C\sum_{k=j_{0}}^{j}\frac{1}{k^{3\epsilon}}\leq\frac{C^{\prime}}{j_{0}^{3\epsilon-1}}.$

Taking $j_{0}$ large, it thus suffices to prove the non-atomicity of the limit
of the positive measures
$\mathscr{M}_{2^{j}}(\theta,s_{\beta})\biggl{|}\biggl{(}\sum_{k=1}^{2^{j}}H_{k}^{\prime}(s_{\beta})-\sqrt{\tfrac{8}{\beta}}j\log
2\biggr{)}\biggr{|}\widehat{\chi}_{j_{0},j}(\theta)d\theta$ for large fixed
$j_{0}$. Toward this end, divide $[0,2\pi]$ to intervals $\Delta_{i}$ of
length $\delta$. Using Lemma 2.20, we can replace
$\varphi_{2^{k}}(\theta)-\varphi_{2^{j_{0}}}(\theta)$ by
$Z_{2^{k}}^{2^{j_{0}}}(\theta)$.

Let
$A_{i,j}=\int_{\Delta_{i}}\mathscr{M}_{2^{j}}(\theta,s_{\beta})\biggl{|}\biggl{(}\sum_{k=1}^{2^{j}}H_{k}^{\prime}(s_{\beta})-\sqrt{\tfrac{8}{\beta}}j\log
2\biggr{)}\biggr{|}\widehat{\chi}_{j_{0},j}(\theta)d\theta$. Using Lemma B.2
we have that

$\mathbb{E}\big{(}A_{i,j}\mid\mathcal{F}_{2^{j_{0}}})=C_{i,j_{0}}\Delta$ (9.7)

where $\max_{i}C_{i,j_{0}}<\infty$ is a random variable independent of
$\Delta$ or $j$. Using Lemmas B.4 and B.5 as in Proposition 2.21, we obtain
that

$\mathbb{E}\big{(}A_{i,j}^{2}\mid\mathcal{F}_{2^{j_{0}}})=C_{i,j_{0}}^{\prime}\Delta
o(\Delta),$ (9.8)

where $\max_{i}C_{i,j_{0}}^{\prime}<\infty$ is a random variable independent
of $\Delta$ or $j$. Hence, for any $\delta>0$,
$\mathbb{P}(A_{i,j}>\delta\mid\mathcal{F}_{2^{j_{0}}})\leq\frac{\max_{i}C_{i,j_{0}}^{\prime}\Delta
o(\Delta)}{\delta^{2}},$ and therefore

$\mathbb{P}(\exists
i:A_{i,j}>\delta\mid\mathcal{F}_{2^{j_{0}}})\leq\frac{\max_{i}C_{i,j_{0}}^{\prime}o(\Delta)}{\delta^{2}}.$

Since this is true uniformly in $j$, taking the limit as first $j\to\infty$
and then $\Delta\to 0$ gives the claim of non-atomiticity of
$\widehat{D}_{\infty}$. ∎

##  10 Decoration process technical estimates

In this section, we collect other estimates about the decoration. In all, we
need a relatively sharp upper estimate on a single ray of the decoration being
high (that is sharp up to constants $c(k_{4})$). We also provide a weak upper
estimate on two rays being high which is sharp up to a power of $(\log
k_{1})$.

We also need a few technical estimates of a different nature. Due to the
technical nature of the two-ray estimates we produce up to level $n_{1}^{+}$,
we show a continuity estimate for the markov kernel
$(x,df)\mapsto\mathfrak{s}(e^{x}df)$ in the parameter $x$. Finally, we show a
mixing estimate of the decoration measure comparing
$\mathfrak{s}(e^{i\alpha}df)$ in the case of $\sigma=1$ to the same with
uniformly random $\alpha$.

###  10.1 Moment estimates

######  Lemma 10.1.

Suppose that $\theta$ is such that

$\sqrt{\tfrac{8}{\beta}}A_{\log
n_{1}^{+}}^{2,-}\leq\varphi_{n_{1}^{+}}(\theta)\leq\sqrt{\tfrac{8}{\beta}}A_{\log
n_{1}^{+}}^{2,+}$ (10.1)

Then, for some $0<c(k_{7},x)<c(k_{7},\infty)$ which is a continuous function
of $k_{7}$,

$\mathbb{P}\biggl{(}\bigl{\\{}\mathfrak{U}^{o}_{T_{+}}(\theta)\in[-k_{7},x]\bigr{\\}}\cap\mathscr{P}_{j}(\theta)~{}\big{|}~{}\mathscr{F}_{n_{1}^{+}}\biggr{)}=(1+o_{k_{4}})\frac{c(k_{7},x)h\sqrt{k_{4}}}{(T_{+}-T_{-})^{3/2}}e^{-(T_{+}-T_{-})}e^{-\sqrt{2}h-h^{2}/2(T_{+}-T_{-})},$
(10.2)

where

$h=-\sqrt{\tfrac{\beta}{4}}\mathfrak{U}^{j}_{T_{-}}(\theta)+\sqrt{2}\log
k_{1}^{+}=-\sqrt{\tfrac{\beta}{4}}\bigg{(}\varphi_{n_{1}^{+}}(\theta)-\sqrt{\tfrac{8}{\beta}}m_{n_{1}^{+}}+o_{n}\bigg{)}.$
(10.3)

Further, the upper bound in (10.2) holds without the restriction (10.1), as
long as $h^{\prime}>0$.
Next, if $\theta_{1},\theta_{2}$ are such that $|\theta_{1}|,|\theta_{2}|\leq
5n^{\delta}$ and

$\sqrt{\tfrac{8}{\beta}}A_{\log
n_{1}^{+}}^{2,-}\leq\varphi_{n_{1}^{+}}(\theta)\leq\sqrt{\tfrac{8}{\beta}}A_{\log
n_{1}^{+}}^{2,+}.$ (10.4)

holds for $\theta\in\\{\theta_{1},\theta_{2}\\}$, then, for some $c(k_{7})>0$,
with $|\theta_{1}-\theta_{2}|e^{T_{*}}=k_{1}$ and $h=(h_{1}\vee h_{2})$,

$\displaystyle\mathbb{P}\biggl{(}\bigcap_{\theta\in\\{\theta_{1},\theta_{2}\\}}\bigl{\\{}\mathfrak{U}^{o}_{T_{+}}(\theta)\in[-k_{7},\infty)\bigr{\\}}\cap\mathscr{P}_{j}(\theta)~{}\big{|}~{}\mathscr{F}_{n_{1}^{+}}\biggr{)}$
(10.5)
$\displaystyle\leq\\!\begin{cases}c(k_{7})h\frac{(T_{+}-T_{*})\wedge(T_{*}-T_{-})}{(T_{*}-T_{-}+1)^{3/2}}\exp\biggl{(}\\!-S(T_{-},T_{+},h)\\!-\\!\Big{(}T_{+}-T_{*}+(T_{+}-T_{*})^{1/10}\Big{)}\\!-\\!{h^{2}/2(T_{+}-T_{-})}\biggr{)},&\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{if
}e^{k_{4}}\leq|\theta_{1}-\theta_{2}|\leq k_{1}e^{-T_{-}},&\\\
c(k_{7})\exp\biggl{(}-S(T_{-},T_{+},h_{1})-S(T_{-},T_{+},h_{2})-(h_{1}^{2}+h_{2}^{2})/2(T_{+}-T_{-})\biggr{)},\\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{if
}|\theta_{1}-\theta_{2}|\geq k_{1}e^{-T_{-}},&\end{cases}$

where

$S(T_{-},T_{+},x)=(T_{+}-T_{-})+\sqrt{2}x.$

We emphasize that in (10.2), we implicitly used that $k_{i+1}\ll k_{i}$ in the
$o$ notation. We also recall that $T_{-}<0$, so that the bottom inequality in
(10.5) represents separation of angles much larger than $k_{1}$.

###### Proof.

We write the proof for $\sigma=1$, the case $\sigma=i$ being similar but
simpler. Also, we omit throughout the superscript $j$.

To alleviate notation, we introduce a useful change of variables. Recall that

$\displaystyle d\mathfrak{L}^{j}_{t}(\theta)=d\mathfrak{L}_{t}(\theta)$
$\displaystyle={i\theta
e^{t}}{k_{1}^{-1}}dt+\sqrt{\tfrac{4}{\beta}}e^{i\Im\mathfrak{L}_{t}(\theta)}d\mathfrak{W}_{t}^{j}.$
(10.6)

Let $(W_{t}^{1},W_{t}^{2})=(\Re\mathfrak{W}_{t},\Im\mathfrak{W}_{t})$ and set

$d\left(\begin{array}[]{l}V_{t}^{1}\\\
V_{t}^{2}\end{array}\right)=\left(\begin{array}[]{cc}\cos(\Im
L_{t}(0))&-\sin(\Im L_{t}(0))\\\ \sin(\Im L_{t}(0))&\cos(\Im
L_{t}(0))\end{array}\right)d\left(\begin{array}[]{l}W_{t}^{1}\\\
W_{t}^{2}\end{array}\right).$ (10.7)

Note that the pair of processes $V_{t}^{1},V_{t}^{2}$ are independent standard
Brownian motions. Let $(R_{t}(\theta),I_{t}(\theta))$ $=$
$(\Re\mathfrak{L}_{t}(\theta),\Im\mathfrak{L}_{t}(\theta))$, and set
$(R_{t},I_{t})=(R_{t}(0),I_{t}(0))$. We then have, with
$\hat{I}_{t}(\theta)=I_{t}(\theta)-I_{t}$, that

$dR_{t}=\sqrt{\tfrac{4}{\beta}}dV_{t}^{1},\quad
dI_{t}=\sqrt{\tfrac{4}{\beta}}dV_{t}^{2},$ (10.8)

and

$\displaystyle dR_{t}(\theta)$ $\displaystyle=$
$\displaystyle\sqrt{\tfrac{4}{\beta}}(\cos(\hat{I}_{t}(\theta))dV_{t}^{1}+\sin(\hat{I}_{t}(\theta))dV_{t}^{2}),$
(10.9) $\displaystyle d\hat{I}_{t}(\theta)$ $\displaystyle=$
$\displaystyle\theta
e^{t}k_{1}^{-1}dt+\sqrt{\tfrac{4}{\beta}}(\sin(\hat{I}_{t}(\theta))dV_{t}^{1}-(1-\cos(\hat{I}_{t}(\theta))dV_{t}^{2}).$

We begin with the proof of (10.2). Note that for any $\theta$, we have
$t\mapsto\mathfrak{U}_{t}(\theta)=R_{t}(\theta)$ has the same law as that of
$t\mapsto R_{t}$, and further $R_{t}$ is a $\sqrt{4/\beta}$ multiple of a
standard Brownian motion. Thus, by standard barrier estimates and the Markov
property, see e.g. [BRZ19, Proof of Lemma 6.3], we obtain, with $h^{\prime}$
as in (10.3), that

$\displaystyle\mathbb{P}\biggl{(}\bigl{\\{}\mathfrak{U}^{j}_{T_{+}}(\theta)\in[-k_{7},x]\bigr{\\}}\cap\mathscr{R}^{\prime}(\theta)~{}\big{|}~{}\mathscr{F}_{n_{1}^{+}}\biggr{)}=\frac{(1+o_{k_{4}})}{\sqrt{2\pi}}\frac{h}{(T_{+}-T_{-}-k_{4})^{3/2}}$
$\displaystyle\qquad\times\int_{-x}^{k_{7}}dy\int_{z\in
J_{k_{4}}}dze^{-(\sqrt{2}(T_{+}-T_{-}-k_{4})+h-z)^{2}/2(T_{+}-T_{-}-k_{4})}z\tfrac{1}{\sqrt{2\pi
k_{4}}}e^{-(z-y+\sqrt{2}k_{4})^{2}/2k_{4}},\ $

where $J_{k_{4}}=[-k_{4}^{9/10},-k_{4}^{1/10}]$. We then obtain (recalling
that $k_{i}\gg k_{i+1}$)

$\displaystyle\mathbb{P}\biggl{(}\bigl{\\{}\mathfrak{U}^{j}_{T_{+}}(\theta)\in[-k_{7},x]\bigr{\\}}\cap\mathscr{R}^{\prime}(\theta)~{}\big{|}~{}\mathscr{F}_{n_{1}^{+}}\biggr{)}$
(10.10)
$\displaystyle=(1+o_{k_{4}})\frac{\sqrt{k_{4}}}{2{\pi}}\frac{h}{(T_{+}-T_{-})^{3/2}}e^{-(T_{+}-T_{-})}e^{-\sqrt{2}h-h^{2}/2(T_{+}-T_{-})}\int_{-x}^{k_{7}}e^{\sqrt{2}y}\,dy.$

We turn to the two ray estimates; note that we are shooting for a rough bound
only, essentially accurate at the exponential scale. We may and will take
$\theta_{1}=0$ and write $\theta=\theta_{2}$. Set
$L_{t}=e^{\lambda(R_{t}+R_{t}(\theta))}$ and $\theta_{t}=\theta
e^{t}k_{1}^{-1}$. We emphasize that the estimates we will obtain will be
uniform with respect to the initial condition $\hat{I}_{T_{-}}(\theta)$. Fix
$\tilde{I}_{t}=\hat{I}_{t}-\theta_{t}$. We have from (10.9) that

$dL_{t}=\sqrt{\tfrac{4\lambda^{2}}{\beta}}L_{t}\big{(}(1+\cos(\theta_{t}+\tilde{I}_{t}))dV_{t}^{1}+\sin(\theta_{t}+\tilde{I}_{t})dV_{t}^{2}\big{)}+\tfrac{4\lambda^{2}}{\beta}(1+\cos(\theta_{t}+\tilde{I}_{t}))L_{t}dt$

and therefore, setting $\ell_{t}=\mathbb{E}L_{t}$,
$\ell_{t}^{c}=\mathbb{E}(\cos(\tilde{I}_{t})L_{t})$,
$\ell_{t}^{s}=\mathbb{E}(\sin(\tilde{I}_{t})L_{t})$, we obtain that

$\frac{d\ell_{t}}{dt}=\tfrac{4\lambda^{2}}{\beta}\big{(}\ell_{t}+\cos(\theta_{t})\ell_{t}^{c}-\sin(\theta_{t})\ell_{t}^{s}\big{)}.$
(10.11)

On the other hand, for some explicit martingale $M_{t}$,

$\displaystyle d(\cos(\tilde{I}_{t})L_{t})$ $\displaystyle=$ $\displaystyle
dM_{t}+\tfrac{4\lambda^{2}}{\beta}(1+\cos(\theta_{t}+\tilde{I}_{t}))\cos(\tilde{I}_{t})L_{t}dt-\tfrac{4}{\beta}\cos(\tilde{I}_{t})(1-\cos(\theta_{t}+\tilde{I}_{t}))L_{t}dt$
$\displaystyle-\tfrac{4\lambda}{\beta}\sin(2(\theta_{t}+\tilde{I}_{t}))\sin(\tilde{I}_{t})L_{t}dt,$

with a similar expression for $d(\sin(\tilde{I}_{t})L_{t})$. Therefore, there
exists a constant $\alpha=\alpha(\lambda,\beta)$ so that

$\left|\tfrac{d\ell_{t}^{c}}{dt}\right|\leq\alpha\ell_{t},\quad\left|\tfrac{d\ell_{t}^{s}}{dt}\right|\leq\alpha\ell_{t}.$
(10.12)

From now on, we take $4\lambda^{2}/\beta=a$ ($a\sim 2$ is the exponent we will
need for Chebychev’s inequality). Define
$\hat{\ell}_{t}=\ell_{t}e^{-a(t-T_{-})}$, and similarly
$\hat{\ell}_{t}^{c},\hat{\ell}_{t}^{s}$. Then,

$\frac{d\hat{\ell}_{t}}{dt}=ae^{-a(t-T_{-})}\big{(}\cos(\theta_{t})\ell_{t}^{c}-\sin(\theta_{t})\ell_{t}^{s}\big{)}=a\big{(}\cos(\theta_{t})\hat{\ell}_{t}^{c}-\sin(\theta_{t})\hat{\ell}_{t}^{s}\big{)},$

and therefore, with $t\geq T_{-}$,

$\hat{\ell}_{t}-\hat{\ell}_{T_{-}}=a\int_{T_{-}}^{t}\big{(}\cos(\theta_{u})\hat{\ell}_{u}^{c}-\sin(\theta_{u})\hat{\ell}_{u}^{s}\big{)}du.$

Using the change of variables $e^{u}=s$, we have that $\theta_{u}=\theta
s/k_{1}$ and thus we obtain that

$\hat{\ell}_{t}-\hat{\ell}_{T_{-}}=a\int_{e^{T_{-}}}^{e^{t}}s^{-1}\big{(}\cos(\theta
s/k_{1})\hat{\ell}^{c}_{\log s}-\sin(\theta s/k_{1})\hat{\ell}^{s}_{\log
s}\big{)}ds.$ (10.13)

Consider first the case $\theta e^{T_{-}}/k_{1}>c$ for some fixed constant
$c>1$, that is $T_{*}\leq T_{-}-\log c$. Set $T_{\ell}=T_{-}+\ell$, $\ell\geq
0$. We consider first $s\in[e^{T_{\ell}},e^{T_{\ell+1}}]:=I_{\ell}$. Let

$F_{c}(s)=\int_{e^{T_{\ell}}}^{s}u^{-1}\cos(\theta u/k_{1})du,\quad
F_{s}(s)=\int_{e^{T_{\ell}}}^{s}u^{-1}\sin(\theta u/k_{1})du.$

Then, for $s\in I_{\ell}$, we have (using integration by parts) that

$|F_{c}(s)|,|F_{s}(s)|\leq 3\tfrac{k_{1}e^{-T_{\ell}}}{\theta}=:b_{\ell}.$

Performing in (10.13) integration by parts and using (10.12) and that
$|\hat{\ell}_{t}^{c}|,|\hat{\ell}_{t}^{s}|\leq\hat{\ell}_{t}$, we obtain
(using that $4ab_{\ell}\ll 1$ for $\lambda$ bounded by say $4$ if $c$ is
chosen large enough) that for $e^{t}\in I_{\ell}$,

$\hat{\ell}_{t}\leq\hat{\ell}_{T_{\ell}}+4ab_{\ell}\hat{\ell}_{t}+4\alpha
ab_{\ell}\int_{e^{T_{\ell}}}^{e^{t}}s^{-1}\hat{\ell}_{\log
s}ds=\hat{\ell}_{T_{\ell}}+4ab_{\ell}\hat{\ell}_{t}+4\alpha
ab_{\ell}\int_{T_{\ell}}^{t}\hat{\ell}_{u}du.$

An application of Gronwall’s inequality then gives that for such $t$,

$\hat{\ell}_{t}\leq\tfrac{1}{1-4ab_{\ell}}\hat{\ell}_{T_{\ell}}e^{4ab_{\ell}\alpha(t-T_{\ell})/(1-4ab_{\ell})}.$

Unravelling the definitions, we have obtained that for $e^{t}\in I_{\ell}$,

$\mathbb{E}\big{(}e^{\lambda(R_{t}+R_{t}(\theta))}\big{)}=\ell_{t}\leq\left(\prod_{j=1}^{\ell}\tfrac{1}{1-4ab_{j}}\right)e^{a(t-T_{-})+4a\sum_{j=1}^{\ell}ab_{j}\alpha(T_{j+1}-T_{j})/(1-4ab_{j})}\leq
Ce^{a(t-T_{-})},$ (10.14)

uniformly in the initial conditions of $\hat{I}_{T_{-}}(\theta)$, where $C$ is
a universal constant independent of $\ell$ and we used that
$\sum_{j=1}^{\ell}ab_{j}\alpha(T_{j+1}-T_{j})/(1-4ab_{j})$ is uniformly
bounded. In particular, for $\theta_{1},\theta_{2}$ with
$|\theta_{1}-\theta_{2}|\geq ck_{1}e^{-T_{-}}$, using Chebycheff’s inequality
with $\lambda\sim 2$ we obtain that

$\displaystyle\mathbb{P}(R_{{T_{+}}}(\theta_{i})\geq\sqrt{8/\beta}((h_{i}-k_{7})/\sqrt{2}+(T_{+}-T_{-})),i=1,2)$
(10.15) $\displaystyle\leq
Ce^{-\sum_{i=1}^{2}\big{(}\sqrt{2}(T_{+}-T_{-})+(h_{1}-k_{7})\big{)}^{2}/2(T_{+}-T_{-})},$

which yields the bottom inequality in (10.5).

We next turn to the case $T_{+}-k_{4}\geq T_{*}>T_{-}-\log c$. In that case,
we apply the barrier estimate for time $(T_{*}-T_{-}-\log c)_{+}$ and the same
computation as above for larger times. That is, assuming without loss of
generality that $h=h_{1}$, we write $T_{*}^{\prime}=(T_{*}\vee T_{-})$ and,
with $\mathscr{P}_{j}^{\prime}(\theta_{i})^{T_{*}^{\prime}}$ the part of the
barrier event $\mathscr{P}_{j}^{\prime}(\theta_{i})$ up to time
$T_{*}^{\prime}$,

$\displaystyle\mathbb{P}(R_{{T_{+}}}(\theta_{i})\geq\sqrt{8/\beta}((h_{i}-k_{7})/\sqrt{2}+(T_{+}-T_{-}))\cap\mathscr{P}_{j}^{\prime}(\theta_{i}),i=1,2)$
(10.16)
$\displaystyle\leq\int\mathbb{P}(\sqrt{\beta/8}R_{{T_{*}^{\prime}}}(\theta_{1})\in
dz\cap\mathscr{P}_{j}^{\prime}(\theta_{1})^{T_{*}^{\prime}})\times$
$\displaystyle\qquad\mathbb{P}\Big{(}R_{T_{+}}(\theta_{1})-R_{T_{*}^{\prime}}(\theta_{1})\geq\sqrt{8/\beta}((h_{1}-k_{7})/\sqrt{2}+(T_{+}-T_{-})-z),$
$\displaystyle\qquad\qquad\qquad
R_{T_{+}}(\theta_{2})-R_{T_{*}^{\prime}}(\theta_{2})\geq\sqrt{8/\beta}(-k_{7}/\sqrt{2}+(T_{+}-T_{*})+T_{*}^{1/10}\Big{)}.$

Note that the integration over the variable $z$ in the right side of (10.16)
is restricted in particular to the range
$z\in[(T_{+}-T_{*}^{\prime})^{1/10},(T_{+}-T_{*}^{\prime})^{9/10}]$. We
control the second probability in the right hand side of (10.16) as in the
case $T_{*}\leq T_{-}-\log c$, while the first probability is controlled by a
standard Gaussian bound. This yields the top inequality in (10.5). Further
details are omitted. ∎

###  10.2 Regularity of the kernel

######  Lemma 10.2.

Let $F$ be the subset of $\mathcal{C}([-2\pi k_{1},0],\mathbb{C})$ for which
$\max_{\theta}|f(\theta)|>e^{-k_{7}}$ (which corresponds to those decorations
in $\Gamma_{k_{7}^{+}}$). Let $\mathfrak{r}$ be the Radon–Nikodym derivative
of the measure $\mathfrak{s}(e^{\sqrt{4/\beta}(\alpha+x)}df)$ with respect to
$\mathfrak{s}(e^{\sqrt{4/\beta}x}df)$ on $F$. Then, uniformly in $|x|\leq(\log
k_{1})^{17/18}$ and $|\alpha|\leq 1$,

$\mathfrak{r}(x,\alpha,f)=e^{\sqrt{2}\alpha}(1+o_{k_{1}}).$

###### Proof.

Recall that $\mathfrak{s}$ is the law of $D_{j}^{o}$. Then $D_{j}^{o}$ can be
decomposed as

$D_{j}^{o}(\theta)=e^{\mathfrak{U}_{T_{\dagger}}^{o}(0)}\times
e^{-\mathfrak{U}_{T_{\dagger}}^{o}(0)}D_{j}^{o}(\theta)\eqqcolon
e^{\mathfrak{U}_{T_{\dagger}}^{o}(0)}\times\widehat{D}(\theta),$ (10.17)

where $\widehat{D}(\theta)$ is independent of the increment of $\mathfrak{U}$.
Let $\widehat{\mathfrak{s}}$ be the law of
$e^{\sqrt{8/\beta}(T_{\dagger}-T_{+})}\widehat{D}(\theta)$. Note to have
$e^{\sqrt{4/\beta}x}D_{j}^{o}\in F$, we must have

$\sqrt{\tfrac{8}{\beta}}\mathcal{A}_{T_{\dagger}}^{+}\leq\sqrt{\tfrac{4}{\beta}}x+\mathfrak{U}_{T_{\dagger}}^{o}(0)\leq\sqrt{\tfrac{8}{\beta}}\mathcal{A}_{T_{\dagger}}^{+},$

or else by definition $D_{j}^{o}\equiv 0$. Let $J-x$ be this interval of
$\mathfrak{U}_{T_{\dagger}}^{o}(0)+\sqrt{8/\beta}(T_{\dagger}-T_{+}),$ which
has width which is at most $O((\log k_{1})^{9/10})$. Thus we have a
representation for $f\in F$, with $T=T_{\dagger}-T_{-}$, (integrating over $y$
which has the law of
$\mathfrak{U}_{T_{\dagger}}^{o}(0)+\sqrt{8/\beta}(T_{\dagger}-T_{+})$,

$\displaystyle\mathfrak{s}(e^{\sqrt{{4}/{\beta}}x}df)$
$\displaystyle=\int_{J-x}e^{-(y+\sqrt{2}T)^{2}/(2T)}\widehat{\mathfrak{s}}(e^{\sqrt{{4}/{\beta}}(y+x)}df)\,\frac{dy}{\sqrt{2\pi
T}}$
$\displaystyle=\int_{J}e^{-(y-x+\sqrt{2}T)^{2}/(2T)}\widehat{\mathfrak{s}}(e^{\sqrt{{4}/{\beta}}y}df)\,\frac{dy}{\sqrt{2\pi
T}},$

after changing $y\to y-x$. Hence this probability is differentiable in $x$ and
in fact

$\partial_{x}\mathfrak{s}(e^{\sqrt{{4}/{\beta}}x}df)=\int_{J}(\sqrt{2}-(x-y)/(2T))e^{-(y-x+\sqrt{2}T)^{2}/(2T)}\widehat{\mathfrak{s}}(e^{\sqrt{{4}/{\beta}}y}df)\,\frac{dy}{\sqrt{2\pi
T}}.$

On the assumptions on $x,y$, the factor $(\sqrt{2}-(x-y)/(2T))$ is bounded
uniformly over the $x$ and $y$ considered by $\sqrt{2}+(\log k_{1})^{-\delta}$
for some $\delta>0,$ and hence we conclude

$\partial_{x}\mathfrak{s}(e^{\sqrt{{4}/{\beta}}x}df)=(\sqrt{2}+o_{k_{1}})\mathfrak{s}(e^{\sqrt{{4}/{\beta}}x}df).$

Hence from Gronwall’s inequality, we conclude the claimed estimate. ∎

###  10.3 Mixing of phase

The final technical estimate that must be done for the decoration, which only
concerns the case $\sigma=1$, is the mixing of phase. Specifically, we must
estimate the statistical difference between

$\mathfrak{s}(e^{i\alpha+x}df)\quad\text{and}\quad\mathbb{E}\mathfrak{s}(e^{i2\pi
U+x}df),$

where $U\overset{\mathscr{L}}{=}\operatorname{Unif}([0,1])$. This is the
measure that appears in $\overline{\mathfrak{m}}$ in (2.69). Once more we let
$F$ be the subset of $\mathcal{C}([-2\pi k_{1},0],\mathbb{C})$ for which
$\max_{\theta}|f(\theta)|>e^{-k_{7}}$.

We use a representation that is similar to (10.17), namely that

$D_{j}^{o}(\theta)\eqqcolon e^{iW}\times\widecheck{D}(\theta),$ (10.18)

where $W$ is a centered Gaussian variable, independent of $\widecheck{D}$ of
variance $\tfrac{4}{\beta}(T_{\dagger}-T_{-})$, corresponding to the imaginary
part of the increment of $\mathfrak{L}_{T_{\dagger}}-\mathfrak{L}_{T_{-}}$.
Now by virtue of the variance being large, we can make a total variation
comparison of $e^{i(\alpha+W)}$ to $e^{iU}$.

######  Lemma 10.3.

With $\operatorname{dtv}$ the total variance distance, for a real Gaussian of
variance $V$ and a uniform variable $U$ on $[0,1]$, there is an absolute
constant $C>0$ so that

$\sup_{\alpha\in{\mathbb{R}}}\operatorname{dtv}(e^{i(\alpha+Z)},e^{i2\pi
U})\leq Ce^{-2\pi^{2}V}.$

###### Proof.

By periodicity, we may take $\alpha\in[0,2\pi]$. We compute the Fourier
coefficients of the law of $e^{i(\alpha+Z)}$. Note that the $k$-th coefficient
is given by

$\mathbb{E}e^{i2\pi k(\alpha+Z)}=e^{i2\pi k\alpha}e^{-2\pi^{2}k^{2}V}.$

As the total variation distance is given by integrating the
$\text{L}^{1}$–distance of the densities, we have

$\sup_{\alpha\in{\mathbb{R}}}\operatorname{dtv}(e^{i(\alpha+Z)},e^{i2\pi
U})\leq\frac{1}{\sqrt{2\pi}}\sum_{k\neq 0}\bigl{|}\mathbb{E}e^{i2\pi
k(\alpha+Z)}\bigr{|}\leq Ce^{-2\pi^{2}V}.$

∎

This leads to:

######  Corollary 10.4.

Over any subset $F\subseteq\mathcal{C}((-2\pi k_{1},0),{\mathbb{C}})$ for
which $e^{i\alpha}F=F$ for any $\alpha\in{\mathbb{R}}$,

$\sup_{\alpha\in{\mathbb{R}}}d_{\operatorname{BL}}(\mathfrak{s}(e^{i\alpha+x}df),\mathbb{E}\mathfrak{s}(e^{i2\pi
U+x}df))\leq Ce^{-(8/\beta)\pi^{2}(T_{\dagger}-T)}\mathfrak{s}(e^{x}F).$

###### Proof.

Rescaling both sides of the equation by $\mathfrak{s}(e^{x}F),$ we then have a
bound for the bounded variation distance of two probability measures,

$\mathfrak{s}(e^{i\alpha+x}df)/\mathfrak{s}(e^{x}F)\quad\text{and}\quad\mathbb{E}\mathfrak{s}(e^{i2\pi
U+x}df)/\mathfrak{s}(e^{x}F).$

From (10.18), we have that the phase $e^{iW}$ is independent of
$\widecheck{D}$, and hence we have a coupling of these two measures that holds
with probability $1-\operatorname{dtv}(e^{i(\alpha+Z)},e^{i2\pi U})$. By the
definition of bounded–Lipschitz metric, the claimed bound follows. ∎

##  Appendix A Point processes

We record some elementary properties about point processes, and in particular
approximation of point processes by Poisson processes. We will let $\Gamma$ be
a complete separable metric space, with a metric $\partial_{0}$ which is
bounded by $1.$ We will use the metrics $\partial_{1}$ and $\partial_{2}$ from
the introduction. Over this space, we also define two metrics for any finite
point measures $\xi_{1}=\sum_{i=1}^{m}\delta_{y_{i}}$ and
$\xi_{2}=\sum_{i=1}^{n}\delta_{z_{i}}$ for $m\neq n$

$d_{1}(\xi_{1},\xi_{2})\coloneqq\begin{cases}0,&\text{if }m=n=0,\\\
\min_{\pi}\frac{1}{n}\sum_{i=1}^{n}\partial_{0}(y_{i},z_{\pi(i)}),&\text{if
}m=n>0,\\\ 1,&\text{if }m\neq n,\\\ \end{cases}$

with the minimum being the distance over all permutations of
$\\{1,2,\dots,n\\}.$ Finally, for two laws on point processes $Q_{1}$ and
$Q_{2}$ we let

$d_{2}(Q_{1},Q_{2})\coloneqq\inf_{(\xi_{1},\xi_{2})}\mathbb{E}(d_{1}(\xi_{1},\xi_{2})).$

We note that it is possible to bound $\partial_{1}(\xi_{1},\xi_{2})\leq
d_{1}(\xi_{1},\xi_{2})\max\\{|\xi_{1}|,|\xi_{2}|\\}$ where $|\xi_{j}|$ is the
cardinality of the point set for $j=1,2$, and hence

$\partial_{2}(Q_{1},Q_{2})\leq\inf_{(\xi_{1},\xi_{2})}\biggl{\\{}\sqrt{\mathbb{E}(d_{1}^{2}(\xi_{1},\xi_{2}))}\sqrt{\vspace{3em}\mathbb{E}(|\xi_{1}|^{2}+|\xi_{2}|^{2})}\biggr{\\}}.$
(A.1)

We will formulate a corollary of [CX11] for Poisson process approximation with
local dependency structure. We let $\left\\{\Xi_{i}:i\in\mathcal{I}\right\\}$
be a set of Bernoulli point processes, so that $\Xi_{i}(\Gamma)\in\\{0,1\\}.$
Let $\Xi$ be the sum $\Xi=\sum_{i\in\mathcal{I}}\Xi_{i}.$ We will suppose that
this set has local dependence, in that for each $i\in\mathcal{I}$ there are
sets $A_{i}\subseteq B_{i}\subseteq\mathcal{I}$ with $i\in A_{i}$ so that

$\forall~{}i\in\mathcal{I}\quad\\{\Xi_{j}:j\in A_{i}\\}\quad\text{is
independent of}\quad\\{\Xi_{j}:j\in\mathcal{I}\setminus B_{i}\\}.$ (A.2)

We let ${\lambda}_{i}$ be the intensity measure of $\Xi_{i}$ for all
$i\in\mathcal{I},$ and we let ${\lambda}\coloneqq\sum_{i}\mathbf{\lambda}_{i}$
and $\Lambda\coloneqq{\lambda}(\Gamma).$

######  Theorem A.1.

Suppose $\left\\{\Xi_{i}:i\in\mathcal{I}\right\\}$ satisfy (A.2), and define
for any $i\in\mathcal{I},$

$T_{i}\coloneqq\sum_{j\in A_{i}\setminus\\{i\\}}\Xi_{i}(\Gamma),\quad
P_{i}\coloneqq\Xi_{i}(\Gamma),\quad\text{and}\quad
L_{i}\coloneqq\sum_{j\in\mathcal{I}\setminus
B_{i}}\mathbb{E}\bigl{(}P_{j}\bigr{)}.$

Then there is a numerical constant $C>0$ so that for a Poisson process $\Pi$
with intensity $\lambda,$

$\partial_{2}(\Xi,\Pi)\leq
C(\operatorname{Var}(\Xi(\Gamma))+3\Lambda)^{3/2}\sum_{i\in\mathcal{I}}\biggl{(}\frac{\mathbb{E}(P_{i})\mathbb{E}(T_{i})+\mathbb{E}(T_{i}P_{i})+(\mathbb{E}(P_{i}))^{2}}{L_{i}^{2}}\biggr{)}.$

###### Proof.

We start by using [CX11, (2.6)] to establish that there is a numerical
constant $C>0$ so that

$d_{2}(\Xi,\Pi)\leq
C\sum_{i\in\mathcal{I}}\biggl{(}\frac{1}{L_{i}}+\frac{\operatorname{Var}(\Xi(\Gamma))}{L_{i}^{2}}\biggr{)}\cdot\bigl{(}\mathbb{E}(P_{i})\mathbb{E}(T_{i})+\mathbb{E}(T_{i}P_{i})+(\mathbb{E}(P_{i}))^{2}\bigr{)}.$
(A.3)

We note that in the notation of that eqution $|V_{i}|=T_{i},$ and their bold
$\lambda$ are our $\lambda.$ Finally, we bound their $\kappa_{i}$ by
$\operatorname{Var}(|\Xi(\Gamma)|)/L_{i},$ and we bound $\sqrt{\kappa_{i}}$ by
$1+\kappa_{i}.$

We also note that [CX11] requires that $\Gamma$ is a locally compact metric
space. As $\lambda$ is a finite Borel measure on $\Gamma,$ it is tight. Hence
there are compact sets $K_{i}$ so that with
$\tilde{\Gamma}=\cup_{i=1}^{\infty}K_{i},$
$\lambda(\Gamma\setminus\tilde{\Gamma})=0.$ The space $\tilde{\Gamma}$ is
locally compact, and both point processes $\Xi$ and $\Pi$ put $0$ mass on
$\Gamma\setminus\tilde{\Gamma}$ with probability $1.$ Hence we may apply the
Theorem of [CX11] to $\Xi$ and $\Pi$ considered as point processes on
$\tilde{\Gamma}$ with the same metrics $\partial_{0},d_{1}$ and $d_{2}$.
Furthermore, coupling the point processes as random measures on
$\tilde{\Gamma}$ gives a coupling of the point processes as random measures on
$\Gamma,$ and hence (A.3) follows.

From (A.1), and using that $d_{1}$ is bounded by $1,$

$\partial_{2}(\Xi,\Pi)\leq
d_{2}(\Xi,\Pi)\sqrt{\operatorname{Var}(\Xi(\Gamma))+3\Lambda}.$

As $L_{i}\leq\Lambda,$ the claim follows. ∎

We also formulate a change-of-intensity lemma that controls the distance
between two Poisson processes of similar intensity. Define for any two finite
Borel measures $\pi$ and $\lambda$ on $\Gamma,$

$d_{\operatorname{BL}}(\pi,\lambda)=\sup_{f}\biggl{|}\int_{\Gamma}fd(\pi-\lambda)\biggr{|},$
(A.4)

with the supremum over all $f$ with both $|f(x)-f(y)|\leq\partial_{0}(x,y)$
for all $x,y\in\Gamma$ and $|f(x)|\leq 1$ for all $x\in\Gamma.$ Then:

######  Theorem A.2.

Let $\pi$ and $\lambda$ be two finite measures on $\Gamma.$ Let
$\alpha=\min\\{\pi(\Gamma),\lambda(\Gamma)\\}.$ Then

$d_{2}(\Pi(\pi),\Pi(\lambda))\leq\tfrac{1-e^{-\alpha}}{\alpha}d_{\operatorname{BL}}(\pi,\lambda)\quad\text{and}\quad\partial_{2}(\Pi(\pi),\Pi(\lambda))\leq\sqrt{2}\max\\{\pi(\Gamma),\lambda(\Gamma)\\}\tfrac{1-e^{-\alpha}}{\alpha}d_{\operatorname{BL}}(\pi,\lambda).$

###### Proof.

The first is nearly a corollary of [BX95, Theorem 1.2]. To derive it, apply
Lemma 2.6 directly to the second line of the displayed equation at the top of
[BX95, p.259]. The second follows from the first by using (A.1). ∎

######  Corollary A.3.

For any compact $\Gamma$, and for any $\epsilon>0$, there is a finite list
$f_{1},f_{2},\dots,f_{n}$ of non-negative functions $f_{j}$ with
$|f_{j}(x)-f_{j}(y)|\leq\partial_{0}(x,y)$ and $|f_{j}(x)|\leq 1$ for all
$x,y\in\Gamma$ and a $\delta$ so that if

$\max_{j=1,\dots,n}\biggl{|}\int_{\Gamma}f_{j}d(\pi-\lambda)\biggr{|}\leq\delta$

then $\partial_{2}(\Pi(\pi),\Pi(\lambda))\leq\epsilon.$

###### Proof.

By Arzelà–Ascoli, we can find a list $\\{g_{j}\\}$ for $j=1,\dots,n$ so that
for all $1$–Lipschitz, 1–bounded $f$,

$\min_{j}\sup_{x\in\Gamma}|f(x)-g_{j}(x)|\leq\epsilon/2.$

Let $\\{f_{j}\\}$ be all the positive and negative parts of the functions in
this list. We may also assume the constant function is in the list. Now
suppose that

$\max_{j=1,\dots,n}\biggl{|}\int_{\Gamma}f_{j}d(\pi-\lambda)\biggr{|}\leq\delta.$

Then $|\pi(\Gamma)-\lambda(\Gamma)|\leq\delta$, and so by Theorem A.2

$\partial_{2}(\Pi(\pi),\Pi(\lambda))\leq\sqrt{2}\max\\{\pi(\Gamma),\lambda(\Gamma)\\}\tfrac{1-e^{-\alpha}}{\alpha}d_{\operatorname{BL}}(\pi,\lambda)\leq\sqrt{2}\tfrac{\max\\{\pi(\Gamma),\lambda(\Gamma)\\}}{\max\\{\pi(\Gamma),\lambda(\Gamma)\\}-{\delta}}(\epsilon/2+\delta).$

If $\max\\{\pi(\Gamma),\lambda(\Gamma)\\}=0,$ there was nothing to prove in
the first place. Otherwise by taking $\delta$ sufficiently small, we conclude
the claim. ∎

##  Appendix B Auxilliary one and two-rays estimates

We collect in this appendix the key two-rays estimates used in the bulk of the
evolution. The argument were developed in details in [CMN18], and in fact even
our tex file is based on theirs. The hurried reader may skip the content of
the appendix, keeping only the statements of Propositions B.2, B.3 and Lemmas
B.4, B.5, B.6, B.8 and B.9 in mind.

Throughout, we adopt exactly, for ease of reference, the notation of [CMN18].

##### Further notation:

For any $n\geq p$, $\theta\in[0,2\pi)$, we define

$\displaystyle
Z_{n}^{(p)}(\theta):=Z_{n}(\theta)-Z_{p}(\theta)=\sum_{j=p}^{n-1}\frac{{\mathcal{N}}_{j}^{\mathbb{C}}}{\sqrt{j+1}}e^{i((j+1)\theta+A_{j}(\theta))}.$

We stress that $Z_{n}^{(p)}$ depends implicitely of $\beta$ because of the
Prüfer phase. Similarly, we define

$\displaystyle
A_{n}^{(p)}(\theta):=\sum_{j=p}^{n-1}a_{j}(\theta)=A_{n}(\theta)-A_{p}(\theta),\
.$

where

$a_{j}(\theta):=A_{j+1}(\theta)-A_{j}(\theta).$

More generally, for any family of quantities depending on an index $k$, we
will denote the difference of the quantities indexed by $k$ and $p$ by the
same notation, with $k$ as an index and $p$ as an upperscript.

In the following, it will be convenient to study the field at times which are
powers of $2$.

###  B.1 The Chhaibi-Madaule-Najnudel coupling

As in [CMN18], we introduce a new process in order to gain more independence.
Recall that $N=\log n$.

##### A more independent field:

For each fixed $\theta$, $(Z_{2^{k}}(\theta))_{k\geq 0}$ is a complex Gaussian
random walk. Moreover we could compute the correlations of $Z_{2^{k}}(\theta)$
and $Z_{2^{k}}(\theta^{\prime})$ and observe that they behave logarithmically
with respect to the distance between $\theta$ and $\theta^{\prime}$ modulo
$2\pi$. However, $Z$ is not globally Gaussian, so we cannot directly apply
known results on the maximum of Gaussian fields, but we will still provide its
approximative branching structure. To achieve this aim we will gain some
independence by making small changes on $Z$.

Let us fix some integer $r$, which will be assumed to be larger than some
suitable universal constant. For $l\geq r$, we denote
$\Delta=\Delta^{(l)}:=e^{\sqrt{\log r}}+100\lfloor\log^{2}l\rfloor$. Observe
that for any $N\geq r$, we can rewrite formula [CMN18, (3.3)] as:

$\displaystyle Z_{2^{N}}^{(2^{r})}(\theta):=$
$\displaystyle\sum_{l=r}^{N-1}\sum_{p=0}^{2^{\Delta}-1}\left(\sum_{j=0}^{2^{l-\Delta}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\Delta}+j}\frac{e^{i\psi_{2^{l}+p2^{l-\Delta}+j}(\theta)}}{\sqrt{2^{l}+p2^{l-\Delta}+j+1}}\right).$
(B.1)

Note that $\Delta^{(l)}$ and $l-\Delta^{(l)}$ are strictly positive if $l\geq
r$ and $r$ is large enough. Now, let $Z^{(2^{r},\Delta)}$ be the process
defined by

$\displaystyle Z_{2^{N}}^{(2^{r},\Delta)}(\theta):=$
$\displaystyle\sum_{l=r}^{N-1}\sum_{p=0}^{2^{\Delta}-1}\left(\sum_{j=0}^{2^{l-\Delta}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\Delta}+j}e^{ij\theta}\right)\frac{e^{i\psi_{2^{l}+p2^{l-\Delta}}(\theta)}}{\sqrt{2^{l}+p2^{l-\Delta}}}.$
(B.2)

Observe that $Z_{2^{N}}^{(2^{r})}(\theta)$ and
$Z_{2^{N}}^{(2^{r},\Delta)}(\theta)$ only differ by the change in the square
root of the denominator, and by the replacement of some increments of the
Prüfer phases by their mean. The following is a slight variation of [CMN18,
Proposition 5.2]; the additional statement (B.4) actually follows from the
proof in [CMN18].

######  Proposition B.1.

For $r$ large enough,

$\displaystyle\sum_{l\geq
r}{\mathbb{P}}\left(\sup_{\theta\in[0,2\pi)}|Z_{2^{l+1}}^{(2^{l},\Delta)}(\theta)-Z_{2^{l+1}}^{(2^{l})}(\theta)|\geq
2l^{-2}\right)<+\infty$

In particular, as $\sum_{l\geq r}2l^{-2}<+\infty$, we have that almost surely,

$\displaystyle\sum_{l\geq
r}\sup_{\theta\in[0,2\pi)}|Z_{2^{l+1}}^{(2^{l},\Delta)}(\theta)-Z_{2^{l+1}}^{(2^{l})}(\theta)|<\infty\
$ (B.3)

and

$\displaystyle\limsup_{r\to\infty}\sum_{l\geq
r}\sup_{\theta\in[0,2\pi)}|Z_{2^{l+1}}^{(2^{l},\Delta)}(\theta)-Z_{2^{l+1}}^{(2^{l})}(\theta)|=0.$
(B.4)

In the proof of Proposition B.1, [CMN18] introduce the variables, for any
$l\geq 0$, $\theta\in[0,2\pi)$,

$\displaystyle\blacksquare_{l}(\theta)$
$\displaystyle:=Z_{2^{l+1}}^{(2^{l},\Delta)}(\theta)-Z_{2^{l+1}}^{(2^{l})}(\theta)$
(B.5)
$\displaystyle=\sum_{p=0}^{2^{\Delta}-1}\frac{e^{i\psi_{2^{l}+p2^{l-\Delta}}(\theta)}}{\sqrt{{2^{l}+p2^{l-\Delta}}}}\left(\sum_{j=0}^{2^{l-\Delta}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\Delta}+j}e^{ij\theta}\square_{j}^{(2^{l}+p2^{l-\Delta})}(\theta)\right)\
,$

with

$\displaystyle\left|\square_{j}^{(2^{l}+p2^{l-\Delta})}(\theta)\right|\leq$
$\displaystyle\left|A_{j+2^{l}+p2^{l-\Delta}}^{(2^{l}+p2^{l-\Delta})}(\theta)\right|+2^{-\Delta}.$
(B.6)

The main steps in the proof consist of the estimates

$\displaystyle{\mathbb{P}}\left(|\blacksquare_{l}(\theta)|\geq
l^{-2}\right)\ll_{\beta}e^{-2^{\frac{1}{4}\Delta}}.$ (B.7)

and, for $\lambda\in{\mathbb{R}}$:

$\displaystyle\sup\left({\mathbb{E}}\left(e^{\lambda\Re\blacksquare_{l}(\theta)}\mathds{1}_{G_{l}(\theta)}\Big{|}\mathcal{G}_{2^{l}}\right),{\mathbb{E}}\left(e^{\lambda\Im\blacksquare_{l}(\theta)}\mathds{1}_{G_{l}(\theta)}\Big{|}\mathcal{G}_{2^{l}}\right)\right)\leq$
$\displaystyle
e^{\lambda^{2}\left(2^{-\frac{1}{4}\Delta}+2^{-\Delta}\right)^{2}}\ .$ (B.8)

(see [CMN18, (5.6), (5.8)].)

###  B.2 One and two ray estimates

In the sequel, for all fields denoted by $Z$ with some indices and
superscripts, we write $R$ with the same indices and superscripts for the real
part of $\sigma$ times the initial field (recall that
$\sigma\in\\{1,i,-i\\}$).

##### An envelope for the paths of $R_{2^{N}}^{(2^{r},\Delta)}(\theta)$:

For $j\geq r$, let

$\displaystyle\tau^{(r)}_{j}$
$\displaystyle:=\sum_{l=r}^{j-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}\frac{2^{l-\Delta^{(l)}}}{2^{l}+p2^{l-\Delta^{(l)}}}=\sum_{l=r}^{j-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}\frac{1}{2^{\Delta^{(l)}}+p}=:(j-r)\log
2+\lambda_{j}^{(r)},$ (B.9)

where $(\lambda_{j}^{(r)})_{j\geq r}$ is a nonnegative and increasing
sequence, tending, when $j\rightarrow\infty$, to a limit
$\lambda_{\infty}^{(r)}$ such that

$\lambda_{\infty}^{(r)}<\sum_{l\geq r}2^{-\Delta^{(l)}}\leq 2^{-e^{\sqrt{\log
r}}}\sum_{l=1}^{\infty}2^{-100\lfloor\log^{2}l\rfloor}\ll 2^{-e^{\sqrt{\log
r}}}\to 0$

when $r$ goes to $\infty$. Let $\alpha_{+}:=1-\frac{1}{10}$ and
$\alpha_{-}:=\frac{1}{10}$, and for $N\geq 2r$, $k\in[|r,N|]$, define

$\displaystyle
u_{k}^{(N)}:=\left\\{\begin{array}[]{ll}-k^{\alpha_{-}},&\text{if
}k\leq\lfloor N/2\rfloor,\\\ -(N-k)^{\alpha_{-}}-\frac{3}{4}\log N,&\text{if
}\lfloor N/2\rfloor<k\leq N,\end{array}\right.$ (B.12)

and

$\displaystyle
l_{k}^{(N)}:=\left\\{\begin{array}[]{ll}-k^{\alpha_{+}},&\text{if
}k\leq\lfloor N/2\rfloor,\\\ -(N-k)^{\alpha_{+}}-\frac{3}{4}\log N,&\text{if
}\lfloor N/2\rfloor<k\leq N.\end{array}\right.$ (B.15)

We then define an envelope by its lower bound and its upper bound at each
$k\in[|r,N|]$:

$\displaystyle U_{k}^{(N)}:=\tau^{(r)}_{k}+u_{k}^{(N)}\quad\text{and }\quad
L_{k}^{(N)}:=\tau^{(r)}_{k}+l_{k}^{(N)}.$

Fix $x<0$ (depending on $k_{2}$), $z<0$ (depending on $k_{1}^{+}$), and
$\upsilon\in(0,1)$. Set $N_{+}=\log n_{1}^{+}$. The basic event we need to
consider is, for $\theta\in[0,2\pi)_{2^{N}}$,

$\displaystyle\mathfrak{I}_{N}(\theta)=\mathfrak{I}_{N}(\theta,x,z):=$
$\displaystyle{\\{\forall k\in[|r,N_{1}^{+}|],\,L_{k}^{(N)}\leq
x+R_{2^{k}}^{(2^{r},\Delta)}(\theta)\leq
U_{k}^{(N)},x+R_{2^{N_{+}}}^{(2^{r},\Delta)}(\theta)\in\tau^{(r)}_{N_{+}}-\frac{3}{4}\log
N+[z,z+\upsilon)\\}}.$

We will consider $x\in[-r^{1/20},-r^{19/20}]$ and
$z\in[-(k_{1}^{+})^{1/20},-(k_{1}^{+})^{19/20}]$. The walk
$(R_{2^{k}}^{(2^{r},\Delta)}(\theta))_{r\leq k\leq N}$ is a Gaussian random
walk whose distribution is the same as
$\sqrt{\frac{1}{2}}(W_{\tau_{j}^{(r)}})_{r\leq k\leq N}$. In the case where an
event $\mathfrak{I}_{N}(\theta)$ occurs for some $\theta\in[0,2\pi)_{2^{N}}$,
it means that $x+R_{2^{k}}^{(2^{r},\Delta)}(\theta)$ is around
$\tau_{k}^{(r)}$ for $r\leq k\leq N$, i.e. the Brownian motion $W$ is roughly
growing linearly with rate $\sqrt{2}$. For this reason, in the sequel of this
part of the paper, we will often estimate the probability of an event $Ev$
concerning the random walk $(R_{2^{k}}^{(2^{r},\Delta)}(\theta))_{r\leq k\leq
N}$ to a the probability of a similar event $GEv$, where a linear function
$t\mapsto t\sqrt{2}$ has been subtracted from the possible trajectories of the
underlying Brownian motion $W$ for which the event $Ev$ is satisfied. If $Ev$
depends only on the trajectory of $W$ up to a certain time $T$, we get, by
using the Girsanov transfomation, an equality of the form

${\mathbb{P}}[Ev((W_{t})_{0\leq t\leq
T})]={\mathbb{P}}[GEv((W_{t}-t\sqrt{2})_{0\leq t\leq
T})]={\mathbb{E}}[e^{-\sqrt{2}W_{T}-T}\mathds{1}_{GEv((W_{t})_{0\leq t\leq
T})}],$

and then the inequality

${\mathbb{P}}[Ev]\leq e^{-T-\sqrt{2}\mu}{\mathbb{P}}[GEv]$

where $\mu$ denotes the smallest possible value of $W_{T}$ for which the event
$GEv$ can occur.

The following proposition gives a lower bound for the first moment of
$\mathfrak{I}_{N}$.

######  Proposition B.2 (First moment of $\mathfrak{I}_{N}$).

For any $\upsilon\in(0,1)$, $r\in{\mathbb{N}}$ large enough and $N$ large
enough depending on $r$:

$\displaystyle{\mathbb{P}}(\mathfrak{I}_{N}(\theta))$ $\displaystyle\geq
e^{2(x-z)}2^{r-N_{+}}e^{-2\upsilon-2\lambda_{\infty}^{(r)}}N^{\frac{3}{2}}{\mathbb{P}}(Event_{r,N}),$
(B.16) $\displaystyle{\mathbb{P}}(\mathfrak{I}_{N}(\theta))$
$\displaystyle\leq
e^{2(x-z)}2^{r-N_{+}}e^{2\upsilon+2\lambda_{\infty}^{(r)}}N^{\frac{3}{2}}{\mathbb{P}}(Event_{r,N}),$
(B.17)

where, with $W$ being a standard Brownian motion,

$\displaystyle Event_{r,N}=Event_{r,N}(x,z):=$ $\displaystyle\left\\{\forall
j\in[|r,N_{+}|],\,l_{j}^{(N)}\leq x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{j}}\leq
u_{j}^{(N)},x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{N_{+}}}\in\tau^{(r)}_{N_{+}}-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\}.$

Further, for $x,z$ as above, the asymptotics of ${\mathbb{P}}(Event_{r,N})$
are given in Lemma C.1 below.

###### Proof.

Since $(R_{2^{j}}^{(2^{r},\Delta)}(\theta))_{r\leq j\leq N_{+}}$ is a Gaussian
random walk whose distribution does not depend on $\theta$, we have

$\displaystyle{\mathbb{P}}\left(\mathfrak{I}_{N}(\theta)\right)=$
$\displaystyle{\mathbb{P}}\left(\forall j\in[|r,N_{+}|],\,L_{j}^{(N)}\leq
x+R_{2^{j}}^{(2^{r},\Delta)}(0)\leq
U_{j}^{(N)},x+R_{2^{N_{+}}}^{(2^{r},\Delta)}(0)\in\tau^{(r)}_{N_{+}}-\frac{3}{4}\log
N+[z,z+\upsilon)\right).$

More precisely, we know that $(R_{2^{j}}^{(2^{r},\Delta)}(0))_{j\geq r}$ is
distributed like $\sqrt{\frac{1}{2}}(W_{\tau^{(r)}_{j}})_{j\geq r}$. By
Girsanov’s transform, with density
$e^{\sqrt{2}W_{\tau^{(r)}_{N_{+}}}-\tau^{(r)}_{N_{+}}}$, we have

$\displaystyle{\mathbb{P}}(\mathfrak{I}_{N}(\theta))$
$\displaystyle={\mathbb{E}}\left(e^{-\sqrt{2}\left(W_{\tau^{(r)}_{N_{+}}}+\sqrt{2}\tau^{(r)}_{N_{+}}\right)+\tau_{N_{+}}^{(r)}}\mathds{1}_{\\{\forall
j\in[|r,N_{+}]|],\,l_{j}^{(N)}\leq x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{j}}\leq
u_{j}^{(N)},x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{N_{+}}}\in-\frac{3}{4}\log
N+[z,z+\upsilon)}\\}\right)$
$\displaystyle=e^{-\tau^{(r)}_{N_{+}}}{\mathbb{E}}\left(e^{-\sqrt{2}W_{\tau^{(r)}_{N_{+}}}}\mathds{1}_{\\{\forall
j\in[|r,N_{+}|],\,l_{j}^{(N)}\leq x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{j}}\leq
u_{j}^{(N)},x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{N_{+}}}\in-\frac{3}{4}\log
N+[z,z+\upsilon)\\}}\right)$ $\displaystyle\geq
2^{r-N_{+}}e^{2(x-z)-2\upsilon-\lambda_{\infty}^{(r)}}N^{\frac{3}{2}}{\mathbb{P}}\left(Event_{r,N}\right),$

with a similar upper bound replacing $-2\upsilon-\lambda_{\infty}^{(r)}$ by
$2\upsilon+\lambda_{\infty}^{(r)}$. Using Lemma C.1 completes the proof of the
right inequality in (B.16). ∎

We will also need a similar estimate for shorter times. Let $t<N/4$ and set

$\displaystyle\mathfrak{I}_{t,f}(\theta)=\mathfrak{I}_{N,t}(\theta,x,z):=$
$\displaystyle{\\{\forall k\in[|r,t|],\,L_{k}^{(N)}\leq
x+R_{2^{k}}^{(2^{r},\Delta)}(\theta)\leq
U_{k}^{(N)},x+R_{2^{t}}^{(2^{r},\Delta)}(\theta)\in\tau^{(r)}_{t}+[z,z+\upsilon)\\}}.$

Note that because of our choice of $t$, only the first part of the barrier is
employed in $\mathfrak{I}_{t,f}$, and therefore $\mathfrak{I}_{t,f}(\theta)$
does not depend on $N$. We have the following analogue of Proposition B.2.

######  Proposition B.3 (First moment of $\mathfrak{I}_{N,t}$).

For any $\upsilon\in(0,1)$, $C>0$, $r\in{\mathbb{N}}$ large enough, $x$ as
above, $z\in[-\sqrt{t}/C,-C\sqrt{t}]$ and $t$ large enough depending on $r$:

$\displaystyle{\mathbb{P}}(\mathfrak{I}_{t,f}(\theta))$ $\displaystyle\geq
e^{2(x-z)}2^{r-N_{+}}e^{-2\upsilon-2\lambda_{\infty}^{(r)}}{\mathbb{P}}(Event_{r,t,f})\gg_{C}\frac{|xz|}{t^{3/2}}e^{2(x-z)}2^{r-t},$
(B.18)

where, with $W$ being a standard Brownian motion,

$\displaystyle Event_{r,t,f}=Event_{r,N,t}(x,z):=$
$\displaystyle\left\\{\forall j\in[|r,t|],\,l_{j}^{(N)}\leq
x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{j}}\leq
u_{j}^{(N)},x+\sqrt{\frac{1}{2}}W_{\tau^{(r)}_{t}}\in\tau^{(r)}_{t}+[z,z+\upsilon)\right\\}.$

The proof is similar to that of Proposition B.2, and is omitted.

###  B.3 Two rays estimate

We will bound here

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})={\mathbb{P}}_{N}(\theta,\theta^{\prime},x,z,x^{\prime},z^{\prime}):={\mathbb{P}}(\mathfrak{I}_{N}(\theta,x,z)\cap\mathfrak{I}_{N}(\theta^{\prime},x^{\prime},z^{\prime})).$
(B.19)

(For the sake of shortness of notation, when no confusion occurs we write
${\mathbb{P}}_{N}(\theta,\theta^{\prime})$, omitting the
$x,z,x^{\prime},z^{\prime}$ from the notation.) This study, which is
technical, is based on the fact that the random walks
$(R_{2^{j}}^{(2^{r},\Delta)}(\theta))_{r\leq j\leq N}$ and
$(R_{2^{j}}^{(2^{r},\Delta)}(\theta^{\prime}))_{r\leq j\leq N}$ are Gaussian
random walks whose increments are approximately independent after some
branching time which is roughly the logarithm in base $2$ of the distance
modulo $2\pi$ between $\theta$ and $\theta^{\prime}$.

The general idea is as follows. For given $\theta,\theta^{\prime}$, we will
consider the integer $\mathtt{k}$ such that
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}<2^{-(\mathtt{k}-1)}$,
$||\cdot||$ denoting the distance on the set ${\mathbb{R}}/2\pi{\mathbb{Z}}$.
One can understand $\mathtt{k}$ as the time of (approximate) branching between
the field at $\theta$ and at $\theta^{\prime}$. We will show that after some
time $\mathtt{k}_{+}=\mathtt{k}_{+}(\mathtt{k})$ “slightly larger” than
$\mathtt{k}$, we are able to bring out independence between the increments of
$(R_{2^{k}}^{(2^{r},\Delta)}(\theta))_{k\geq r}$ and
$(R_{2^{k}}^{(2^{r},\Delta)}(\theta^{\prime}))_{k\geq r}$. By analogy with the
Gaussian field, we will see this time as a time of decorrelation. It is
defined as follows:

  * •

For $\mathtt{k}\leq N/2$, the time of decorrelation is
$\mathtt{k}_{+}:=\mathtt{k}+3\Delta^{(\mathtt{k})}$. Recall that:

$\Delta^{(\mathtt{k})}:=e^{\sqrt{\log r}}+100\lfloor\log^{2}\mathtt{k}\rfloor\
.$

In particular, for $\mathtt{k}\leq r/2$ and $r$ large enough, the walks
$(R_{2^{k}}^{(2^{r},\Delta)}(\theta))_{k\geq r}$ and
$(R_{2^{k}}^{(2^{r},\Delta)}(\theta^{\prime}))_{k\geq r}$ will have “almost
independent increments” from the starting time $r$, since
$r\leq\mathtt{k}_{+}$.

  * •

For $N/2<\mathtt{k}\leq N_{+}$, we will require a faster decorrelation . We
take $\mathtt{k}_{+}:=\mathtt{k}+3\kappa^{(\mathtt{k})}$, where

$\kappa^{(\mathtt{k})}:=\lfloor
r/100\rfloor+100\lfloor\log(N-\mathtt{k})\rfloor^{2}.$

However, the price to pay is that we will have to modify our field
$(R_{2^{j}}^{(2^{r},\Delta)}(\theta))_{r\leq j\leq N}$ in the spirit of
subsection B.1.

Our two rays estimates are divided to three lemmas, numbered B.4, B.5 and B.6.
The statement of each of these lemmas gives a suitable majorization of
${\mathbb{P}}_{N}(\theta,\theta^{\prime})$, for a given range of values of
$\mathtt{k}$.

######  Lemma B.4 (Time of branching $\mathtt{k}\leq N/2$).

For any $\upsilon\in(0,1)$, $r$ large enough, $N$ large enough depending on
$r$, $\mathtt{k}\leq\frac{N}{2}$ and
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}<2^{-(\mathtt{k}-1)}$,
we have

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})$
$\displaystyle\ll\left\\{\begin{array}[]{ll}|zz^{\prime}xx^{\prime}|2^{2r-2N_{+}+2(x-z+x^{\prime}-z^{\prime})},&\text{when
}\mathtt{k}_{+}\leq r,\\\
|zz^{\prime}x|e^{2(x-z-z^{\prime})}2^{r-N_{+}}2^{\mathtt{k}_{+}-N_{+}}e^{-\mathtt{k}_{+}^{\alpha_{-}}},&\text{when
}\mathtt{k}_{+}\geq r.\end{array}\right.$ (B.22)

The following lemma studies pairs whose time of branching happen before $r/2$.
It refines the estimate obtained in the previous lemma.

######  Lemma B.5 (Time of branching $\mathtt{k}\leq r/2$).

For any $\upsilon\in(0,1)$, $r\in{\mathbb{N}}$ large enough,
$\mathtt{k}\leq\frac{r}{2}$,
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}\leq
2^{-(\mathtt{k}-1)}$ and $N$ large enough depending on $r$, we have

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})$
$\displaystyle\leq(1+\eta_{r,\upsilon})2^{2r-2N_{+}}e^{2(x-z+x^{\prime}-z^{\prime})}N^{3}{\mathbb{P}}(Event_{r,N}(x,z)){\mathbb{P}}(Event_{r,N}(x^{\prime},z^{\prime})),$

where

$\underset{\upsilon\rightarrow
0}{\limsup}\,\underset{r\rightarrow\infty}{\limsup}\,\eta_{r,\upsilon}=0.$

######  Lemma B.6 (Time of branching $N/2<\mathtt{k}\leq N_{+}$).

For $\upsilon\in(0,1)$, $r\in{\mathbb{N}}$ large enough, $N$ large enough
depending on $r$, $N/2<\mathtt{k}\leq N_{+}$ and
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}<2^{-(\mathtt{k}-1)}$,
we have $\mathtt{k}_{+}<N$ and

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})\ll|xz|e^{2(x-z-z^{\prime})}2^{r-N_{+}}2^{\mathtt{k}_{+}-N_{+}}e^{-\frac{1}{2}(N-\mathtt{k}_{+})^{\alpha_{-}}},$
(B.23)

######  Remark B.7.

The proof of the Lemma B.6 is the unique place where we use
$(l_{j}^{(N)})_{r\leq j\leq N}$, the lower part of the envelope.

####  B.3.1 Dyadic case

We start by assuming $\frac{||\theta-\theta^{\prime}||}{2\pi}=2^{-\mathtt{k}}$
and prove Lemmas B.4 and B.6 in this dyadic case. This part is mainly for
pedagogical purposes while laying the ground for the general case. It
illustrates perfectly the machinery of the proof in a simpler setting.

It will be convenient to denote, for any $l\in[|r,N_{+}|]$,
$p\in[|0,2^{\Delta^{(l)}}-1|]$,

$\displaystyle
I_{l,p}(\theta):=\frac{\sum_{j=0}^{2^{l-\Delta^{(l)}}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\Delta^{(l)}}+j}e^{i\theta
j}}{\sqrt{{2^{l}+p2^{l-\Delta^{(l)}}}}}\operatorname{\stackrel{{\scriptstyle{\mathcal{L}}}}{{=}}}{\mathcal{N}}^{{\mathbb{C}}}\left(0,(2^{\Delta^{(l)}}+p)^{-1}\right).$
(B.24)

Recall that $R_{2^{N}}^{(2^{r},\Delta)}(\theta)$ and
$R_{2^{N}}^{(2^{r},\Delta)}(\theta^{\prime})$ can be written as

$R_{2^{N}}^{(2^{r},\Delta)}(\theta)=\Re\left(\sigma\sum_{l=r}^{N-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}I_{l,p}(\theta)e^{i\psi_{2^{l}+p2^{l-\Delta^{(l)}}}(\theta)}\right),$
$R_{2^{N}}^{(2^{r},\Delta)}(\theta^{\prime})=\Re\left(\sigma\sum_{l=r}^{N-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}I_{l,p}(\theta^{\prime})e^{i\psi_{2^{l}+p2^{l-\Delta^{(l)}}}(\theta^{\prime})}\right).$

The crucial observation is that for any $l\in[|\mathtt{k}_{+},N_{+}|]$ (we
easily check that $\mathtt{k}_{+}<N_{+}$ if $r$ is large enough, $N\geq r$ and
$\mathtt{k}\leq N/2$) and any $p\leq 2^{\Delta^{(l)}}-1$, the random variables
$I_{l,p}(\theta)$ and $I_{l,p}(\theta^{\prime})$ are independent and
identically distributed. Indeed, they form a complex Gaussian vector, and they
are uncorrelated, since for $l\geq\mathtt{k}_{+}$, one has
$l-\Delta^{(l)}\geq\mathtt{k}$ if $l\geq r$ and $r$ is large enough, and then

$\sum_{j=0}^{2^{l-\Delta^{(l)}}}e^{i(\theta-\theta^{\prime})j}=\sum_{j=0}^{2^{l-\Delta^{(l)}}}e^{\pm
2ij\pi/2^{\mathtt{k}}}=0.$

We deduce that the increments of $R_{2^{j}}^{(2^{r},\Delta)}(\theta)$ and
$R_{2^{j}}^{(2^{r},\Delta)}(\theta^{\prime})$ after the time $\mathtt{k}_{+}$
are independent and identically distributed. Recalling the definition of
$\tau^{(r)}$ in (B.9) and (B.24), it follows that

$\displaystyle(R_{2^{j}}^{(2^{r},\Delta)}(\theta))_{j\geq
r}\operatorname{\stackrel{{\scriptstyle{\mathcal{L}}}}{{=}}}(R_{2^{j}}^{(2^{r},\Delta)}(\theta^{\prime}))_{j\geq
r}\operatorname{\stackrel{{\scriptstyle{\mathcal{L}}}}{{=}}}\sqrt{\frac{1}{2}}(W_{\tau_{j}^{(r)}})_{j\geq
r}.$

For any $k\in[|r,N_{+}|]$, $x,z<0$, we introduce the events

$\displaystyle Ev(k,x,z)$ $\displaystyle:=\left\\{\forall
j\in[|k,N_{+}|],\,l_{j}^{(N)}\leq\sqrt{\frac{1}{2}}W_{\tau_{j}^{(k)}}-\tau^{(k)}_{j}+x\leq
u_{j}^{(N)},\right.$
$\displaystyle\qquad\left.\sqrt{\frac{1}{2}}W_{\tau_{N_{+}}^{(k)}}-\tau^{(k)}_{N_{+}}+x\in-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\},$ $\displaystyle GEv(k,x,z)$
$\displaystyle:=\left\\{\forall
j\in[|k,N_{+}|],\,l_{j}^{(N)}\leq\sqrt{\frac{1}{2}}W_{\tau_{j}^{(k)}}+x\leq
u_{j}^{(N)},\right.$
$\displaystyle\qquad\left.\sqrt{\frac{1}{2}}W_{\tau_{N_{+}}^{(k)}}+x\in-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\},$ (B.25)

Note that $GEv(r,x,z)=Event_{r,N}(x,z)$ from Proposition B.2, we only keep the
separate notation for compatibility with [CMN18]. Furthermore note that
$GEv(k,x,z)$ is equal to the event obtained from $Ev(k,x,z)$ after Girsanov
transform with density
$\exp\left(\sqrt{2}W_{\tau^{(k)}_{N_{+}}}-\tau^{(k)}_{N_{+}}\right)$.
Performing the transform yields:

$\displaystyle{\mathbb{P}}\left(Ev(k,x,z)\right)$
$\displaystyle=e^{-\tau_{N_{+}}^{(k)}}{\mathbb{E}}\left(e^{-\sqrt{2}W_{\tau^{(k)}_{N_{+}}}}\mathds{1}_{GEv(k,x,z)}\right)\leq
2^{k-N_{+}}e^{2(x-z+\upsilon)}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(k,x,z)\right),$
(B.26)

where in the last inequality we used the definition (B.9) of
$\tau_{N_{+}}^{(k)}$ and the fact that
$e^{-\sqrt{2}W_{\tau^{(k)}_{N_{+}}}}\leq N^{\frac{3}{2}}e^{2(x-z+\upsilon)}$
on $GEv(k,x,z)$.

In order to allow for more flexibility and for later use, let us record the
following analogous events. For $k\in[|r,N_{+}|]$, $a\geq 0$,
$E=(E_{j})_{j\geq k}$ a sequence of reals such that
$(\tau_{j}^{(k)}-E_{j})_{j\geq k}$ is positive and nondecreasing,
$x,z\in{\mathbb{R}}$, define

$\displaystyle Ev(k,a,E,x.z)$ $\displaystyle:=\left\\{\forall
j\in[|k,N_{+}|],\,l_{j}^{(N)}-a\leq\sqrt{\frac{1}{2}}W_{\tau_{j}^{(k)}-E_{j}}+x-\tau^{(k)}_{j}\leq
u_{j}^{(N)}+a,\right.$
$\displaystyle\quad\left.\sqrt{\frac{1}{2}}W_{\tau_{N_{+}}^{(k)}-E_{N_{+}}}+x-\tau^{(k)}_{N_{+}}\in-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\},$ (B.27) $\displaystyle GEv(k,a,E,x,z)$
$\displaystyle:=\left\\{\forall
j\in[|k,N_{+}|],\,l_{j}^{(N)}-a\leq\sqrt{\frac{1}{2}}W_{\tau_{j}^{(k)}-E_{j}}+x\leq
u_{j}^{(N)}+a,\right.$
$\displaystyle\quad\left.\sqrt{\frac{1}{2}}W_{\tau_{N_{+}}^{(k)}-E_{N_{+}}}+x\in-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\}.$ (B.28)

Again, the event $GEv(k,a,E,x,z)$ is, up to an error due to the time shift
$E$, ”quasi equal” to what we obtain when we apply the Girsanov’ transform
with density

$\exp\left(\sqrt{2}W_{\tau_{N_{+}}^{(k)}-E_{N_{+}}}-(\tau_{N_{+}}^{(k)}-E_{N_{+}})\right)$

to the event $Ev(k,a,E,x,z)$. This time, the inequality takes the form:

$\displaystyle{\mathbb{P}}\left(Ev(k,a,E,x,z)\right)$ $\displaystyle\leq
2^{k-N_{+}}e^{-E_{N_{+}}+2(x-z+a+\upsilon)}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(k,a+\sup_{k\leq
j\leq N}|E_{j}|,E,x,z)\right).$ (B.29)

Indeed, by the Girsanov transform and then using the barrier at time $N_{+}$:

$\displaystyle{\mathbb{P}}\left(Ev(k,a,E,x,z)\right)$
$\displaystyle=e^{-\tau_{N_{+}}^{(k)}+E_{N_{+}}}{\mathbb{E}}\left(e^{-\sqrt{2}W_{\tau^{(k)}_{N_{+}}-E_{N_{+}}}}\mathds{1}_{\left\\{\forall
j\in[|k,N_{+}|],\,l_{j}^{(N)}-a\leq\sqrt{\frac{1}{2}}W_{\tau_{j}^{(k)}-E_{j}}+x-E_{j}\leq
u_{j}^{(N)}+a\right\\}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left.\mathds{1}_{\left\\{\sqrt{\frac{1}{2}}W_{\tau_{N_{+}}^{(k)}-E_{N_{+}}}+x\in-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\}}\right)$ $\displaystyle\leq$ $\displaystyle
2^{k-N_{+}}e^{E_{N_{+}}+2(x-z-
E_{N_{+}}+a+\upsilon)}N^{\frac{3}{2}}{\mathbb{P}}\left(\forall
j\in[|k,N_{+}|],l_{j}^{(N)}\\!-\\!a\leq\sqrt{\frac{1}{2}}W_{\tau_{j}^{(k)}-E_{j}}+x-E_{j}\leq
u_{j}^{(N)}\\!+\\!a,\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\left.\left\\{\sqrt{\frac{1}{2}}W_{\tau_{N_{+}}^{(k)}-E_{N_{+}}}+x\in-\frac{3}{4}\log
N+[z,z+\upsilon)\right\\}\right)$ $\displaystyle\leq$ $\displaystyle
2^{k-N_{+}}e^{-E_{N_{+}}+2(x-z+a+\upsilon)}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(k,a+\sup_{k\leq
j\leq N_{+}}|E_{j}|,E,x,z)\right).$

###### Proof of Lemma B.4 in dyadic case.

When $\mathtt{k}_{+}\leq r$: The increments of
$R_{2^{j}}^{(2^{r},\Delta)}(\theta)$ and
$R_{2^{j}}^{(2^{r},\Delta)}(\theta^{\prime})$ after the time $\mathtt{k}_{+}$
are independent and identically distributed, thus we have

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta)=\quad$
$\displaystyle{\mathbb{P}}\left(Ev(r,x,z)\right){\mathbb{P}}\left(Ev(r,x^{\prime},z^{\prime})\right)$
$\displaystyle\stackrel{{\scriptstyle Eq.\eqref{eq:girsanov1}}}{{\ll_{v}}}$
$\displaystyle
2^{2r-2N_{+}}e^{2(x-z+x^{\prime}-z^{\prime})}\left(N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(r,x,z)\right)\right)\left(N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(r,x^{\prime},z^{\prime})\right)\right).$

Finally by applying [CMN18, (A.15)] (with $E_{j-r}=\lambda_{j}^{(r)}/\log 2$,
which implies $||E||\leq 1$ for $r$ large enough), we obtain (B.22).

When $\mathtt{k}_{+}\geq r$: The increments of
$R_{2^{j}}^{(2^{r},\Delta)}(\theta)$ and
$R_{2^{j}}^{(2^{r},\Delta)}(\theta^{\prime})$ after time $\mathtt{k}_{+}$ are
independent and identically distributed. Moreover, all these increments are
independent of those of $R_{2^{j}}^{(2^{r},\Delta)}(\theta)$ for $j$ between
$r$ and $\mathtt{k}_{+}$ (we see this fact by first conditioning with respect
to the $\sigma$-algebra $\mathcal{G}_{2^{\mathtt{k}_{+}}}$). We then have:

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime}x,z,x^{\prime},z^{\prime})\leq{\mathbb{P}}\left(Ev(r,x,z)\right)\sup_{x^{\prime}+l_{\mathtt{k}_{+}}^{(N)}\leq
w\leq
x^{\prime}+u_{\mathtt{k}_{+}}^{(N)}}{\mathbb{P}}(Ev(\mathtt{k}_{+},w,z^{\prime}))$
$\displaystyle\stackrel{{\scriptstyle Eq.\eqref{eq:girsanov1}}}{{\ll}}$
$\displaystyle
2^{r-N_{+}}e^{2(x-z)}\left(N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(r,x,z)\right)\right)$
$\displaystyle\qquad\qquad\qquad
2^{\mathtt{k}_{+}-N_{+}}\sup_{l_{\mathtt{k}_{+}}^{(N)}\leq w+x^{\prime}\leq
u_{\mathtt{k}_{+}}^{(N)}}\left(N^{\frac{3}{2}}e^{2(w+x^{\prime}-z)}{\mathbb{P}}\left(GEv(\mathtt{k}_{+},w+x^{\prime},z^{\prime})\right)\right)$
$\displaystyle\ll\quad$
$\displaystyle|xzz^{\prime}|2^{r-N_{+}}2^{\mathtt{k}_{+}-N_{+}}e^{2(x-z-z^{\prime})}\sup_{l_{\mathtt{k}_{+}}^{(N)}\leq
w^{\prime}\leq
u_{\mathtt{k}_{+}}^{(N)}}\left(N^{\frac{3}{2}}e^{2(w^{\prime})}{\mathbb{P}}\left(GEv(\mathtt{k}_{+},w^{\prime},z^{\prime})\right)\right).$

If $\mathtt{k}_{+}\leq N/4$, according to [CMN18, (A.15], for any
$w^{\prime}\in{\mathbb{R}}$, we have

$\displaystyle{\mathbb{P}}\left(GEv(\mathtt{k}_{+},w^{\prime},z^{\prime})\right)\ll|w^{\prime}||z^{\prime}|(N-\mathtt{k}_{+})^{-\frac{3}{2}}\
.$

Recalling that $w^{\prime}\leq
u_{\mathtt{k}_{+}}^{(N)}=-(\mathtt{k}_{+})^{\alpha_{-}}$, we have

$\displaystyle\sup_{l_{\mathtt{k}_{+}}^{(N)}\leq w^{\prime}\leq
u_{\mathtt{k}_{+}}^{(N)}}\left(N^{\frac{3}{2}}e^{2(w^{\prime})}{\mathbb{P}}\left(GEv(\mathtt{k}_{+},w^{\prime},z^{\prime})\right)\right)\ll
e^{-\mathtt{k}_{+}^{\alpha_{-}}}.$ (B.30)

On the other hand, for $\mathtt{k}_{+}>N/4$, we can crudely bound
${\mathbb{P}}\left(GEv(\mathtt{k}_{+},w^{\prime},z;)\right)$ by $1$, and using
the fact that $w^{\prime}\leq-\mathtt{k}_{+}^{\alpha_{-}}$ if
$\mathtt{k}_{+}<N/2$ and
$w^{\prime}\leq-(N-\mathtt{k}_{+})^{\alpha_{-}}\leq-0.9\mathtt{k}_{+}^{\alpha_{-}}$
if $\mathtt{k}\leq N/2\leq\mathtt{k}_{+}$, we obtain that

$\displaystyle\sup_{l_{\mathtt{k}_{+}}^{(N)}\leq w^{\prime}\leq
u_{\mathtt{k}_{+}}^{(N)}}\left(N^{\frac{3}{2}}e^{2w^{\prime}}{\mathbb{P}}\left(GEv(\mathtt{k}_{+},w^{\prime},z)\right)\right)\leq
e^{2w^{\prime}}N^{\frac{3}{2}}\ll
e^{-1.8\mathtt{k}_{+}^{\alpha_{-}}}N^{3/2}\ll
e^{-\mathtt{k}_{+}^{\alpha_{-}}},$ (B.31)

since $\mathtt{k}_{+}\geq N/5$ for $N$ large. (Recall that we take
$N\to\infty$ before we take $k_{1}\to\infty$.)

In all cases, by combining Eq. (B.30) and (B.31), we deduce:

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})\ll|xzz^{\prime}|2^{r-N_{+}}2^{\mathtt{k}_{+}-N_{+}}e^{2(x-z-z^{\prime})}e^{-\mathtt{k}_{+}^{\alpha_{-}}}.$

It concludes the proof of Lemma B.4 when
$\frac{||\theta-\theta^{\prime}||}{2\pi}$ is a negative power of $2$. ∎

###### Proof of Lemma B.6 in the dyadic case.

Now we shall study ${\mathbb{P}}_{N}(\theta,\theta^{\prime})$, when the
branching between the field in $\theta$ and the field in $\theta^{\prime}$
appears after the time $N/2$. This time we shall prove that when one restricts
to the paths which are in the envelope, the increments of the path of the
field at $\theta$ and at $\theta^{\prime}$ are approximately independent after
the time of decorrelation $k_{+}$. We recall that for this range
$\mathtt{k}_{+}=\mathtt{k}+3\kappa^{(\mathtt{k})}$ where
$\kappa^{(\mathtt{k})}:=\lfloor\frac{r}{100}\rfloor+100\lfloor\log(N-\mathtt{k})\rfloor^{2}$,
if $\mathtt{k}\leq N-1$.

We need to exhibit the independence between the increments of the processes
$(R_{2^{j}}^{(2^{\mathtt{k}_{+}},\Delta)}(\theta))_{j\geq\mathtt{k}_{+}}$ and
$(R_{2^{j}}^{(2^{\mathtt{k}_{+}},\Delta)}(\theta^{\prime}))_{j\geq\mathtt{k}_{+}}$.
The crucial observation we used in case of $\mathtt{k}\leq N/2$ does not work
anymore for such a short decorrelation time. We first need to modify our field
using similar arguments to those used for the proof of Proposition B.1.

In the following we shall use the quantity

$\displaystyle
J_{l,p}^{(\mathtt{k})}(\theta):=\frac{\sum_{j=0}^{2^{l-\kappa^{(\mathtt{k})}}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+j}e^{i\theta
j}}{\sqrt{{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}}}\operatorname{\stackrel{{\scriptstyle{\mathcal{L}}}}{{=}}}{\mathcal{N}}^{{\mathbb{C}}}\left((0,2^{\kappa^{(\mathtt{k})}}+p)^{-1}\right)$

Let $\mathtt{k}\in[N/2,N_{+}]$ and
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}<2^{-(\mathtt{k}-1)}$.
We have, for $r$ large enough, since $N-\mathtt{k}\geq r/2$ is large,

$\mathtt{k}_{+}\leq\mathtt{k}+(r/33)+300\log^{2}(N-\mathtt{k})\leq\mathtt{k}+(N-\mathtt{k})/16<N.$

Recall that for $\tilde{\theta}\in\\{\theta,\theta^{\prime}\\}$, $r\leq k\leq
N$, $R_{2^{k}}^{(2^{r},\Delta)}(\tilde{\theta})=\Re(\sigma
Z_{2^{k}}^{(2^{r},\Delta)}(\tilde{\theta}))$. Since for $l\geq\mathtt{k}_{+}$,
$\kappa^{(\mathtt{k})}\leq\Delta^{(l)}$, we can write for all
$\mathtt{k}_{+}\leq k\leq N$, $\tilde{\theta}\in\\{\theta,\theta^{\prime}\\}$:

$\displaystyle Z_{2^{k}}^{(2^{\mathtt{k}_{+}},\Delta)}(\tilde{\theta})$
$\displaystyle=\sum_{l=\mathtt{k}_{+}}^{k-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}I_{l,p}(\tilde{\theta})e^{i\psi_{2^{l}+p2^{l-\Delta^{(l)}}}(\tilde{\theta})}$
$\displaystyle=\sum_{l=\mathtt{k}_{+}}^{k-1}\sum_{p=0}^{2^{\Delta^{(l)}}-1}\frac{\sum_{j=0}^{2^{l-\Delta^{(l)}}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\Delta^{(l)}}+j}e^{i\tilde{\theta}j}}{\sqrt{{2^{l}+p2^{l-\Delta^{(l)}}}}}e^{i\psi_{2^{l}+p2^{l-\Delta^{(l)}}}(\tilde{\theta})}$
$\displaystyle=$
$\displaystyle\sum_{l=\mathtt{k}_{+}}^{k-1}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}\left(\sum_{j=0}^{2^{l-\kappa^{(\mathtt{k})}}-1}\frac{{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+j}e^{i\tilde{\theta}\left\\{j-\mathfrak{j}\right\\}}}{\sqrt{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+\mathfrak{j}}}e^{i\psi_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+\mathfrak{j}}(\tilde{\theta})}\right),\text{(with
$\mathfrak{j}=\lfloor\frac{j}{2^{l-\Delta^{(l)}}}\rfloor
2^{l-\Delta^{(l)}}$)}$
$\displaystyle=Z_{2^{k}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\tilde{\theta})+\mathfrak{E}_{2^{k}}(\tilde{\theta})$

where

$\displaystyle
Z_{2^{k}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\tilde{\theta})$
$\displaystyle:=\sum_{l=\mathtt{k}_{+}}^{k-1}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}J_{l,p}^{(\mathtt{k})}(\tilde{\theta})e^{i\psi_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}(\tilde{\theta})},$
$\displaystyle\mathfrak{E}_{2^{k}}(\tilde{\theta})$
$\displaystyle:=\sum_{l=\mathtt{k}_{+}}^{k-1}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}\frac{e^{i\psi_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}(\tilde{\theta})}}{\sqrt{{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}}}\left(\sum_{j=0}^{2^{l-\kappa^{(\mathtt{k})}}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+j}e^{i\tilde{\theta}j}\lozenge_{j}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta})\right),$

with

$\displaystyle\left|\lozenge_{j}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta})\right|\leq$
$\displaystyle\left|A_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+\mathfrak{j}}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta}))\right|+2^{-\kappa^{(\mathtt{k})}}.$
(B.32)

Indeed:

$\displaystyle\lozenge_{j}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta})=$
$\displaystyle-1+\sqrt{\frac{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+\mathfrak{j}}}e^{iA_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+\mathfrak{j}}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta})}$
$\displaystyle=$
$\displaystyle-1+\left(1+\Theta\right)e^{iA_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+\mathfrak{j}}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta})},$

with $|\Theta|\leq 2^{-\kappa^{(\mathtt{k})}}$ which implies inequality
(B.32). In the following, we shall denote:

$\displaystyle\blacklozenge^{(\mathtt{k})}_{l}(\tilde{\theta}):=\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}\frac{e^{i\psi_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}(\tilde{\theta})}}{\sqrt{{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}}}\left(\sum_{j=0}^{2^{l-\kappa^{(\mathtt{k})}}-1}{\mathcal{N}}^{\mathbb{C}}_{2^{l}+p2^{l-\kappa^{(\mathtt{k})}}+j}e^{i\tilde{\theta}j}\lozenge_{j}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta})\right).$

Notice that, on the contrary of the proof of Proposition B.1, where
$\Delta^{(l)}$ varies with $l$, here we fix $\kappa^{(\mathtt{k})}$ as soon as
we know that
$2^{-\mathtt{k}}\leq\frac{||\theta-\theta^{\prime}||}{2\pi}<2^{-(\mathtt{k}-1)}$.
By using the same arguments used to prove (B.7) and (B.8), one can show
similarly that for any $l\geq\mathtt{k}_{+}$,
$\tilde{\theta}\in\\{\theta,\theta^{\prime}\\}$,

$\displaystyle{\mathbb{P}}\left(|\blacklozenge^{(\mathtt{k})}_{l}(\tilde{\theta})|\geq
2^{-\frac{\kappa^{(\mathtt{k})}}{8}},\cap_{l\geq\mathtt{k}_{+}}\widetilde{G}_{l}(\tilde{\theta})\Big{|}{\mathcal{G}}_{2^{\mathtt{k}_{+}}}\right)\ll
e^{-2^{\frac{\kappa^{(\mathtt{k})}}{8}}},$ (B.33)
$\displaystyle\qquad{\mathbb{P}}\left((\cap_{l\geq\mathtt{k}_{+}}\widetilde{G}_{l}(\tilde{\theta}))^{c}\Big{|}{\mathcal{G}}_{2^{\mathtt{k}_{+}}}\right)\ll_{\beta}\exp\left(-\frac{\beta}{33}2^{\frac{1}{2}\kappa^{(\mathtt{k})}}\right).$
(B.34)

with

$\displaystyle\widetilde{G}_{l}(\tilde{\theta}):=$ $\displaystyle\
\bigcap_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}\left\\{\sup_{0\leq j\leq
2^{l-{\kappa^{(\mathtt{k})}}}-1}\left|A_{j+2^{l}+p2^{l-\kappa^{(\mathtt{k})}}}^{(2^{l}+p2^{l-\kappa^{(\mathtt{k})}})}(\tilde{\theta})\right|\leq
2^{-\frac{1}{4}\kappa^{(\mathtt{k})}}\right\\}\ .$

Moreover it is plain to observe that for any $r$ large enough, $N$ large
enough depending on $r$, and $\mathtt{k}\in[|N/2,N-r/2|]$, and under the
complement of the two events just above,

$\displaystyle\sum_{l=\mathtt{k}_{+}}^{N-1}|\blacklozenge^{(\mathtt{k})}_{l}(\tilde{\theta})|\leq\sum_{l=\mathtt{k}_{+}}^{N-1}2^{-\frac{\kappa^{(\mathtt{k})}}{8}}\leq(N-\mathtt{k}_{+})2^{-\frac{1}{8}\lfloor\frac{r}{100}\rfloor-\frac{25}{2}\lfloor\log(N-\mathtt{k})\rfloor^{2}}\leq
1.$ (B.35)

So for any $\tilde{\theta}\in\\{\theta,\theta^{\prime}\\}$, we can replace
$(R_{2^{k}}^{(2^{\mathtt{k}_{+}},\Delta)}(\tilde{\theta}))_{\mathtt{k}_{+}\leq
k\leq N}$ by
$(R_{2^{k}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\tilde{\theta}))_{\mathtt{k}_{+}\leq
k\leq N}$ with an error at most $1$. Thus we have

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})$
$\displaystyle\leq{\mathbb{P}}\left(\forall
j\in[|r,N_{+}|],\,L_{j}^{(N)}-1\leq
x+R_{2^{j\wedge\mathtt{k}_{+}}}^{(2^{r},\Delta)}(\theta)+R_{2^{j}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta)\mathds{1}_{{\\{j>k_{0}\\}}}\leq
U_{j}^{(N)}+1,\,\right.$ $\displaystyle\left.\qquad\qquad\qquad\forall
j\in[|\mathtt{k}_{+},N_{+}|],\,L_{j}^{(N)}-1\leq
x^{\prime}+R_{2^{\mathtt{k}_{+}}}^{(2^{r},\Delta)}(\theta^{\prime})+R_{2^{j}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta^{\prime})\leq
U_{j}^{(N)}+1,\right.$
$\displaystyle\qquad\qquad\qquad\left.x+R_{2^{\mathtt{k}_{+}}}^{(2^{r},\Delta)}(\theta)+R_{2^{j}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta)\in-\frac{3}{4}\log
N+[z,z+v],\right.$
$\displaystyle\qquad\qquad\qquad\left.x^{\prime}+R_{2^{\mathtt{k}_{+}}}^{(2^{r},\Delta)}(\theta^{\prime})+R_{2^{j}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta^{\prime})\in-\frac{3}{4}\log
N+[z,z+v]\right)$ $\displaystyle+$
$\displaystyle\sum_{\tilde{\theta}\in\\{\theta,\theta^{\prime}\\}}{\mathbb{E}}\Bigg{(}\mathds{1}_{\\{\forall
j\in[|r,\mathtt{k}_{+}|],\,L_{j}^{(N)}\leq\tilde{x}+R_{2^{j}}^{(2^{r},\Delta)}(\tilde{\theta})\leq
U_{j}^{(N)}\\}}\Big{(}\sum_{l=\mathtt{k}_{+}}^{N_{+}}{\mathbb{P}}\left(|\blacklozenge^{(\mathtt{k})}_{l}(\tilde{\theta})|\geq
2^{-\frac{\kappa^{(\mathtt{k})}}{8}},\cap_{l\geq\mathtt{k}_{+}}\tilde{G}_{l}(\tilde{\theta})\Big{|}{\mathcal{G}}_{2^{\mathtt{k}_{+}}}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+{\mathbb{P}}\left((\cap_{l\geq\mathtt{k}_{+}}\tilde{G}_{l}(\tilde{\theta}))^{c}\Big{|}{\mathcal{G}}_{2^{\mathtt{k}_{+}}}\right)\Big{)}\Bigg{)},$
$\displaystyle=:{\mathbb{P}}_{N}^{(\Delta,\kappa)}(\theta,\theta^{\prime})+Q_{N}(\theta,\theta^{\prime}),$

where $\tilde{x}=x^{\prime}$ if $\tilde{\theta}=\theta^{\prime}$ and
$\tilde{x}=x$ otherwise.

We first deal with the sum in $\tilde{\theta}\in\\{\theta,\theta^{\prime}\\}$,
that is with $Q_{N}(\theta,\theta^{\prime})$. By using (B.33), then the
Girsanov transfom with density
$e^{\sqrt{2}W_{\tau^{(r)}_{\mathtt{k}_{+}}}-\tau_{\mathtt{k}_{+}}^{(r)}}$, and
[CMN18, Corollary A.6] (when $\mathtt{k}_{+}\geq\frac{2N}{3}$), the sum is

$\displaystyle\ll
2^{r-\mathtt{k}_{+}}N^{\frac{3}{2}}e^{2(N-\mathtt{k}_{+})^{\alpha_{+}}}\left((N-\mathtt{k}_{+})e^{-2^{\frac{\kappa^{(\mathtt{k})}}{8}}}+\exp\left(-\frac{\beta}{33}2^{\frac{1}{2}\kappa^{(\mathtt{k})}}\right)\right),\;\text{when
}\,\mathtt{k}_{+}\leq\frac{2N}{3},$ $\displaystyle\ll
2^{r-\mathtt{k}_{+}}\frac{N^{\frac{3}{2}}(N-\mathtt{k}_{+})^{2\alpha_{+}}}{(\mathtt{k}_{+}-r)^{\frac{3}{2}}}e^{2(N-\mathtt{k}_{+})^{\alpha_{+}}}\left((N-\mathtt{k}_{+})e^{-2^{\frac{\kappa^{(\mathtt{k})}}{8}}}+\exp\left(-\frac{\beta}{33}2^{\frac{1}{2}\kappa^{(\mathtt{k})}}\right)\right),\;\text{when
}\,\mathtt{k}_{+}\geq\frac{2N}{3}$

which are both dominated by
$|xz|e^{-(x-z-z^{\prime})}2^{-2N_{+}+\mathtt{k}_{+}+r}e^{-\frac{1}{2}(N-\mathtt{k}_{+})^{\alpha_{-}}}$.
It remains to bound the expression
${\mathbb{P}}_{N}^{(\Delta,\kappa)}(\theta,\theta^{\prime})$. Let
$\tau_{j}^{(r,\mathtt{k}_{+})}:=\tau_{j}^{(r)}$ if $j\leq\mathtt{k}_{+}$ and
$\tau_{j}^{(r,\mathtt{k}_{+})}:=\tau_{\mathtt{k}_{+}}^{(r)}+\sum_{l=\mathtt{k}_{+}}^{j-1}\sum_{p=0}^{2^{\kappa^{(\mathtt{k})}}-1}(2^{\kappa^{(\mathtt{k})}}+p)^{-1}$
if $j\geq\mathtt{k}_{+}$ and
$E^{(\mathtt{k}_{+})}_{j}:=\tau_{j}^{(r)}-\tau_{j}^{(r,\mathtt{k}_{+})}$, for
any $r\leq j\leq N_{+}$ . It is plain to check that
$\sum_{j=r}^{N_{+}}|E^{(\mathtt{k}_{+})}_{j+1}-E^{(\mathtt{k}_{+})}_{j}|\ll
2^{-\frac{r}{100}}$ and

$\displaystyle(R_{2^{j\wedge\mathtt{k}_{+}}}^{(2^{r},\Delta)}(\theta^{\prime})+R_{2^{j}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta^{\prime})\mathds{1}_{{\\{j>k_{0}\\}}})_{r\leq
j\leq
N}\operatorname{\stackrel{{\scriptstyle{\mathcal{L}}}}{{=}}}\sqrt{\frac{1}{2}}(W_{\tau_{j}^{(r,\mathtt{k}_{+})}})_{j\geq
r}\ .$

Now it suffices to reproduce the proof of Lemma B.4. In this first part, we
assume $\frac{||\theta-\theta^{\prime}||}{2\pi}=2^{-\mathtt{k}}$. In this
case, we check the independence of $J_{l,p}^{(\mathtt{k})}(\theta)$ and
$J_{l,p}^{(\mathtt{k})}(\theta^{\prime})$ for $l\geq\mathtt{k}_{+}$, since
$\mathtt{k}_{+}-\kappa^{(\mathtt{k})}\geq\mathtt{k}$. We then show, by doing
the suitable conditionings, that the increments of
$(R_{2^{k}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta))_{\mathtt{k}_{+}\leq
k\leq N_{+}}$,
$(R_{2^{k}}^{(2^{\mathtt{k}_{+}},\kappa^{(\mathtt{k})})}(\theta^{\prime}))_{\mathtt{k}_{+}\leq
k\leq N_{+}}$, $(R_{2^{k}}^{(2^{r},\Delta)}(\theta^{\prime}))_{r\leq
k\leq\mathtt{k}_{+}}$ are independent. Thus we have

$\displaystyle{\mathbb{P}}_{N}^{(\Delta,\kappa)}(\theta,\theta^{\prime})\leq{\mathbb{P}}\left(Ev(r,1,E^{(\mathtt{k}_{+})},x,z)\right)\max_{l_{\mathtt{k}_{+}}^{(N)}-1\leq
w\leq
u_{\mathtt{k}_{+}}^{(N)}+1}{\mathbb{P}}(Ev(\mathtt{k}_{+},1,E^{(\mathtt{k}_{+})},w,z^{\prime})).$

By the same arguments as in the proof of Lemma B.4 in the dyadic case we have

$\displaystyle\max_{l_{\mathtt{k}_{+}}^{(N)}-1\leq w\leq
u_{\mathtt{k}_{+}}^{(N)}+1}{\mathbb{P}}(Ev(\mathtt{k}_{+},1,E^{(\mathtt{k}_{+})},w,z^{\prime}))\ll_{\upsilon}$
$\displaystyle|z^{\prime}|2^{-(N_{+}-\mathtt{k}_{+})}\max_{l_{\mathtt{k}_{+}}^{(N)}-1\leq
w\leq u_{\mathtt{k}_{+}}^{(N)}+1}|e^{2(w-z^{\prime})}$
$\displaystyle\leq|z^{\prime}|2^{-(N_{+}-\mathtt{k}_{+})}e^{-2z^{\prime}-(N-\mathtt{k}_{+})^{\alpha}_{-}},$

where we used that in the stated range,
$|w^{\prime}|\geq(N-\mathtt{k}_{+})^{\alpha}_{-}/2$, and

${\mathbb{P}}\left(Ev(r,1,E^{(\mathtt{k}_{+})},x,z)\right)\ll|xz|2^{-N_{+}+r}.$

Finally one gets

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})\ll|xzz^{\prime}|2^{r-N_{+}}2^{\mathtt{k}_{+}-N_{+}}e^{-(N-\mathtt{k}_{+})^{\alpha_{-}}}e^{2(x-z-z^{\prime})}\
.$

This concludes the proof of Lemma B.6, when
$\frac{||\theta-\theta^{\prime}||}{2\pi}$ is a negative power of $2$. ∎

####  B.3.2 General case

###### Proof of Lemma B.4 in general case.

Fix $\upsilon>0$, $r$ large enough, $N$ large enough depending on $r$, and
$\mathtt{k}$ such that $\mathtt{k}\leq\frac{N}{2}$. Unlike the previous dyadic
case, for $\mathtt{k}_{+}\leq l\leq N-1$ and $p\leq 2^{\Delta^{(l)}}-1$, the
random variables $I_{l,p}(\theta)$ and $I_{l,p}(\theta^{\prime})$ are not
rigorously independent. However observe that for any $\mathtt{k}_{+}\leq l\leq
N-1$, the absolute value of their correlations decreases exponentially with
$l$. Indeed, if
$C_{l,p}(\theta,\theta^{\prime}):={\mathbb{E}}\left(I_{l,p}(\theta)\overline{I_{l,p}(\theta^{\prime})}\right)$,
then

$\displaystyle\left|C_{l,p}(\theta,\theta^{\prime})\right|$
$\displaystyle=\frac{1}{{2^{l}+p2^{l-\Delta^{(l)}}}}\left|\sum_{j=0}^{2^{l-\Delta^{(l)}}-1}e^{i(\theta-\theta^{\prime})j}\right|\leq\frac{4}{2^{l}||\theta-\theta^{\prime}||}\ll
2^{\mathtt{k}-l}.$

Since ${\mathbb{E}}\left(I_{l,p}(\theta)I_{l,p}(\theta^{\prime})\right)=0$,
and $(I_{l,p}(\theta),I_{l,p}(\theta^{\prime}))$ is a centered complex
Gaussian vector, one checks, by computing covariances, that it is possible to
write

$I_{l,p}(\theta)=\frac{C_{l,p}(\theta,\theta^{\prime})}{C_{l,p}(\theta^{\prime},\theta^{\prime})}I_{l,p}(\theta^{\prime})+I^{ind}_{l,p}(\theta,\theta^{\prime}),$

where the two terms of the sums are independent, with an expectation of the
square equal to zero. Note that

$C_{l,p}:=C_{l,p}(\theta^{\prime},\theta^{\prime})=\frac{2^{l-\Delta^{(l)}}}{2^{l}+p2^{l-\Delta^{(l)}}}=(2^{\Delta^{(l)}}+p)^{-1}$

does not depend on $\theta^{\prime}$. Moreover, we have by Pythagoras’
theorem:

${\mathbb{E}}[|I_{l,p}^{ind}(\theta,\theta^{\prime})|^{2}]={\mathbb{E}}[|I_{l,p}(\theta)|^{2}]-\left|\frac{C_{l,p}(\theta,\theta^{\prime})}{C_{l,p}}\right|^{2}{\mathbb{E}}[|I_{l,p}(\theta^{\prime})|^{2}]=C_{l,p}-\frac{|C_{l,p}(\theta,\theta^{\prime})|^{2}}{C_{l,p}}$

Using this decomposition of $I_{l,p}(\theta)$ and the measurability of the
different quantities with respect to the $\sigma$-algebras of the form
$\mathcal{G}_{j}$, we deduce that one can write:

$\displaystyle(R_{2^{l}}^{(2^{\mathtt{k}_{+}},\Delta)}(\theta))_{l\geq\mathtt{k}_{+}}=(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)}+E_{2^{l}}^{(2^{\mathtt{k}_{+}})})_{l\geq\mathtt{k}_{+}}\
.$ (B.36)

Here $(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)})_{l\geq\mathtt{k}_{+}}$ is a
Gaussian process, independent of
$(R_{2^{l}}^{(2^{\mathtt{k}_{+}},\Delta)}(\theta^{\prime}))_{l\geq\mathtt{k}_{+}}$,
and distributed as
$(\sqrt{\frac{1}{2}}W_{\tau^{(\mathtt{k}_{+})}_{l}-\mathtt{C}_{l}})_{l\geq\mathtt{k}_{+}}$
with

$\mathtt{C}_{l}:=\sum_{t=\mathtt{k}_{+}}^{l-1}\sum_{p=0}^{2^{\Delta^{(t)}}-1}\frac{|C_{t,p}(\theta,\theta^{\prime})|^{2}}{C_{t,p}}\
.$

Notice that $\mathtt{C}=\mathtt{C}^{(\mathtt{k}_{+})}$ implicitly depends of
$\mathtt{k}_{+}$. $(E_{2^{l}}^{(2^{r})})_{l\geq\mathtt{k}_{+}}$ on the other
hand is defined by

$\displaystyle
E_{2^{l+1}}^{(2^{l})}=\Re\left(\sigma\sum_{p=0}^{2^{\Delta^{(l)}}-1}e^{i\psi_{2^{l}+p2^{l-\Delta^{(l)}}}(\theta)}\frac{C_{l,p}(\theta,\theta^{\prime})}{C_{l,p}}I_{l,p}(\theta^{\prime})\right).$
(B.37)

Furthermore notice that

##### Fact 1:

For any $l\geq\mathtt{k}_{+}$,

$\displaystyle|E_{2^{l+1}}^{(2^{l})}|\leq|E|_{2^{l+1}}^{(2^{l})}:=\sum_{p=0}^{2^{\Delta^{(l)}}-1}|C_{l,p}(\theta,\theta^{\prime})|(2^{\Delta^{(l)}}+p)|I_{l,p}(\theta^{\prime})|$

is measurable with respect to the sigma field
$\sigma\left({\mathcal{N}}_{t}^{\mathbb{C}},\,t\in[|2^{l},2^{l+1}-1|]\right)$.

##### Fact 2:

The process $(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)})_{l\geq\mathtt{k}_{+}}$ is
independent of the couple
$(R_{2^{l}}^{(2^{\mathtt{k}_{+}},\Delta)}(\theta^{\prime}),|E|_{2^{l}}^{(2^{r})})_{l\geq\mathtt{k}_{+}}$

##### Fact 3:

$\sup_{l\geq\mathtt{k}_{+}}|\mathtt{C}_{l}|\ll 2^{-\Delta^{(\mathtt{k})}}$ if
$r$ is large enough. Indeed, in this case, since $\mathtt{k}_{+}\geq
3\Delta^{(\mathtt{k})}\geq 3e^{\sqrt{\log r}}$ is also large, we have

$\displaystyle\sup_{l\geq\mathtt{k}_{+}}|\mathtt{C}_{l}|$
$\displaystyle\ll\sum_{l=\mathtt{k}_{+}}^{\infty}\sum_{p=0}^{2^{\Delta^{(l)}}-1}(2^{\mathtt{k}-l})^{2}(2^{\Delta^{(l)}}+p)\ll\sum_{l=\mathtt{k}_{+}}^{\infty}2^{2\mathtt{k}-2l+2\Delta^{(l)}}\ll\sum_{l=\mathtt{k}_{+}}^{\infty}2^{2\mathtt{k}-2l+2e^{\sqrt{\log
r}}+200\log^{2}l}$ $\displaystyle\ll\sum_{\mathtt{k}_{+}\leq l\leq
100\mathtt{k}}2^{2\mathtt{k}-2l+2e^{\sqrt{\log
r}}+300\log^{2}\mathtt{k}}+\sum_{l\geq\max(100\mathtt{k},\mathtt{k}_{+})}2^{2\mathtt{k}-2l+0.8\mathtt{k}_{+}+200\log^{2}l}$
$\displaystyle\ll\sum_{\mathtt{k}_{+}\leq l\leq
100\mathtt{k}}2^{2\mathtt{k}-2l+3\Delta^{(\mathtt{k})}}+\sum_{l\geq\max(100\mathtt{k},\mathtt{k}_{+})}2^{2\mathtt{k}+0.8\mathtt{k}_{+}-1.99l}$
$\displaystyle\ll
2^{2\mathtt{k}-2\mathtt{k}_{+}+3\Delta^{(\mathtt{k})}}+2^{2\mathtt{k}+0.8\mathtt{k}_{+}-1.99[(0.05)(100\mathtt{k})+0.95(\mathtt{k}_{+})]}\ll
2^{-3\Delta^{(\mathtt{k})}}+2^{-\mathtt{k}_{+}}\ll
2^{-3\Delta^{(\mathtt{k})}}.$

It means (see [CMN18, Lemma A.5]) that the process
$(R_{2^{l}}^{(2^{\mathtt{k}_{+}},ind)})_{l\geq\mathtt{k}_{+}}$ is very
”similar” to the process
$\sqrt{\frac{1}{2}}(W_{\tau_{l}^{(\mathtt{k}_{+})}})_{l\geq\mathtt{k}_{+}}$.
Moreover, if for $\mathtt{k}_{+}\leq r\leq l$,
$\mathtt{C}_{l}^{(r)}:=\mathtt{C}_{l}-\mathtt{C}_{r}$, then we have for $r$
large enough:

$\sup_{l\geq
r}|\mathtt{C}_{l}^{(r)}|\leq\sup_{l\geq\mathtt{k}_{+}}|\mathtt{C}_{l}|\leq
2^{-\Delta^{(\mathtt{k})}}\leq 2^{-e^{\sqrt{\log r}}}\leq 1/2.$

##### Fact 4:

$|E|$ is small. For any $m\geq 0$, $l\geq\mathtt{k}_{+}$, we introduce the
event

$\mathtt{E}_{m}^{(l)}:=\\{|E|_{2^{l+1}}^{(2^{l})}\geq
2^{-\frac{\Delta^{(l)}}{4}}m\\}\ .$

For some universal constants $c,c^{\prime}>0$, and $m\geq 1/2$, the
probability of $\mathtt{E}_{m}^{(l)}$ is smaller than

$\displaystyle{\mathbb{P}}\left(|E|_{2^{l+1}}^{(2^{l})}\geq
2^{-\frac{\Delta^{(l)}}{4}}m\right)\leq
2^{\Delta^{(l)}}{\mathbb{P}}\left(c2^{\mathtt{k}-l}|I_{l,0}(\theta^{\prime})|\geq
m2^{-\frac{9}{4}\Delta^{(l)}}\right)$ $\displaystyle\ll
2^{\Delta^{(l)}}e^{-c^{\prime}m^{2}2^{2(l-\mathtt{k}-\frac{7}{4}\Delta^{(l)})}}$
$\displaystyle\ll e^{-c^{\prime}m^{2}2^{2(l-\mathtt{k}-2\Delta^{(l)})}}.$
(B.38)

For $r$ (and then $\mathtt{k}_{+}$ and $l$) large enough and $l\leq
100\mathtt{k}$, $2\Delta^{(l)}\leq
3\Delta^{(\mathtt{k})}+(\log(c^{\prime})/\log 4)$, and then

${\mathbb{P}}\left(|E|_{2^{l+1}}^{(2^{l})}\geq
2^{-\frac{\Delta^{(l)}}{4}}m\right)\ll e^{-m^{2}2^{2(l-\mathtt{k}_{+})}}.$

If $r$ is large enough and $l\geq\sup(100\mathtt{k},\mathtt{k}_{+})$ (and
therefore, also large), we use that

$\displaystyle 2(l-\mathtt{k}-2\Delta^{(l)})$ $\displaystyle=$ $\displaystyle
l-\mathtt{k}_{+}+(l-2\mathtt{k}+\mathtt{k}_{+}-4\Delta^{(l)})=l-\mathtt{k}_{+}+(l-k+3\Delta^{(\mathtt{k})}-4\Delta^{(l)})$
$\displaystyle\geq$ $\displaystyle l-\mathtt{k}+(l-\mathtt{k}-400(\log
l)^{2})\geq l-\mathtt{k}+(0.99l-400(\log l)^{2})$ $\displaystyle\geq$
$\displaystyle l-\mathtt{k}-\log c^{\prime},$

and then, using (B.38),

${\mathbb{P}}\left(|E|_{2^{l+1}}^{(2^{l})}\geq
2^{-\frac{\Delta^{(l)}}{4}}m\right)\ll
e^{-c^{\prime}m^{2}2^{2(l-\mathtt{k}-2\Delta^{(l)})}}\leq
e^{-m^{2}(l-\mathtt{k}_{+})}.$

Hence in any case, for $r$ large and $l\geq\mathtt{k}_{+}$,

${\mathbb{P}}\left(\mathtt{E}_{m}^{(l)}\right)={\mathbb{P}}\left(|E|_{2^{l+1}}^{(2^{l})}\geq
2^{-\frac{\Delta^{(l)}}{4}}m\right)\ll e^{-m^{2}2^{l-\mathtt{k}_{+}}}.$

When $\mathtt{k}_{+}\leq r$: Using the decomposition (B.36) and Fact 2, and
noticing that $\sum_{l=r}^{+\infty}m2^{-\frac{\Delta^{(l)}}{4}}\leq
m2^{-e^{\sqrt{\log r}}}$, for $r$ large enough, we can affirm that

$\displaystyle{\mathbb{P}}_{N}(\theta,\theta^{\prime})\leq{\mathbb{P}}\left(Ev(r,0,0,x,z)\right){\mathbb{P}}\left(Ev(r,2^{-e^{\sqrt{\log
r}}},\mathtt{C}^{(r)},x^{\prime},z^{\prime})\right)$
$\displaystyle\qquad+\sum_{m\geq
1}{\mathbb{P}}\left(Ev(r,0,0,x,z),\,\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{m}^{(j)}\right){\mathbb{P}}\left(Ev(r,(m+1)2^{-e^{\sqrt{\log
r}}},\mathtt{C}^{(r)},x^{\prime},z^{\prime})\right),$ (B.39)

where the Brownian motion involved in the event $Ev(r,0,0,x,z)$ is suitably
coupled with the complex Gaussian random walk whose increments are of the form
$I_{l,p}(\theta^{\prime})e^{i\psi_{2^{l}+p2^{l-\Delta^{(l)}}}(\theta^{\prime})}$
for $r\leq l\leq N-1$ and $0\leq p\leq 2^{\Delta^{(l)}}-1$.

By using Eq. (B.29) and then the fact that $\sup_{r\leq j\leq
N}\mathtt{C}^{(r)}_{j}\leq 2^{1-e^{\sqrt{\log r}}}$ if $r$ is large enough, we
have for any $m\geq 0$:

$\displaystyle{\mathbb{P}}\left(Ev(r,(m+1)2^{-e^{\sqrt{\log
r}}},\mathtt{C}^{(r)},x^{\prime},z^{\prime})\right)$ $\displaystyle\leq
e^{-2(x^{\prime}-z^{\prime})}2^{r-N_{+}}e^{2\upsilon+2(m+1)2^{-e^{\sqrt{\log
r}}}-\mathtt{C}^{(r)}_{N}}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(r,(m+1)2^{-e^{\sqrt{\log
r}}}+\sup_{r\leq j\leq
N}\mathtt{C}^{(r)}_{j},\mathtt{C}^{(r)},x^{\prime},z^{\prime})\right)$
$\displaystyle\leq
e^{-2(x^{\prime}-z^{\prime})}2^{r-N_{+}}e^{2\upsilon+2(m+1)2^{-e^{\sqrt{\log
r}}}-\mathtt{C}^{(r)}_{N}}N^{\frac{3}{2}}{\mathbb{P}}\left(GEv(r,2(m+2)2^{-e^{\sqrt{\log
r}}},\mathtt{C}^{(r)},x^{\prime},z^{\prime})\right)\ .$ (B.40)

Now, we invoke [CMN18, Corollary A.6] as, by the Fact 3, with the notation of
the corollary, $||E||_{1}\leq 1$ if $r$ is large enough. Thus, we deduce, for
$N$ large enough depending on $r$, that

$\displaystyle{\mathbb{P}}\left(Ev(r,(m+1)2^{-e^{\sqrt{\log
r}}},\mathtt{C}^{(r)},x^{\prime},z^{\prime})\right)\ll_{\upsilon}|x^{\prime}z^{\prime}|e^{2(x^{\prime}-z^{\prime})}2^{r-N}e^{2m2^{-\frac{r}{400}}}(1+(m+1)2^{-e^{\sqrt{\log
r}}})^{3}\ .$ (B.41)

Similarly, to compute
${\mathbb{P}}\left(Ev(r,0,0,x,z),\,\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{m}^{(j)}\right)$
we will apply the Girsanov transform with the density
$e^{\sqrt{2}W_{\tau_{N_{+}}^{(r)}}-\tau_{N_{+}}^{(r)}}$. It requires to study
what is the effect of this density on the event
$\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{m}^{(j)}$. The increments of the complex
random walk which were
$I_{j,p}(\theta^{\prime})e^{i\psi_{2^{j}+p2^{j-\Delta^{(j)}}}(\theta^{\prime})}$
before the Girsanov transform, increase by $\sigma^{-1}C_{j,p}$ afterwards.
Hence, between the two situations, before and after the Girsanov transform,
$|E|_{2^{j+1}}^{(2^{j})}$, defined as the sum, for $0\leq p\leq
2^{\Delta^{(j)}}-1$, of the absolute value of the increments of the random
walk multiplied by $|C_{j,p}(\theta,\theta^{\prime})|/C_{j,p}$, vary, for $r$
large enough, at most by $2^{2\Delta^{(j)}+\mathtt{k}-j}$, since

$\sum_{p=0}^{2^{\Delta^{(j)}}-1}|C_{j,p}(\theta,\theta^{\prime})|\ll
2^{\Delta^{(j)}+\mathtt{k}-j}.$

Now, for $j\geq r\geq\mathtt{k}_{+}$ and $r$ large enough, we have:

$2^{2\Delta^{(j)}+\mathtt{k}-j}<\frac{1}{2}2^{-\frac{1}{4}\Delta^{(j)}}.$

Indeed, for $j\leq 100\mathtt{k}$,

$2^{2\Delta^{(j)}+\mathtt{k}-j}\leq
2^{2\Delta^{(j)}+\mathtt{k}-\mathtt{k}_{+}}\leq
2^{2\Delta^{(j)}-3\Delta^{(\mathtt{k})}}\leq 2^{-0.9\Delta^{(j)}},$

and for $j\geq\max(\mathtt{k}_{+},100\mathtt{k})$ (necessarily large, since
$r$ is large),

$\displaystyle 2^{2\Delta^{(j)}+\mathtt{k}-j}$ $\displaystyle\leq
2^{2\Delta^{(j)}-0.99j}\leq 2^{e^{\sqrt{\log r}}+200\log^{2}j-0.99j}\leq
2^{e^{\sqrt{\log j}}+200\log^{2}j-0.99j}$ $\displaystyle\leq 2^{-0.96j}\leq
2^{-e^{\sqrt{\log r}}-0.94j}\leq 2^{-\Delta^{(j)}}.$

Hence, if for $m\geq 1$, before (respectively after) the Girsanov transform,
$\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{m}^{(j)}$ occurs, then
$\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{\frac{1}{2}m}^{(j)}$ still occurs after
(respectively before) the transform. Finally we get, for any $m\geq 1$,

$\displaystyle{\mathbb{P}}\left(Ev(r,0,0,x,z),\,\cup_{j\in[|r,N_{+}|]}\mathtt{E}_{m}^{(j)}\right)$
$\displaystyle\leq
e^{2(x-z)}2^{r-N}{\mathbb{E}}\left(e^{-\sqrt{2}W_{\tau_{N}^{(r)}}}\mathds{1}_{\\{\forall
j\in[|r,N_{+}|],\,l_{j}^{(N)}\leq x+\sqrt{\frac{1}{2}}W_{\tau_{j}^{(r)}}\leq