# Fluctuations and Localization Length for Random Band GOE Matrix.

Michael Goldstein  Dept. of Mathematics, University of Toronto, Toronto,
Ontario, Canada M5S 1A1<EMAIL_ADDRESS>
###### Abstract.

We prove that GOE random band matrix localization length is $\leq C\left(\log
W\right)^{3}W^{2}$, where $W$ is the width of the band and $C$ is an absolute
constant. Our method consists of Green function edge-to-edge vector action
approach to the Schenker method. That allows to split and decouple the action,
so that it becomes transparent that _the magnitudes of two consecutive Schur
complements vector actions can not be both larger than an absolute constant_.
That is the central technological ingedient of the method. It comes from
rather involved estimates $($ the main estimates of the metod $)$, in
combination with an equation relating two magnitudes in question. We call the
latter _recurrence equation_. The method results in the _lower bound of the
variance of the $\log$–norm of the vector action at $\gtrsim NW^{-1}$_, where
$N$ is the total number of GOE blocks, condition $N\lesssim W^{D}$ with an
absolute constant $D\gg 1$ applies.

Supported in part by NSERC

###### Contents

  1. 1 Introduction
  2. 2 Variables Change.
  3. 3 Vector Action Decoupling-Splitting. Recurrence Equation.
  4. 4 Main Estimates.
  5. 5 Concentration Estimates and Log-Variance of Spherical Radius Against Special Super-Exponential Densitities.
  6. 6 Proof of Theorem B.
  7. 7 Proofs of Theorem C.
  8. A Matrix Variables Orthogonal Change.

##  1\. Introduction

We consider the following random band matrix

(1.1) $H_{[1,N]}=\begin{bmatrix}V_{1}&T_{1}&0&\cdots&\cdots&0\\\
T_{1}^{t}&V_{2}&T_{2}&0&\cdots&0\\\ 0&T^{t}_{2}&V_{3}&T_{3}&0\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&\cdots&\cdots&0&T^{t}_{N-1}&V_{N}\end{bmatrix},$

Here $V_{k},T_{\ell}$ are independent $W\times W$ random matrices. The
matrices $V_{k}=(v_{k,p,q})_{1\leq p,q\leq W}$ are real symmetric from the
GOE, i.e.

$\displaystyle d\mathbb{P}\big{(}V_{k}\big{)}=\prod_{1\leq p\leq
W}\sqrt{\frac{W}{2\pi}}e^{-\frac{Wv_{k,p,p}^{2}}{4}}dv_{k,p,p}\prod_{1\leq
p<q\leq W}\sqrt{\frac{W}{2\pi}}e^{-\frac{Wv_{k,p,q}^{2}}{2}}dv_{k,p,q}:=$
(1.2)
$\displaystyle\phi(V_{k})dV_{k},\qquad\qquad\qquad\qquad\qquad\qquad\qquad$

where

$dV=\prod_{1\leq p\leq q\leq W}dv_{p,q}$

stands for the Lebesgue measure in ${\mathbb{R}}^{{W(W+1)\over 2}}$, as we
identify every real symmetric matrix $V=(v_{p,q})_{1\leq p,q\leq W}$ with the
vector $(v_{p,q})_{1\leq p\leq q\leq W}\in{\mathbb{R}}^{{W(W+1)\over 2}}$. The
matrices $T_{k}=(\xi_{k,p,q})_{1\leq p,q\leq W}$ are $W\times W$ real random
Gaussian non-symmetric, i.e.

(1.3) $\displaystyle d\mathbb{P}\big{(}T_{k}\big{)}=\prod_{1\leq p,q\leq
W}\sqrt{\frac{W}{2\pi}}e^{-\frac{W\xi_{k,p,q}^{2}}{2}}d\xi_{k,p,q}:=\rho(T_{k})dT_{k}$

where

$dT=\prod_{1\leq p,q\leq W}d\xi_{p,q}$

stands for the Lebesgue measure in ${\mathbb{R}}^{W^{2}}$, as we identify
every real matrix $T=(\xi_{p,q})_{1\leq p,q\leq W}$ with the vector
$(\xi_{p,q})_{1\leq p,q\leq W}\in{\mathbb{R}}^{W^{2}}$,

(1.4)
$d\mathbb{P}_{[1,N]}{\big{(}V_{1},V_{2},\ldots,T_{1},T_{2},\dots\big{)}}=\prod_{k=1}^{N}d\mathbb{P}\big{(}V_{k}\big{)}\prod_{k=1}^{N-1}d\mathbb{P}\big{(}T_{k}\big{)}$

We use also the equations

$d\mathbb{P}\big{(}V\big{)}=\mu_{W}e^{-{W\over 4}TrV^{2}}dV,$
$d\mathbb{P}\big{(}T\big{)}=\nu_{W}e^{-{W\over 2}TrT^{t}T}dT,$

where $\mu_{W},\nu_{W}$ are the normalizing factors.

Fix $E\in{\mathbb{R}}$. Denote $G_{[1,N]}=\big{[}H_{[1,N]}-E\big{]}^{-1}$ the
Green function. Denote ${\mathcal{M}}_{W}$ the space of all $W\times W$
matrices. For $1\leq m,n\leq N$ introduce also the following notations

$G_{[1,N]}(m;n):=\left(G_{[1,N]}(p,q)\right)_{(m-1)W+1\leq p\leq
mW;(n-1)W+1\leq q\leq nW}\in{\mathcal{M}}_{W}$

We call $G_{[1,N]}(1;N)$ _Green function edge–to–edge matrix_.

######  Theorem A.

There exists a large absolute constant $D_{0}\gg 1$ such that for $N\geq
W^{D_{0}}$ the following statement holds. There exists a set
${\mathcal{B}}_{N}$ such that

$\mathbb{P}\big{[}{\mathcal{B}}_{N}\big{]}\leq\exp\left[-c_{0}{N\over
W^{D_{0}}}\right]$

where $c_{0}>0$ is an absolute constant, and off this set set

(1.5) $\displaystyle\big{\|}G_{[1,N]}(1;N)\big{\|}<e^{-{cN\over W\left(\log
W\right)^{3}}},$

where $c>0$ is an absolute constant.

######  Theorem B.

There exists a large absolute constant $D_{0}\gg 1$ such that for $W^{2}\leq
N\leq W^{D_{0}}$ the following statement holds. Take arbitrary unit vector
$f\in{\mathbb{R}}^{W}$. Then

(1.6)
$\displaystyle{\mathrm{Var}}\left[\log\big{\|}G_{[1,N]}(1;N)f\big{\|}\right]\geq{cN\over
W}$

where $c>0$ is an absolute constant.

######  Remark 1.1.

$(1)$ We would like to remark here on why we need more estimates on top of the
variance in Theorem B. The reason for that is a trivial objection for such
general conclusion. This objection is possible to eliminate for the setting in
question. The objection is as follows. Take _a non-positive random variable
$\xi$_. Assume that ${\mathrm{Var}}\xi=\mathfrak{S}^{2}\gg 1$. It is _not
true, in general_ , that

$\displaystyle\mathbb{E}\left[e^{\xi}\right]\leq e^{-c\mathfrak{S}^{2}},$

with an absolute constant $c>0$. A trivial example of a random variable taking
two values only

$\displaystyle\begin{cases}\mathbb{P}\left\\{\xi=-\mathfrak{S}^{{3\over
2}}\right\\}=p:=\mathfrak{S}^{-1},\\\
\mathbb{P}\left\\{\xi=0\right\\}=q:=1-p\end{cases}$

results

$\displaystyle{\mathrm{Var}}\xi=\left(\mathfrak{S}^{2}-\mathfrak{S}\right)\sim\mathfrak{S}^{2},$
$\displaystyle\mathbb{E}\left[e^{\xi}\right]=pe^{-\mathfrak{S}^{{3\over 2}}}$

The reason for this kind of outcome is _the lack of a ”classical” large
deviation estimate_ for $\xi$. The latter means that the estimate in question
comes from the projection of one with $\xi=\sum_{1\leq j\leq N}\xi_{k}$ and
with $\xi_{k}$ being independent, not necessarily equaly distributed, with
”classical” Bernstein condition for the moments, see for instance [Pe-75]:

$\displaystyle\left|\mathbb{E}\left[\xi_{k}-\mathbb{E}\xi_{k}\right]^{n}\right|\leq
n!\sigma^{2}_{k}H^{n-2},$ $\displaystyle\sigma_{k}^{2}:={\mathrm{Var}}\xi_{k}$

with $H$ being the same for all $k$. As we have mentioned in the beginning of
this remark, there is no such problem for the variable in question in the
current work setup. In fact the random variable in Theorem B, can be
conditioned so, that it is exactly as in ”classical” Bernstein condition and
$H=C\log W$ with absolute constant $C$. That is exactly the _junction where
the localization lengths acquires the factor $\left(\log W\right)^{3}$_.

$(2)$ Theorem $B$ _does have a version for arbitrary $N$_. Unfortunately, _it
does not seem possible to fit Bernstein condition for the latter seting
without loosing the sharpness of the estimate_. This objection leads to
application of _multi-scale analysis method_ from the random Schrodinger
operators, introduced in the seminal work of Frohlich and Spencer, [FS83].
Verification reduces to validation of what is called in multiscale analysis
_initial scale estimates_. Once these estimates established and the Wegner
estimate is known, the technology explicitely evaluates the localization
length. Theorem C below states all the estimates needed for this objective.
The statement of Theorem A is exactly the result of application of multiscale
analysis with estimates of Theorem C and well-known Wegner estimate in the
following form

(1.7)
$\displaystyle\mathbb{E}\left[\left\|G_{[1,N]}\left(1;N\right)\right\|^{t}\right]\leq
C_{1}W^{C_{2}},\quad\text{ for any $0<t<{1\over 2}$}$

where $C_{1},C_{2}$ are absolute constant. _We do not discuss the derivation
of Theorem A from Theorem C, for it is well-known to the experts how this
works_.

######  Theorem C.

There exists a large absolute constant $D_{0}\gg 1$ such that for $W^{2}\leq
N\leq W^{D_{0}}$ the following statement holds. There exists a set
${\mathcal{B}}_{N}$ such that

$\mathbb{P}\big{[}{\mathcal{B}}_{N}\big{]}\leq N^{-C_{0}D_{0}},$

where $C_{0}\gg 1$ is an absolute constant, and off this set set

(1.8) $\displaystyle\big{\|}G_{[1,N]}(1;N)\big{\|}<e^{-{cN\over W\left(\log
W\right)^{3}}}$

##  2\. Variables Change.

Take

(2.1) $H_{[1,N]}=\begin{bmatrix}V_{1}&T_{1}&0&\cdots&\cdots&0\\\
T_{1}^{t}&V_{2}&T_{2}&0&\cdots&0\\\ 0&T^{t}_{2}&V_{3}&T_{3}&0\cdots&0\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\\
0&\cdots&\cdots&0&T^{t}_{N-1}&V_{N}\end{bmatrix},$

where $V_{n}\in\textsf{M}_{W,\textsf{s}}$, $T_{n}\in\textsf{M}_{W}$. Assume
$V_{n}$ and $T_{m}$ are random matrices. Given unit vector
$f\in{\mathbb{R}}^{W}$ we call the map
$\left((V_{1},...,V_{N},T_{1},...T_{N-1}\right)\to G_{[1,N]}\left(1;N\right)f$
_Green function edge–to–edge vector action_. The method we develop in this
work targets the random variable

(2.2) $\displaystyle\gamma:=\log\left\|G_{[1,N]}\left(1;N\right)f\right\|$

with $V_{n}$ being random GOE matrices, $T_{n}$ random from gaussian matrices
ensemble, and all matrices being independent. As it was mentioned in the
Introduction to study this object we employ changes of variabes. In this
section we discuss these new variables and changes of variables in question

The most important change of variables targets the Green function edge–to–edge
matrix factorization. The idea of effectiveness of this approach was
introduced and demonstrated in the Schenker’s work [Sch09]. We call all these
new variables _Schenker variables_. It starts with application of Schur
complement formula to the $W\times W$ diagonal blocks of $H_{[1,N]}-E$. After
that it proceeds with a change of variables involved in Schur complement
formula, which allows to see ”Markovian features” of the random matrix
$G_{[1,N]}\left((V_{1},...,V_{N},T_{1},\dots,T_{N-1};1;N\right)$. For
convenience of the reader we state in the next proposition the details of
application of Schur complement formula for non-random setting.

######  Proposition 2.1.

Using the above notations the following equations hold

$(i)$ For $n\geq 2$

$\displaystyle G_{[1,1]}(1;1)=U_{1}^{-1},\quad U_{1}=V_{1}-E,$ $\displaystyle
G_{[1,n]}(1;n)=-G_{[1,n-1]}(1;n-1)T_{n-1}U_{n}^{-1},\quad n\geq 2$
$\displaystyle
U_{n}=\left(V_{n}-E-T^{t}_{n-1}G_{[1,n-1]}(n-1;n-1)T_{n-1}\right)$

$(ii)$ Using the notations is $(i)$

$\displaystyle G_{[1,N]}(1;N)=(-1)^{N-1}U_{1}^{-1}\times T_{1}\times...\times
U^{-1}_{N-1}\times T_{N-1}\times U_{N}^{-1}$

Next we state Schenker variables basic features. The key observation by
Schenker is of course that the Jacobian of the variables change is identically
1. That allows for explicit joint distribution equation. All features,
including ”Markovian” ones, follow straight from the definitions combined with
this fact. The verification of all features is short. Since in the current
setup $T_{k}$ are random we make detailed statements and discuss the proofs
for completeness. We consider not identically distributed variables just
because the features in question do not require this condition. We discuss
also Schenker variables _marginal and conditioned distributions_. That makes
it more convenient to explain validity of certain details in the main
technological estimates in Section 3. Those estimates enter _spherical radius
concentration and fluctuations estimates against certain super-exponential
densities_ , see next Section 5. Similar distribution features were introduced
independently in the recent work of N. Chen and C.Smart [CS22] In this
section, and later in what follows, out of convenience, we suppress the
condition variables from the notations of conditioned distributions, unless it
causes umbiguity.

######  Proposition 2.2.

$(i)$ Using the notations in Proposition 2.1, assume $T_{k}\in\textsf{M}_{W}$,
$V_{\ell}\in\textsf{M}_{\textsf{s},W}$, are random independent with

$\displaystyle\quad
d\mathbb{P}\left(V_{k}\right)=\phi_{k}\left(V_{k}\right)dV_{k},\quad
dV_{k}=\prod_{1\leq p\leq q\leq W}dv_{k,p,q},$ $\displaystyle
V_{k}=\left(v_{k,p,q}\right)_{1\leq p,q\leq W},\quad
v_{k,p,q}=v_{k,q,p},\qquad$ (2.3) $\displaystyle
d\mathbb{P}\left(T_{k}\right)=\rho_{k}\left(T_{k}\right)dT_{k},\quad
dT_{k}=\prod_{1\leq p,q\leq W}d\xi_{k,p,q},$ $\displaystyle
T_{k}=\left(\xi_{k,p,q}\right)_{1\leq p,q\leq W}\qquad\qquad\qquad$

Given $T_{1},\dots,T_{N-1}$, define new variables $u_{k,p,q}$, $1\leq p\leq
q\leq W$ in $\prod_{k=1}^{N}{\mathbb{R}}^{{W(W+1)\over 2}}$ via

$\displaystyle U_{1}=V_{1}-E,\quad U_{k}=\left(V_{k}-E-T^{t}_{k-1}\times
U_{k-1}^{-1}\times T_{k-1}\right),$ $\displaystyle
U_{k}=\left(u_{k,p,q}\right)_{1\leq p,q\leq W},\quad u_{k,p,q}=u_{k,q,p}$

The Jacobian of the transformation

$\displaystyle{\mathbb{R}}^{{NW(W+1)\over 2}}\ni\left(v_{k,p,q}\right)_{1\leq
k\leq N;1\leq p\leq q\leq
W}\to\mathfrak{F}_{T_{1},\dots,T_{N-1}}\left(\left(v_{k,p,q}\right)_{1\leq
k\leq N;1\leq p\leq q\leq W}\right):=$
$\displaystyle\left(u_{k,p,q}\right)_{1\leq k\leq N;1\leq p\leq q\leq
W}\in{\mathbb{R}}^{{NW(W+1)\over 2}}$

is $=1$.

$(ii)$ The joint distribution of
$U_{1},U_{2},\ldots,U_{N},T_{1},\dots,T_{N-1}$ is as follows

$d\mathbb{P}_{N}\Big{(}U_{1},U_{2},\ldots,U_{N},T_{1},\dots,T_{N-1}\Big{)}=$
$\phi_{1}(U_{1})\times\prod_{k=2,\ldots
N}\rho_{k-1}\left(T_{k-1}\right)\phi_{k}\left(U_{k}+T^{t}_{k-1}\times
U_{k-1}^{-1}\times T_{k-1}\right)\times dU_{k}\times dT_{k-1}$

$(iii)$ The variables $U_{1},\dots,U_{n-1},T_{1},\dots,T_{n-1}$ are
independent from the variables $U_{n+1},\dots,U_{N},T_{n},\dots,T_{N-1}$, when
conditioned on $U_{n}$. In particular, the variables $T_{n}$, $n=1,...,N-1$
are independent, when conditioned on $U_{n}$, $n=1,...$.

$(iv)$ The distribution of $T_{n-1}$, conditioned on the rest of the variables
is as follows

$\displaystyle d\mathbb{P}_{N}\Big{(}T_{n-1}\big{|}\Big{)}=$
$\displaystyle\mu_{T,n-1}\Big{(}U_{n-1},U_{n}\Big{)}\rho_{n-1}\left(T_{n-1}\right)\times\phi_{n}\left(U_{n}+T^{t}_{n-1}\times
U_{n-1}^{-1}\times T_{n-1}\right)\times dT_{n-1},$

where $\mu_{T,n-1}\Big{(}U_{n-1},U_{n}\Big{)}$ is the normalization factor.
The joint distribution of $T_{n}$, conditioned on $U_{n}$, $n=1,...,N$ is as
follows

$\displaystyle d\mathbb{P}_{N}\Big{(}T_{1},...,T_{N-1}\big{|}\Big{)}=$
$\displaystyle\prod_{1\leq n\leq
N-1}\mu_{T,n}\Big{(}U_{n},U_{n+1}\Big{)}\rho_{n}\left(T_{n}\right)\times\phi_{n+1}\left(U_{n+1}+T^{t}_{n}\times
U_{n}^{-1}\times T_{n}\right)\times dT_{n}$

$(v)$ The $\sigma$–algebras
${\mathcal{F}}\left(U_{1},T_{1},...,U_{n-1},T_{n-1}\right)$ and
${\mathcal{F}}\left(V_{n}\right)$ are independent.

###### Proof.

$(i)$ From the transformation definition,

$\displaystyle v_{k,p,q}=u_{k,p,q}+s_{k-1,p,q},\quad p\leq
q,\quad\big{(}s_{k-1,p,q}\big{)}_{1\leq p,q\leq k}=U_{k-1}^{-1}$

That implies the statement.

$(ii)$ Combine the change of variables distribution density rule with part
$(i)$, get the statement in $(ii)$.

$(iii)$ Combine conditional distribution density rule with the the joint
distribution equation from $(ii)$, get the statement.

$(iv)$ Combine the conditioned distribution rule with the joint distribution
equation from part $(ii)$, get the statement.

$(v)$ From the definitions
${\mathcal{F}}\left(U_{1},T_{1},...,U_{n-1},T_{n-1}\right)\subset{\mathcal{F}}\left(V_{1},T_{1},...,V_{n-1},T_{n-1}\right)$
$($ actually those are the same $)$. Since
$V_{1},T_{1},...,V_{n-1},T_{n-1},V_{n}$ are independent random variables, the
statement is correct.

∎

##  3\. Vector Action Decoupling-Splitting. Recurrence Equation.

In this section we discuss the setup of the method, which is instrumental to
evaluate the random variable (2.2) $($the central object of the method$)$.
Recall that the random variable is as follows

(3.1) $\displaystyle\gamma:=\log\left\|G_{[1,N]}\left(1;N\right)f\right\|,$

where $G_{[1,N]}\left(1;N\right)$ stands for the Green function edge–to–edge
matrix and $f\in{\mathbb{R}}^{W}$ is a fixed unit vector. Recall that using
the notations in Proposition 2.1

(3.2) $\displaystyle G_{[1,N]}(1;N)f=(-1)^{N-1}U_{1}^{-1}\times
T_{1}\times...\times U^{-1}_{N-1}\times T_{N-1}\times U_{N}^{-1}f$

The statements in this section address the distribution of _this equation
factors_ vector action. In other words, we look into the following random
vector–functions

$\displaystyle\textsf{v}_{g}\left(T_{n-1}\big{|}\right):=T_{n-1}\times
U_{n}^{-1}g,\qquad\qquad\qquad\qquad\qquad$ (3.3) with variables
$U_{n},U_{n-1}$, being fixed and $g$, being an arbitrary fixed unit vector

We setup a technology which allows for a relay from a given factor to the next
one on its left, so that _this one also acts on a fixed unit vector_ , i.e.
vector action starts over with a completely similar setup. The technology also
_”decouples” the vector action outcome from the rest of components of the
Schenker variable in question_. In other words it provides an explicit
equation for the vector action outcome distribution conditioned on the rest of
the components.

We do not discuss in this section how to combine the estimates with
$n=1,2,...$, we just look into each given factor action only. Moreover, here
we do not address the most important question concerning the random
vector–functions in (3.3),– _what is the magnitue of fluctuations, of
$\log\|\textsf{v}_{n}\|$, conditioned on_

(3.4)
$\displaystyle\left\|\textsf{v}_{n}\left(T_{n-1}\big{|}\right)\right\|^{-1}\textsf{v}_{n}\left(T_{n-1}\big{|}\right)=h$

_with a given unit vector $h\in{\mathbb{R}}^{W}$ $($ the one to relay to the
neighbour factor$)$_ We address these questions in Section 6. It turns out
that the magnitude of the fluctuations in question is _completely defined by
magnitude of either of two specific vectors in the splitting setup_. We call
these vectors _splitting principal vectors_ and denote them
$\textsf{B},\textsf{Z}$. We will show in Section 6 that _the magnitude of the
fluctuations in question is of order $\sim W^{-{1\over
2}}\left\|\textsf{Z}\right\|^{-1}\sim W^{-{1\over
2}}\left\|\textsf{B}\right\|^{-1}$_.

The principal vector Z naturally indexed by $n-1$ from (3.4). The number
$\left\|\textsf{Z}\right\|$ itself is _of magnitude of the Schur complement
factor vector action $U_{n-2}^{-1}h$_, see (3.2) and (3.4). In the current
section we _identify a recurrence equation relating the vectors
$\textsf{Z}_{n}$ and $\textsf{B}_{n+1}$_. This is the key technological piece.
In Section 4 we show that with high probability

(3.5)
$\displaystyle\min\left(\left\|\textsf{Z}_{n}\right\|,\left\|\textsf{B}_{n+1}\right\|\right)\sim
1$

This equation in the display does not hold eveywhere,- it is much more subtle.
The definition of the objects needed to state it properly is pretty lengthy.
This is the crucial technological estimate because it implies that at least
for every second $n$ the magnitude of the fluctuations in question is $\sim
W^{-{1\over 2}}$ exactly. The details missing in the display equation are
delicate. To incorporate these details a long series of involved estimates is
needed. In Remark 3.2 we comment on _elementary nature of the recurrence
equation itself, but still being effective_.

We state the main results of the current section in Proposition 3.1 below.
Recurrence equation appears in part $(4)$ of the proposition. We assume that
the random matrices $V_{k}$ are GOE random matrices, i.e. $\phi_{k}=\phi$ as
in (1.2). The random matrices $T_{k}$ are Gaussian non-symmetric, i.e.
$\rho_{k}=\rho$, as in (1.3).

######  Proposition 3.1.

Use the notations in Proposition 2.2 with $\phi_{k}=\phi$, $\rho_{k}=\rho$.

$(1)$ Use the above notations in this section. Take unit vector
$g\in{\mathbb{R}}^{W}$. Take arbitrary orthogonal matrix
$Q_{1}=Q_{1}\left(U_{n},g\right)\in\mathbb{O}_{W}$, such that

(3.6) $\displaystyle\left\|U_{n}^{-1}g\right\|^{-1}Q_{1}U_{n}^{-1}g=e_{1},$

where $e_{1}$ is the first standard basis in ${\mathbb{R}}^{W}$ vector. Change
the variable $T_{n-1}=R_{n-1}Q_{1}$. With this change of variable the
distribution of the random vector $\textsf{v}_{g}\left(T_{n-1}\big{|}\right)$
is the the push–forward of the following distribution

$\displaystyle d\mathbb{P}\left(R_{n-1}\big{|}\right)=\kappa e^{-{W\over
2}Tr\left[R_{n-1}^{t}R_{n-1}\right]}\times e^{-{W\over
4}Tr\left[U_{n}+Q_{1}^{t}R_{n-1}^{t}\times U_{n-1}^{-1}\times
R_{n-1}Q_{1}\right]^{2}}dR_{n-1}$

under the map

(3.7) $\displaystyle
R_{n-1}\to\textsf{v}\left(R_{n-1}\big{|}\right):=\left\|U_{n}^{-1}g\right\|R_{n-1}e_{1},$

Here $\kappa=\kappa\left(U_{n-1},U_{n}\right)$ is the normalization factor.

$(2)$ Use notations in part $(1)$. There exists orthogonal matrix
$Q_{2}=Q_{2}\left(U_{n-1}\right)\in\mathbb{O}_{W}$, such that with new
variable

$S_{n-1}:=S=Q^{t}_{2}R_{n-1}=Q^{t}_{2}T_{n-1}Q^{t}_{1},$

holds

(3.8)
$\displaystyle\textsf{v}\left(R_{n-1}\big{|}\right)=\left\|U_{n}^{-1}g\right\|Q_{2}Se_{1},$

and

$\displaystyle d\mathbb{P}\left(S\big{|}\right)=\kappa e^{-{W\over
2}Tr\left[S^{t}S\right]}\times e^{-{W\over
4}Tr\left[Q_{1}U_{n}Q_{1}^{t}+S^{t}\times C\times S\right]^{2}}\times dS\times
dU_{n},$ $\displaystyle Tr\left[Q_{1}U_{n}Q_{1}^{t}+S^{t}\times C\times
S\right]^{2}=Tr\left[\lambda_{1}\tilde{Y}^{t}_{1}\times\tilde{Y}_{1}+\check{S}^{t}\check{C}\check{S}+\check{B}\right]^{2}$
$\displaystyle+\left(\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle+b_{1,1}\right)^{2}+2\left\|\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}+\tilde{B}_{1}\right\|^{2},$

where

$\displaystyle S=\left(\xi_{i,j}\right)_{1\leq i,j\leq
W},\quad\check{S}=\left(\xi_{i,j}\right)_{2\leq i,j\leq W},$
$\displaystyle\tilde{Y}_{1}=\left(\xi_{1,j}\right)_{2\leq j\leq
W},\quad\tilde{X}_{1}=\left(\xi_{i,1}\right)^{t}_{2\leq i\leq W},$
$\displaystyle Q_{1}U_{n}Q_{1}^{t}:=B:=\left(b_{i,j}\right)_{1\leq i,j\leq
W},\quad B_{1}=Be_{1}=\left(b_{i,1}\right)^{t}_{1\leq i\leq W},$
$\displaystyle\check{B}=\left(b_{i,j}\right)_{2\leq i,j\leq
W},\quad\tilde{B}_{1}=\left(b_{i,1}\right)^{t}_{2\leq i\leq W},$
$\displaystyle C:=Q^{t}_{2}U^{-1}_{n-1}Q_{2},$ $\displaystyle
C=\left(\lambda_{i}\delta_{i,j}\right)_{1\leq i,j\leq
W},\quad\check{C}=\left(\lambda_{i}\delta_{i,j}\right)_{2\leq i,j\leq W},$

Here $\left\|X\right\|$ stands for the standard Euclidean norm of the vector.
Furthermore, with $X_{1}^{t}:=\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)$ holds

$\displaystyle
S^{t}CX_{1}=\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\end{bmatrix},$

$(3)$ Use the notations in part $(2)$. The conditional distribution of the
random vector $X_{1}^{t}$ is as follows

$\displaystyle
d\mathbb{P}\left(\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t}\big{|}\right)=\mu\Big{(}U_{n-1},U_{n}\Big{)}e^{-{W\over
2}\phi\left(X_{1}\big{|}\right)}\times dX_{1},$
$\displaystyle\phi\left(X_{1}\big{|}\right)=\left\|X_{1}\right\|^{2}+{1\over
2}\left(\langle CX_{1},X_{1}\rangle+b_{1,1}\right)^{2}+$
$\displaystyle\left\|\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}+\tilde{B}_{1}\right\|^{2}$

Here $\mu\Big{(}U_{n-1},U_{n}\Big{)}$ is the normalization factor. The vector
function $\textsf{v}\left(T_{n-1}\big{|}\right)$ from (3.3) and the
conditioning equation (3.4) are as follows

$\displaystyle\textsf{v}\left(T_{n-1}\big{|}\right)=Q_{2}\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t},\qquad$
(3.9)
$\displaystyle\quad\left\|\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)\right\|^{-1}Q_{2}\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t}=h$

$(4)$ Use the notations in part $(2)$, attach the index $(n-1)$ to $S$ back
i.e. use $S_{n-1}=Q^{t}_{2}R_{n-1}$. Add the index $(n-1)$ to some of the
notations in parts $(2)$, $(3)$:

(3.10) $\displaystyle B_{1,n-1}:=B_{1},\quad C_{n-1}:=C,\quad
X_{1,n-1}^{t}:=\left(\xi_{1,1,n-1},\tilde{X}^{t}_{1,n-1}\right)$

Introduce the site $(n-1)$–principal vectors:

(3.11)
$\displaystyle\textsf{B}_{n-1}=B_{1,n-1},\quad\textsf{Z}_{n-1}=C_{n-1}X_{1,n-1}$

Define all these objects for the site $(n-2)$ with $h$ as in part $(3)$ in the
role of $g$. Then the following recurrence equation holds

(3.12)
$\displaystyle\left\|\textsf{B}_{n-2}\right\|=\left\|X_{1,n-1}\right\|\left\|\textsf{Z}_{n-1}\right\|^{-1}$

######  Remark 3.2.

It is completely clear, from common sense point, that the recurrence equation
(3.12) must be nothing, but a simple computation, which follows
straightforward from the the Green function edge–to–edge matrix structure in
(3.2). That is true, but is hard see it in the original setup. The main point
is that it is not so clear why this helps to evaluate the fluctuations in
question. It becomes transparent in the splitting-decoupling setup development
though.

Before we prove of Proposition 3.1 we want to recall one equation from
Proposition 2.2, to which we refer repeatedly in this section and the next one

$\displaystyle
d\mathbb{P}_{N}\Big{(}T_{n-1}\big{|}\Big{)}=\mu_{T,n-1}\Big{(}U_{n-1},U_{n}\Big{)}\times\qquad\qquad$
(3.13)
$\displaystyle\rho_{n-1}\left(T_{n-1}\right)\times\phi_{n}\left(U_{n}+T^{t}_{n-1}\times
U_{n-1}^{-1}\times T_{n-1}\right)\times dT_{n-1}$

where $\mu_{T,n-1}\Big{(}U_{n-1},U_{n}\Big{)}$ is the normalization factor. We
also need the following elementary linear algebra statement.

######  Lemma 3.3.

Take arbitrary matrices $A\in\textsf{M}_{W,\textsf{s}}$ and
$S=\left(\xi_{i,j}\right)_{1\leq i,j\leq W}\in\textsf{M}_{W}$. Then there
exists orthogonal matrix $Q\in\mathbb{O}_{W}$, such that

$\displaystyle S^{t}Q^{t}\times A\times
QS=\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle&\lambda_{1}\xi_{1,1}\tilde{Y}_{1}+\left(\check{S}^{t}\check{C}\tilde{X}_{1}\right)^{t}\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}&\lambda_{1}\tilde{Y}^{t}_{1}\times\tilde{Y}_{1}+\check{S}^{t}\check{C}\check{S}\end{bmatrix}$

where $\check{S}\in\textsf{M}_{W-1}$, and the matrices $\check{S}$,
$\tilde{X}_{1}$, $\tilde{Y}_{1}$ come from the following block-matrix
representation

$\displaystyle S=\begin{bmatrix}\xi_{1,1}&\tilde{Y}_{1}\\\
\tilde{X}_{1}&\check{S}\end{bmatrix},$

$\lambda_{1},\lambda_{2},\dots,\lambda_{W}$ are arbitrary enumerated
eigenvalues of $A$, $\check{C}=\left(\delta_{i,j}\lambda_{j}\right)_{2\leq
i,j\leq W}$. Furthermore, with
$X_{1}^{t}:=\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)$ holds

$\displaystyle
S^{t}CX_{1}=\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\end{bmatrix},$

###### Proof.

Find orthogonal matrix $Q\in\mathbb{O}_{W}$ such that $Q^{t}\times A\times Q$
is a diagonal matrix with $\lambda_{1},\dots,\lambda_{W}$ on its diagonal. Use
block matrices mutiplication rule, get

$\displaystyle S^{t}\times Q^{t}\times A\times Q\times
S=\begin{bmatrix}\xi_{1,1}&\tilde{X}^{t}_{1}\\\
\tilde{Y}^{t}_{1}&\check{S}^{t}\end{bmatrix}\times\begin{bmatrix}\lambda_{1}&0\\\
0&\check{C}\end{bmatrix}\times\begin{bmatrix}\xi_{1,1}&\tilde{Y}_{1}\\\
\tilde{X}_{1}&\check{S}\end{bmatrix}=$
$\displaystyle\begin{bmatrix}\xi_{1,1}&\tilde{X}^{t}_{1}\\\
\tilde{Y}^{t}_{1}&\check{S}^{t}\end{bmatrix}\times\begin{bmatrix}\lambda_{1}\xi_{1,1}&\lambda_{1}\tilde{Y}_{1}\\\
\check{C}\tilde{X}_{1}&\check{C}\check{S}\end{bmatrix}=\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle&\lambda_{1}\xi_{1,1}\tilde{Y}_{1}+\left(\check{S}^{t}\check{C}\tilde{X}_{1}\right)^{t}\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}&\lambda_{1}\tilde{Y}^{t}_{1}\times\tilde{Y}_{1}+\check{S}^{t}\check{C}\check{S}\end{bmatrix},$

as claimed. Take $X_{1}^{t}:=\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)$, get

$\displaystyle S^{t}CX_{1}=\begin{bmatrix}\xi_{1,1}&\tilde{X}^{t}_{1}\\\
\tilde{Y}^{t}_{1}&\check{R}^{t}\end{bmatrix}\times\begin{bmatrix}\lambda_{1}\xi_{1,1}\\\
\check{C}\tilde{X}_{1}\end{bmatrix}=\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{R}^{t}\check{C}\tilde{X}_{1}\end{bmatrix},$

as claimed. ∎

###### Proof of Proposition 3.1.

$(1)$ Use the proposition notations. Take unit vector $g\in{\mathbb{R}}^{W}$.
Take arbitrary orthogonal matrix
$Q_{1}=Q_{1}\left(U_{n},g\right)\in\mathbb{O}_{W}$, such that

$\displaystyle\left\|U_{n}^{-1}g\right\|^{-1}Q_{1}U_{n}^{-1}g=e_{1},$

where $e_{1}$ is the first standard basis in ${\mathbb{R}}^{W}$ vector. Change
the variable $T_{n-1}=S_{n-1}Q_{1}$ with the rest of the variables being
intact. Then

$\displaystyle\textsf{v}_{g}\left(T_{n-1}\big{|}\right)=T_{n-1}\times
U_{n}^{-1}g=R_{n-1}Q\times U_{n}^{-1}g=$ (3.14)
$\displaystyle\left\|U_{n}^{-1}g\right\|^{-1}S_{n-1}e_{1}=\textsf{v}\left(S_{n-1}\big{|}\right),\qquad\qquad$

as in (3.7) in part $(1)$, see the definition equation (3.3). The Jacobian of
the variable change is $=1$. Combine the change of variables distribution
density rule with Jaconian $=1$, with the second equation in (3.13), and with
invariance of $\rho_{n-1}$ under the change of variable, get the statement in
$(1)$.

$(2)$ Apply Lemma 3.3 with $S_{n-1}$ in the role of $T$ and $U_{n-1}^{-1}$ in
the role of $A$, find $Q:=Q_{2}=Q_{2}\left(U_{n-1}\right)\in\mathbb{O}_{W}$ as
stated in the lemma. Change the variable $S=Q^{t}_{2}S_{n-1}$ as stated in
$(2)$, write

$\displaystyle S_{n-1}^{t}\times U_{n-1}^{-1}\times S_{n-1}=S^{t}\times
Q_{2}^{t}U_{n-1}^{-1}Q_{2}\times S=$
$\displaystyle\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle&\lambda_{1}\xi_{1,1}\tilde{Y}_{1}+\left(\check{T}^{t}\check{C}\tilde{X}_{1}\right)^{t}\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{T}^{t}\check{C}\tilde{X}_{1}&\lambda_{1}\tilde{Y}^{t}_{1}\times\tilde{Y}_{1}+\check{T}^{t}\check{C}\check{T}\end{bmatrix},$
$\displaystyle
S^{t}CX_{1}=\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{R}^{t}\check{C}\tilde{X}_{1}\end{bmatrix},$

see the notations in the statement of part $(2)$. Change the variable
$S_{n-1}=Q_{2}S$ in the equation (3.13), get

(3.15)
$\displaystyle\textsf{v}_{g}\left(S_{n-1}\big{|}\right)=\left\|U_{n}^{-1}g\right\|^{-1}Q_{2}Se_{1},$

as in (3.8) in part $(2)$. The Jacobian of the variable change is $=1$. Use
the distribution equation from part $(1)$, get

$\displaystyle
d\mathbb{P}_{N}\Big{(}S\big{|}\Big{)}=\mu\Big{(}U_{n-1},U_{n}\Big{)}\times
e^{-{W\over 2}TrS^{t}S}\times e^{-{W\over
4}Tr\left[U_{n}+Q_{1}^{t}S^{t}Q_{2}^{t}\times U_{n-1}^{-1}\times
Q_{2}SQ_{1}\right]^{2}}dS$

Use that and

$\displaystyle Tr\left[Q^{t}\times X\times
Q\right]^{2}=Tr\left[X^{2}\right],\quad Q^{t}Q=I$

for any $X\in\textsf{M}_{W,\textsf{s}}$ and $Q\in\mathbb{O}_{W}$, rewrite

$\displaystyle Tr\left[U_{n}+Q_{1}^{t}S^{t}Q_{2}^{t}\times U_{n-1}^{-1}\times
Q_{2}SQ_{1}\right]^{2}=Tr\left[Q_{1}U_{n}Q_{1}^{t}+S^{t}Q_{2}^{t}\times
U_{n-1}^{-1}\times Q_{2}S\right]^{2},$ $\displaystyle
d\mathbb{P}_{N}\Big{(}S\big{|}\Big{)}=\hat{\mu}\Big{(}U_{n-1},U_{n}\Big{)}\times
e^{-{W\over 2}TrS^{t}S}\times e^{-{W\over
4}Tr\left[Q_{1}U_{n}Q_{1}^{t}+S^{t}Q_{2}^{t}\times U_{n-1}^{-1}\times
Q_{2}S\right]^{2}}dS$

Combine, use the notations in part $(2)$ statement, get

$\displaystyle Q_{1}U_{n}Q_{1}^{t}+S_{n-1}^{t}\times U_{n-1}^{-1}\times
S_{n-1}=B+S_{n-1}^{t}\times U_{n-1}^{-1}\times S_{n-1}=$ (3.16)
$\displaystyle\left(b_{i,j}\right)_{1\leq i,j\leq
W}+\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle&\lambda_{1}\xi_{1,1}\tilde{Y}_{1}+\left(\check{T}^{t}\check{C}\tilde{X}_{1}\right)^{t}\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{T}^{t}\check{C}\tilde{X}_{1}&\lambda_{1}\tilde{Y}^{t}_{1}\times\tilde{Y}_{1}+\check{T}^{t}\check{C}\check{T}\end{bmatrix}$

Split the matrix $B$ in blocks, use

$\displaystyle Tr\begin{bmatrix}x&\check{X}^{t}_{1}\\\
\check{X}_{1}&\check{X}\end{bmatrix}^{2}=x^{2}+2\left\|\check{X}_{1}\right\|^{2}+Tr\left[\check{X}^{2}\right]$

for any symmetric block matrix, get the distribution statement in part $(2)$.

$(3)$ Use the notations in part $(2)$. Use conditional distribution rule, get
the first statement in $(3)$. Combine the definition equation (3.3),
respectively, the conditioning equation (3.4), with equations (3.14) and
(3.15), get the second statement in $(3)$

$(4)$ We derive an equation for the site $(n-1)$, which clearly projects onto
a statement for the site $(n-2)$, and which implies easily the statement in
$(4)$. This way we do not need to introduce additional notations. Use equation
(3.6), combine with equations in parts $(2)$, write

$\displaystyle\left\|U_{n}^{-1}g\right\|^{-1}Q_{1}U_{n}^{-1}g=e_{1},$
$\displaystyle\left\|U_{n}^{-1}g\right\|^{-1}U_{n}^{-1}g=Q_{1}^{t}e_{1},$
$\displaystyle\textsf{B}_{n-1}=B_{1,n-1}=Be_{1}=Q_{1}U_{n}Q_{1}^{t}e_{1}=\left\|U_{n}^{-1}g\right\|^{-1}Q_{1}g,$
$\displaystyle\left\|\textsf{B}_{n-1}\right\|=\left\|U_{n}^{-1}g\right\|^{-1}$

The last equation here clearly projects on the $(n-2)$ site:

$\displaystyle\left\|\textsf{B}_{n-2}\right\|=\left\|U_{n-1}^{-1}h\right\|^{-1}$

Recall that $h$ here is as in part $(3)$, i.e.

$\displaystyle
h=\left\|\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)\right\|^{-1}Q_{2}\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t}=\left\|X_{1,n-1}\right\|^{-1}Q_{2}X_{1,n-1}$

Write

$\displaystyle\left\|U_{n-1}^{-1}h\right\|=\left\|X_{1,n-1}\right\|^{-1}\left\|U_{n-1}^{-1}Q_{2}X_{1,n-1}\right\|=\left\|X_{1,n-1}\right\|^{-1}\left\|Q_{2}^{t}U_{n-1}^{-1}Q_{2}X_{1,n-1}\right\|=$
$\displaystyle\left\|X_{1,n-1}\right\|^{-1}\left\|C_{n-1}X_{1,n-1}\right\|=\left\|X_{1,n-1}\right\|^{-1}\left\|\text{Z}_{n-1}\right\|$

Combine, get the recurrence equation in the statement.

∎

##  4\. Main Estimates.

We state the main results of the current section in Proposition 4.1 below.

######  Proposition 4.1.

Use the notations in Proposition 3.1.

$(0)$ Use the notations in Proposition 3.1, part $(3)$. Introduce the
spherical variables

$\displaystyle\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t}:=X=r\textsf{x},$
$\displaystyle
r=\left\|X\right\|,\quad\textsf{x}=\left(\textsf{x}_{1},...,\textsf{x}_{W}\right)=r^{-1}X^{t}\in\mathbb{S}^{W-1}$

The joint conditional distribution
$d\mathbb{P}\left(r,\textsf{x}\big{|}\right)$ and the conditional distribution
$d\mathbb{P}\left(r\big{|}\right)$ are, respectively, as follows

$\displaystyle
d\mathbb{P}\left(r,x\big{|}\right)=\mu\Big{(}U_{n-1},U_{n}\Big{)}r^{W-1}e^{-{W\over
2}\phi\left(r,\textsf{x}|\right)}drd\textsf{x},$ $\displaystyle
d\mathbb{P}\left(r\big{|}\right)=\mu\Big{(}U_{n-1},U_{n},\textsf{x}\Big{)}r^{W-1}e^{-{W\over
2}\phi\left(r|\right)}dr,$
$\displaystyle\phi\left(r,\textsf{x}|\right)=\phi\left(r\textsf{x}|\right),\quad\text{see
Proposition~{}\ref{prop:condecoupling1}, part $(3)$},$
$\displaystyle\phi\left(r|\right)=\textsf{a}^{2}r^{4}+\textsf{b}r^{2}+\textsf{c}r,$
$\displaystyle\textsf{a}^{2}=\textsf{a}^{2}\left(\textsf{x}\right)={1\over
2}\langle C\textsf{x},\textsf{x}\rangle^{2},$
$\displaystyle\textsf{b}=\textsf{b}\left(\textsf{x}\right)=1+\beta^{2}+\gamma,$
$\displaystyle\beta^{2}=\beta^{2}\left(\textsf{x}\right)=\left\|\lambda_{1}\textsf{x}_{1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{\textsf{x}}\right\|^{2},\quad\tilde{\textsf{x}}=\left(\textsf{x}_{2},...,\textsf{x}_{W}\right)^{t},$
$\displaystyle\gamma=\gamma\left(\textsf{x}\right)=b_{1,1}\langle
C\textsf{x},\textsf{x}\rangle,$
$\displaystyle\textsf{c}=\textsf{c}\left(\textsf{x}\right)=2\langle\lambda_{1}\textsf{x}_{1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{\textsf{x}},\tilde{B}_{1}\rangle,$

Here $\mu\Big{(}U_{n-1},U_{n}\Big{)}$,
$\mu\Big{(}U_{n-1},U_{n},\textsf{x}\Big{)}$ are the normalizing factors.

$(1)$ There exists a set
$\widehat{{\mathcal{B}}}_{U,n-2,n-1,D}\in\mathcal{F}\left(U_{n-2},U_{n-1}\right)$,
such that

$\displaystyle\mathbb{P}_{N}\left[\widehat{{\mathcal{B}}}_{U,n-1,n,D,A}\right]\leq
CW^{D-1},$

and a set ${\mathcal{B}}_{S,n-1,A}\in\mathcal{F}\left(S\right)$ with

$\displaystyle\mathbb{P}_{N}{\mathcal{B}}_{S,n-1,A}\leq C_{1}e^{-c_{1}A^{2}W}$

such that for any
$\left(U_{n-1},U_{n}\right)\notin\widehat{{\mathcal{B}}}_{U,n-1,n,D,A}$,
$S\notin{\mathcal{B}}_{S,n-1,A}$ and any x holds

$\displaystyle\max\left(\textsf{a}^{2}\left(\textsf{x}\right),\left|\textsf{b}\left(\textsf{x}\right)\right|,\left|\textsf{c}\left(\textsf{x}\right)\right|\right)\leq
2A^{2}W^{2D}$

Furthermore,

$\displaystyle\mathbb{P}_{N}\left\\{\textsf{a}^{2}\left(\textsf{x}\right)=0\right\\}=0$

$(2)$ There exists a set
$\widehat{{\mathcal{B}}}_{U,n-1,n,A}\in\mathcal{F}\left(U_{n-1},U_{n}\right)$,
such that

$\displaystyle\mathbb{P}_{N}\left[\widehat{{\mathcal{B}}}_{U,n-1,n,A}\right]\leq
Ce^{-cA^{2}W},$

with absolute constants $c,C$, and for any
$\left(U_{n-1},U_{n}\right)\notin\widehat{{\mathcal{B}}}_{U,n-1,n,A}$ there
exists a set
${\mathcal{B}}_{X,n-1,n,A}\in\mathcal{F}\left(r,\textsf{x}\right)$, such that

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{X,n-1,n,A}\big{|}U_{n-1},U_{n}\right]\leq
Ce^{-cA^{2}W},$

such that for any $\left(r,\textsf{x}\right)\notin{\mathcal{B}}_{X,n-1,n,A}$
the following equations hold

$\displaystyle r\leq A,$
$\displaystyle\big{|}\gamma\left(\textsf{x}\right)r^{2}+2\textsf{a}^{2}\left(\textsf{x}\right)r^{4}\big{|}\leq\sqrt{2}A\textsf{a}\left(\textsf{x}\right)r^{2},$
$\displaystyle\left|\textsf{c}\left(\textsf{x}\right)r+2\beta^{2}\left(\textsf{x}\right)r^{2}\right|\leq
2A\beta\left(\textsf{x}\right)r,$

and also

$\displaystyle\textsf{a}r^{2}<A+\left\|B_{1}\right\|,$
$\displaystyle|\gamma|r^{2}\leq\left\|B_{1}\right\|\left(A+\left\|B_{1}\right\|\right),$
$\displaystyle\beta^{2}r^{2}\leq\left(A+\left\|B_{1}\right\|\right)^{2},$
$\displaystyle|b|r^{2}\leq
A^{2}+\left(A+\left\|B_{1}\right\|\right)^{2}+\left\|B_{1}\right\|\left(A+\left\|B_{1}\right\|\right),$
$\displaystyle|\textsf{c}|r\leq
2\left\|B_{1}\right\|\left(A+\left\|B_{1}\right\|\right)$

$(3)$ Use the notations in $(2)$. For any
$\left(U_{n-1},U_{n}\right)\notin\widehat{{\mathcal{B}}}_{U,n-1,n,A}$ there
exists a set
${\mathcal{B}}_{\texttt{x},n-1,n,A}\in\mathcal{F}\left(\textsf{x}\right)$,
such that

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{\texttt{x},n-1,n,A}\big{|}U_{n-1},U_{n}\right]\leq
C_{0}e^{-c_{0}A^{2}W},$

with absolute constants $c_{0},C_{0}$, and for any
$\textsf{x}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-1,n,A}$ there exists a
set ${\mathcal{B}}_{\texttt{r},\textsf{x},n-1,n,A}\subset(0,+\infty)$, such
that

$\displaystyle\mathbb{P}_{N}\left[\left\\{r\in{\mathcal{B}}_{\texttt{r},n-1,n,A}\right\\}\big{|}U_{n-1},U_{n}\right]\leq
C_{0}e^{-cA^{2}W},$

such that for any $r\notin{\mathcal{B}}_{\texttt{r},n-1,n,A}$ the equations in
part $(2)$ hold.

$(4)$ Take $\textsf{x}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-1,n,A}$, see
$(3)$ with no condition $r\notin{\mathcal{B}}_{\texttt{r},\textsf{x},n-1,n,A}$
though. Take one of the $(n-1)$–principal vectors
$\textsf{Z}=\textsf{Z}_{n-1}(\textsf{x})=CX$, defined in (3.11), set
$\zeta=\left\|\textsf{X}\right\|^{-1}\left\|\textsf{Z}\right\|$. Then

$\displaystyle\textsf{a}^{2}\leq{1\over 2}\zeta^{2},\quad\beta^{2}\leq
A^{2}\zeta^{2},$ $\displaystyle\left|\textsf{b}\right|\leq
1+A\zeta+2A^{2}\zeta^{2},$ $\displaystyle\left|\textsf{c}\right|\leq
2A^{3}\zeta^{2}+2A\zeta$

$(5)$ In the above notation attach the index $(n-1)$ to

$\textsf{a},\textsf{b},\textsf{c},\beta,\gamma,\zeta,B_{1},$

i.e. use

$\textsf{a}_{n-1},\textsf{b}_{n-1},\textsf{c}_{n-1},\beta_{n-1},\gamma_{n-1},\zeta_{n-1},B_{1,n-1}$

instead. Define

$\textsf{a}_{n-2},\textsf{b}_{n-2},\textsf{c}_{n-2},\beta_{n-2},\gamma_{n-2},\zeta_{n-2},B_{1,n-2}$

in a completely similar way. There exists an absolute constant C, such that
the following statement holds. Take

$\displaystyle\textsf{y}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-2,n-1,A},\quad\rho\notin{\mathcal{B}}_{\texttt{r},\textsf{x},n-2,n-1,A}$

If

$\zeta_{n-1}\left(\textsf{x}\right)>\textsf{C},$

then the following equations hold

$\displaystyle\textsf{a}_{n-2}\left(\textsf{y}\right)\rho^{2}<A+\textsf{C}^{-1},$
$\displaystyle|\gamma_{n-2}\left(\textsf{y}\right)|\rho^{2}\leq\textsf{C}^{-1}\left(A+\textsf{C}^{-1}\right),$
$\displaystyle\beta^{2}_{n-2}\left(\textsf{y}\right)\rho^{2}\leq\left(A+\textsf{C}^{-1}\right)^{2},$
$\displaystyle|\textsf{c}_{n-2}\left(\textsf{y}\right)|\rho\leq
2\textsf{C}^{-1}\left(A+\textsf{C}^{-1}\right)$

Furthermore, either

(4.1) $\displaystyle\textsf{b}_{n-2}\left(\textsf{y}\right)>0,$

or

(4.2)
$\displaystyle|\textsf{b}_{n-2}\left(\textsf{y}\right)|<2\sqrt{2}\textsf{C}^{-1}|\textsf{a}_{n-2}\left(\textsf{y}\right)|$

In other words, using the above notations, the following dichotomy holds:
There exists an absolute constant A, such that for any $n$ either

(4.3)
$\displaystyle\max\left(\textsf{a}_{n-1}^{2}(\textsf{x}),\left|\textsf{b}_{n-1}(\textsf{x})\right|,\left|\textsf{c}_{n-1}(\textsf{x})\right|\right)\leq\textsf{A}$

for any $\textsf{x}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-1,n,A}$, or

$\displaystyle\max\left(\textsf{a}_{n-2}^{2}(\textsf{y})\rho^{4},\left|\textsf{b}_{n-2}(\textsf{y})\right|\rho^{2},\left|\textsf{c}_{n-2}(\textsf{x})\right|\rho\right)\leq\textsf{A}$

for any $\textsf{y}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-2,n-1,A}$,
$\rho\notin{\mathcal{B}}_{\texttt{r},\textsf{x},n-2,n-1,A}$. Furthermore, in
this case, for any
$\textsf{y}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-2,n-1,A}$, either

(4.4) $\displaystyle\textsf{b}_{n-2}\left(\textsf{y}\right)>0,$

or

(4.5)
$\displaystyle|\textsf{b}_{n-2}\left(\textsf{y}\right)|<2\sqrt{2}\textsf{C}^{-1}|\textsf{a}_{n-2}\left(\textsf{y}\right)|$

######  Remark 4.2.

$(1)$ We want to remark here on equation (4.1). This one is crucial for the
dichotomy second case. It is a key to the log-variance of spherical radius
against special super-exponential densitities estimate, which we establish in
section 5. The latter, in its turn, is one of the main ingredients in the
proof of Theorem B. The reader will note that in the proof of the log-variance
estimate in section 5, we use other estimates in the display above (4.1). This
is confusing. The delicate detail, which explains the confusion is as follows.
The estimates in the display hold _only under condition
$\rho\notin{\mathcal{B}}_{\texttt{r},\textsf{x},n-2,n-1,A}$_. The set
${\mathcal{B}}_{\texttt{r},\textsf{x},n-2,n-1,A}$ is negligible, for its
probability is very small. However, one needs to know _that the estimates hold
for the value $\rho_{1}$, at which the exponent in the density in question
assumes its maximum_. It is very natural to expect that the bulk of the
distribution sits around $\rho_{1}$ and most of the points there must be out
of the set ${\mathcal{B}}_{\texttt{r},\textsf{x},n-2,n-1,A}$. This is a
delicate problem though. The equation (4.1) resolves this problem by making
all terms of a central piece inequality relating _the quantities in question
positive, which implies that each term obeys the inequality_ , i.e. there is
no cancelations hidden.

$(2)$ Write the vector action from Proposition 3.1 in spherical coordinates
$\textsf{v}\left(T_{n-1}\big{|}\right)=r\textsf{y}$. As we had mentioned in
the Introduction the spherical variable $\textsf{y}\in S^{W-1}$ distribution
is hard to evaluate. That is the reason there is no statement addressing this
distribution in Proposition 4.1.

The proof of Proposition 4.1 uses two statements which we discuss separately.
We use extensively Proposition 3.1 with the same notations. Out of
convenience, we write up below the equation we mainly use

$\displaystyle
d\mathbb{P}_{N}\Big{(}T_{n-1}\big{|}\Big{)}=\qquad\qquad\qquad\qquad$ (4.6)
$\displaystyle\qquad\qquad\rho_{n-1}\left(T_{n-1}\right)\times\phi_{n}\left(U_{n}+T^{t}_{n-1}\times
U_{n-1}^{-1}\times T_{n-1}\right)\times dT_{n-1}\times dU_{n}$

$\blacktriangle$ We use Gaussian matrix operator norm upper tail estimate,
which must be well-known. We follow the method in Corollary 2.3.5, from the
monohraph [TAO13]. The statement requires $M=\left(\xi_{p,q}\right)_{1\leq
p,q\leq W}$ with _random variables $\xi_{p,q}$ being i.i.d. and uniformly
bounded by one_. The derivation goes via _$\epsilon$ –net argument_ introduced
in the proof of Corollary 2.3.5 and based on the following fixed vector
estimate from [TAO13], Lemma 2.3.1:

$\displaystyle\mathbb{P}\left\\{\left\|Mx\right\|>A\sqrt{W}\right\\}\leq
Ce^{-cAW},$

for any unit vector $x\in{\mathbb{R}}^{W}$, with absolute constants $c,C$. The
statement then follows from unit sphere $\mathbb{S}^{W-1}$ $\epsilon$-net
cardinality estimate. Nothing else is needed for the $\epsilon$–net argument.
For Gaussian non-symmetric matrices $T$, normalized as in this work, the
estimate in the last display has the following version

$\displaystyle\mathbb{P}\left\\{\left\|Mx\right\|>A\right\\}\leq
Ce^{-cA^{2}W}$

The same is true for GOE random matrix. In fact _the latter one, i.e. the GOE
matrix operator norm upper tail estimateis, is what we use the most_. For
Gaussian random non-symmetric matrix _we use vector action concentration
estimate in the last display applied to the first standard basis vector_.
Since we use the estimate repeatedly, it is convenient to state the estimate
as a separate proposition. We discuss the proof of the estimate for
completeness.

######  Proposition 4.3.

$(0)$ For Gaussian random vector $x=(\xi_{1},...,\xi_{W})$, normalized as in
this work, holds

$\displaystyle\mathbb{P}\left\\{\left|\left\|x\right\|-1\right|>AW^{-{1\over
2}}\right\\}\leq Ce^{-cA^{2}}$

for any $A>0$ with absolute constants $c,C$.

$(i)$ For Gaussian non-symmetric matrix $T$ holds

$\displaystyle\mathbb{P}\left\\{\left|\left\|Tx\right\|-1\right|>AW^{-{1\over
2}}\right\\}\leq Ce^{-cA^{2}}$

for any unit vector $x\in{\mathbb{R}}^{W}$ and any $A>0$. In particular,

$\displaystyle\mathbb{P}\left\\{\left|\left\|Tx\right\|-1\right|>a\right\\}\leq
Ce^{-ca^{2}W}$

for any $a>0$.

$(ii)$ For Gaussian random matrix $T$ holds

$\displaystyle\mathbb{P}\left\\{\left\|T\right\|>A\right\\}\leq Ce^{-cA^{2}W}$

for any $A>0$.

$(iii)$ For GOE matrix $V$ holds

$\displaystyle\mathbb{P}\left\\{\left|\left\|Vx\right\|-1\right|>A\right\\}\leq
Ce^{-cA^{2}W}$

for any unit vector $x\in{\mathbb{R}}^{W}$ and any $A>0$. In particular,

$\displaystyle\mathbb{P}\left\\{\left\|Vx\right\|>1+a\right\\}\leq
Ce^{-ca^{2}W}$

for any $a>0$.

$(iv)$ For GOE matrix $V$ holds

$\displaystyle\mathbb{P}\left\\{\left\|V\right\|>A\right\\}\leq Ce^{-cA^{2}W}$

for any $A>0$.

###### Proof.

$(0)$ Write

$\displaystyle\mathbb{P}\left\\{\left\|x\right\|>1+AW^{-{1\over
2}}\right\\}=\mathbb{P}\left\\{\sum_{1\leq j\leq W}\xi_{i}^{2}>1+2AW^{-{1\over
2}}+A^{2}W^{-1}\right\\}=$
$\displaystyle\mathbb{P}\left\\{\chi^{2}_{W}>W+2AW^{{1\over
2}}+A^{2}\right\\},$

where $\chi^{2}_{n}$ stands for chi–squared distributed with $n$ degrees of
freedom. The Laurent-Massart concentration estimate reads

$\displaystyle\mathbb{P}\left\\{\chi^{2}_{n}-n>2\sqrt{nx}+2x\right\\}<e^{-x},$
$\displaystyle\mathbb{P}\left\\{-\chi^{2}_{n}>2\sqrt{nx}\right\\}<e^{-x}$

for any $x>0$, see [LM00]. Take $n=W$, use the estimate, get

$\displaystyle\mathbb{P}\left\\{\chi^{2}_{W}>W+2AW^{{1\over
2}}+A^{2}\right\\}\leq\mathbb{P}\left\\{\chi^{2}_{W}-W>2\sqrt{W\times{A^{2}\over
2}}+2\times{A^{2}\over 2}\right\\}<e^{-{A^{2}\over 4}}$

Similarly,

$\displaystyle\mathbb{P}\left\\{\left\|x\right\|<1-AW^{-{1\over
2}}\right\\}=\mathbb{P}\left\\{\chi^{2}_{W}<W-2AW^{{1\over
2}}+A^{2}\right\\}\leq\mathbb{P}\left\\{\chi^{2}_{W}<W-AW^{{1\over
2}}\right\\}\leq$
$\displaystyle\mathbb{P}\left\\{W-\chi^{2}_{W}>2\sqrt{W\times{A^{2}\over
4}}\right\\}<e^{-{A^{2}\over 4}},$

provided $A<W^{1\over 2}$. That finishes the statement in $(0)$ for any $A>0$.

$(i)$ Use orthogonal right multiplication invariance of
$d\mathbb{P}\big{(}T\big{)}$, replace vector $x$ by the standard basis in
${\mathbb{R}}^{W}$ first vector $e_{1}=(1,0,\dots,0)^{t}$, write

$\displaystyle Tx=Te_{1}=(\xi_{1,1},\xi_{2,1},\dots,\xi_{W,1})^{t}$

Use part $(0)$, get the statement.

The estimate in $(ii)$ follows from $(i)$ via $\epsilon$–net argument, see
[TAO13], p. 129.

$(iii)$ Use the argument in the proof of $(i)$, get the statement.

The estimate in $(iv)$ follows from $(iii)$ via $\epsilon$–net argument. ∎

$\blacktriangle$ As we mentioned in the Introduction, we need a version of
Wegner estimate for GOE perturbations of arbitrary symmetric matrix $A$. Any
version which is not ”too weak” would do. The latter means that the estimate
eliminates the _inverse matrix magnitudes which are polynomial in size of the
matrix_. On the other hand _an optimal Wegner estimate was established_ in
[APSSS17]. Out of convenience we state this estimate here. Take GOE matrix
$V$, normalized as in this work. Take arbitrary real symmetric matrix $A$.
Denote $H=V+A$. Take arbitrary unit vector $f$. Theorem 1 in [APSSS17] says
that the following estimate holds for any $K>0$

(4.7) $\mathbb{P}\left\\{\big{\|}H^{-1}\big{\|}>KW\right\\}\leq{C\over K},$

where $C$ is an absolute constant. We use this estimate with $K=W^{D-1}$, i.e.

(4.8) $\mathbb{P}\left\\{\big{\|}H^{-1}\big{\|}>W^{D}\right\\}\leq
CW^{-\left(D-1\right)},$

where $D\gg 1$ is an absolute constant.

$\blacktriangle$ Next proposition discusses straightforward applications of
the estimates from Proposition 4.3 and (4.8) to Schenker variables.

######  Proposition 4.4.

Use the notations in Proposition 2.2 with $\phi_{k}=\phi$, $\rho_{k}=\rho$.

$(0)$ For any $n$ and $A>0$

$\displaystyle\mathbb{P}_{N}\left\\{\left\|T_{n-1}\right\|>A\right\\},\mathbb{P}_{N}\left\\{\left\|V_{n}\right\|>A\right\\}\leq
Ce^{-cA^{2}W}$

with absolute constants $c,C$.

$(1)$ For any $U_{n-2}$ holds

$\displaystyle\mathbb{P}_{N}\left(\left\\{\big{\|}U^{-1}_{n-1}\big{\|}>W^{D}\right\\}|U_{n-2}\right)\leq
CW^{-\left(D-1\right)},$

where $C$ is an absolute constant.

$(2)$ Take arbitrary orthogonal matrices $Q_{1},Q_{2}\in\mathbb{O}_{W}$, which
may depend on the variables $U_{n-1},U_{n}$. Change the variable
$T_{n-1}=Q_{2}S_{n-1}Q_{1}$ with the rest of the variables being intact. Then
$S_{n-1}\in\mathcal{F}\left(U_{n-1},T_{n-1},U_{n}\right)$,

$\displaystyle
d\mathbb{P}_{N}\Big{(}S_{n-1}\Big{)}=\rho_{n-1}\left(S_{n-1}\right)\times
dS_{n-1},\qquad\qquad\quad\qquad\qquad$ (4.9) $\displaystyle
d\mathbb{P}_{N}\Big{(}S_{n-1},U_{n}\big{|}\Big{)}=\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle\qquad\qquad\rho_{n-1}\left(S_{n-1}\right)\times\phi_{n}\left(U_{n}+Q_{1}^{t}S^{t}_{n-1}Q_{2}^{t}\times
U_{n-1}^{-1}\times Q_{2}S_{n-1}Q_{1}\right)\times dS_{n-1}\times dU_{n}$

$(3)$ For any $A>0$

$\displaystyle\mathbb{P}_{N}\left\\{\left\|S_{n-1}\right\|>A\right\\}\leq
Ce^{-c\kappa^{2}W},$

$(4)$ Denote:

$\displaystyle\textsf{V}_{n}=\textsf{V}_{n}\left(S_{n-1};U_{n-1},U_{n}\right)=U_{n}+Q_{1}^{t}S^{t}_{n-1}Q_{2}^{t}\times
U_{n-1}^{-1}\times Q_{2}S_{n-1}Q_{1},$
$\displaystyle{\mathcal{B}}_{S,n-1,A}=\left\\{\left\|S_{n-1}\right\|>A\right\\},\quad{\mathcal{B}}_{V,n,A}=\left\\{\left\|V_{n}\right\|>A\right\\},$
$\displaystyle{\mathcal{B}}_{S,n-1,n,A}:={\mathcal{B}}_{S,n-1,n,A,\big{|}U_{n-1}}={\mathcal{B}}_{S,n-1,A}\cup\left\\{\left(S_{n-1},U_{n}\right):\textsf{V}_{n}\in{\mathcal{B}}_{V,n,A}\right\\}$

Then

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{S,n-1,n,A,\big{|}U_{n-1}}\big{|}U_{n-1}\right]\leq
2Ce^{-cA^{2}W}$

$(5)$ There exists a set
${\mathcal{B}}_{U,n-1,n,A}\in\mathcal{F}\left(U_{n-1},U_{n}\right)$ such that

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{U,n-1,n,A}\right]\leq
C_{1}e^{-c_{1}A^{2}W},$
$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{S,n-1,n,A}\big{|}U_{n-1},U_{n}\right]\leq
C_{1}e^{-c_{1}A^{2}W},\quad\text{for any
$\left(U_{n-1},U_{n}\right)\notin{\mathcal{B}}_{U,n-1,n,A}$},$

with absolute constants $c_{1},C_{1}$.

$(6)$ Use the notations in $(5)$. Denote:

$\displaystyle\widetilde{\textsf{V}}_{n}:=\widetilde{\textsf{V}}_{n}\left(S_{n-1};U_{n-1},U_{n}\right)=Q_{1}U_{n}Q_{1}^{t}+S^{t}_{n-1}Q_{2}^{t}\times
U_{n-1}^{-1}\times Q_{2}S_{n-1},$
$\displaystyle\widetilde{{\mathcal{B}}}_{S,n-1,n,A}={\mathcal{B}}_{S,n-1,A}\cup\left\\{\left\|\widetilde{\textsf{V}}_{n}\right\|>A\right\\},$

Then

$\displaystyle\mathbb{P}_{N}\left[\widetilde{{\mathcal{B}}}_{S,n-1,n,A}\big{|}U_{n-1},U_{n}\right]\leq
C_{1}e^{-c_{1}A^{2}W},\quad\text{for any
$\left(U_{n-1},U_{n}\right)\notin{\mathcal{B}}_{U,n-1,n,A}$}$

###### Proof.

$(0)$ The estimates follow from Proposition 4.3 parts $(ii)$ and $(iv)$
respectively.

$(1)$ By the definition $U_{n-1}=V_{n-1}-T^{t}_{n-2}\times U_{n-2}^{-1}\times
T_{n-2}$. Proposition 3.1 says that the randdom variables $V_{n-1}$ and
$T^{t}_{n-2}\times U_{n-2}^{-1}\times T_{n-2}$ are independent. Take $H=V+A$,
where $V$ is GOE distributed, and $A=-T^{t}_{n-2}\times U_{n-2}^{-1}\times
T_{n-2}$. Apply (4.8), get the statement.

$(2)$ Lemma A.2 says that the Jacobian of the variable change
$T_{n-1}=S_{n-1}Q$ is $=1$, including the case when $Q$ depends on the rest of
variables. From the definitions in $(3)$ follows
$S_{n-1}\in\mathcal{F}\left(U_{n-1},T_{n-1},U_{n}\right)$. Combine the change
of variables distribution density rule with Jacobian $=1$, with the equations
in (4.6), and with invariance of $\rho_{n-1}$ under the change of variable in
question, get the statements in $(3)$.

$(3)$ From the definition $T_{n-1}=Q_{2}S_{n-1}Q_{1}$. Therefore
$\left\|T_{n-1}\right\|=\left\|S_{n-1}\right\|$ and the statement follows from
part $(1)$.

$(4)$ Use (4.9), write

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}\big{|}U_{n-1}\right]=\int_{{\mathcal{B}}}d\mathbb{P}_{N}\Big{(}S_{n-1},U_{n}\big{|}U_{n-1}\Big{)}=$
$\displaystyle\int_{{\mathcal{B}}}\rho_{n-1}\left(S_{n-1}\right)\times\phi_{n}\left(U_{n}+Q_{1}^{t}S^{t}_{n-1}Q_{2}^{t}\times
U_{n-1}^{-1}\times Q_{2}S_{n-1}Q_{1}\right)\times dT_{n-1}\times dU_{n}$

In this dispaly ${\mathcal{B}}:={\mathcal{B}}_{U,S,n,A,Q\big{|}U_{n-1}}$ to
save space. In the display second line integral change back the variable
$S_{n-1}=Q_{2}^{t}T_{n-1}Q_{1}^{t}$. Combine the integration change of
variables rule with Jacobian $=1$ and with invariance of $\rho_{n-1}$, get

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{U,S,n,A,Q\big{|}U_{n-1}}\big{|}U_{n-1}\right]=$
$\displaystyle\int_{{\mathcal{B}}_{U,T,n,A\big{|}U_{n-1}}}\rho_{n-1}\left(T_{n-1}\right)\times\phi_{n}\left(U_{n}+T^{t}_{n-1}\times
U_{n-1}^{-1}\times T_{n-1}\right)\times dT_{n-1}\times dU_{n}$

Denote:

$\displaystyle{\mathcal{B}}_{U,T,n,A,Q\big{|}U_{n-1}}=\left\\{\Big{(}T_{n-1},U_{n}\Big{)}:U_{n}+T^{t}_{n-1}\times
U_{n-1}^{-1}\times T_{n-1}\in{\mathcal{B}}_{V,n,\kappa}\right\\}$

In the last integral change the variable $U_{n}+T^{t}_{n-1}\times
U_{n-1}^{-1}\times T_{n-1}=V_{n}$ and keep the variable $T_{n-1}$ intact. The
Jacobian of this change of the variables is $=1$. From the definitions the
image of ${\mathcal{B}}_{U,T,n,\kappa,Q\big{|}U_{n-1}}$ under this change of
variables is ${\mathcal{B}}_{V,n,\kappa}$. Write

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{U,S,n,A,Q\big{|}U_{n-1}}\big{|}U_{n-1}\right]=\int_{{\mathcal{B}}_{V,n,A}}\rho_{n-1}\left(T_{n-1}\right)\times\phi_{n}\left(V_{n}\right)\times
dT_{n-1}\times dV_{n}$
$\displaystyle\leq\mathbb{P}_{N}\left[{\mathcal{B}}_{S,n-1,A}\right]+\mathbb{P}_{N}\left[{\mathcal{B}}_{V,n,A}\right]\leq
2Ce^{-cA^{2}W}$

Here the last inequality in the display comes from part $(1)$.

$(5)$ From the definitions

$\displaystyle\left[{\mathcal{B}}_{S,n-1,n,A}\right]_{\big{|}U_{n-1},U_{n}}={\mathcal{B}}_{S,n-1,n,A,\big{|}U_{n-1}},$

where ${\mathcal{B}}_{\big{|}\cdot}$ stands for the cross-section of the set
${\mathcal{B}}$ at a given point $\cdot$ from the respective variable domain.
Combine, use the standard disintegration argument with help of Chebyshev
inequality, get the statement.

$(6)$ Use

$\displaystyle\widetilde{\textsf{V}}_{n}=Q_{1}\textsf{V}_{n}Q^{t}_{1},\quad\big{\|}\widetilde{\textsf{V}}_{n}\big{\|}=\left\|\textsf{V}_{n}\right\|,$

combine with $(5)$, get the statements in $(6)$.

∎

###### Proof of Proposition 4.1.

$(0)$ Use the notations in part $(3)$ of Proposition 3.1. The spherical
variables change Jacobian is $=r^{W-1}$. Combine the density change of
variables rule with distribution
$d\mathbb{P}\left(\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t}\big{|}\right)$
equations from part $(3)$ of Proposition 3.1, compute the joint distribution
$d\mathbb{P}\left(r,\textsf{x}\right)$. Use conditional distribution rule, get
the statements in $(0)$.

$(1)$ Proposition 3.1, $(1)$ says that for any $U_{n-2}$ holds

$\displaystyle\mathbb{P}_{N}\left(\left\\{\big{\|}U^{-1}_{n-1}\big{\|}>W^{D}\right\\}|U_{n-2}\right)\leq
CW^{-\left(D-1\right)},$

It is convenient to denote
$\widehat{{\mathcal{B}}}_{U,n-2,n-1,D,A}\in\mathcal{F}\left(U_{n-1},U_{n}\right)$
the set of pairs $\left(U_{n-2},U_{n-1}\right)$ in the display, though
$U_{n-2}$ is actually arbitrary . Use the definitions, get

$\displaystyle\left\|C\right\|=\left\|Q_{2}^{t}\times U_{n-1}^{-1}\times
Q_{2}\right\|=\left\|U_{n-1}^{-1}\right\|\leq CW^{-\left(D-1\right)},$

for any
$\left(U_{n-2},U_{n-1}\right)\notin\widehat{{\mathcal{B}}}_{U,n-2,n-1,D,A}$.
Use Proposition 4.4, $(6)$

$\displaystyle\mathbb{P}_{N}\left\\{\left\|S_{n-1}\right\|>A\right\\}\leq
C_{1}e^{-c_{1}A^{2}W}$

with absolute constants $c_{1},C_{1}$. Use the definitions of
$\textsf{a}^{2}\left(\textsf{x}\right),\textsf{b}\left(\textsf{x}\right),\textsf{c}\left(\textsf{x}\right)$,
combine, get the first statement in $(1)$. For any symmetric matrix $M\neq 0$
holds

$\displaystyle{\rm mes}_{n-1}\left\\{\textsf{x}\in\mathbb{S}^{n-1}:\langle
Mx,x\rangle=0\right\\}=0,$

where ${\rm mes}_{n-1}$ stands for the normalized Lebesgue measure on the unit
sphere $\mathbb{S}^{n-1}$. It follows from the definitions that $C$ is
actually invertible with probability $1$. That implies the second statement in
$(1)$.

$(2)$ Use Proposition 4.4, $(6)$: for any $A>0$ there exists a set
${\mathcal{B}}_{U,n-1,n,A}\in\mathcal{F}\left(U_{n-1},U_{n}\right)$, such that

$\displaystyle\mathbb{P}_{N}\left[{\mathcal{B}}_{U,n-1,n,A}\right]\leq
C_{1}e^{-c_{1}A^{2}W},$
$\displaystyle\mathbb{P}_{N}\left[\widetilde{{\mathcal{B}}}_{S,n-1,n,A}\big{|}U_{n-1},U_{n}\right]\leq
C_{1}e^{-c_{1}A^{2}W},\quad\text{for any
$\left(U_{n-1},U_{n}\right)\notin{\mathcal{B}}_{U,n-1,n,A}$},$

with absolute constants $c_{1},C_{1}$, where

$\displaystyle\widetilde{{\mathcal{B}}}_{S,n-1,n,A}:=\left\\{\left\|S_{n-1}\right\|>A\right\\}\cup\left\\{\left\|\widetilde{\textsf{V}}_{n}\right\|>A\right\\},\qquad\qquad\qquad\qquad$
(4.10)
$\displaystyle\widetilde{\textsf{V}}_{n}:=\widetilde{\textsf{V}}_{n}\left(S;U_{n-1},U_{n}\right)=Q_{1}U_{n}Q_{1}^{t}+S^{t}Q_{2}^{t}\times
U_{n-1}^{-1}\times Q_{2}S$

Redenote

(4.11)
$\displaystyle{\mathcal{B}}_{{\mathcal{X}},n-1,n,A}=\widetilde{{\mathcal{B}}}_{\big{|}U_{n-1},U_{n}}$

Here, as before, ${\mathcal{B}}_{\big{|}\cdot}$ stands for the cross-section
of the set ${\mathcal{B}}$ at a given point $\cdot$ from the respective
variable domain. The probability estimates in part $(6)$ hold for these sets.
Verify the equation in part $(5)$ of the proposition with these sets in place
and with

$\left(U_{n-1},U_{n}\right)\notin{\mathcal{B}}_{U,n-1,n,A},\quad
X\notin{\mathcal{B}}_{{\mathcal{X}},n-1,n,A}$

From the definition the vector
$X:=\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t}$ is the first column of $S$.
Therefore,

(4.12) $\displaystyle
r=\left\|\left(\xi_{1,1},\tilde{X}^{t}_{1}\right)^{t}\right\|\leq\left\|S\right\|\leq
A,$

as claimed. Similarly,

(4.13)
$\displaystyle\left\|\widetilde{\textsf{V}}_{n}e_{1}\right\|\leq\left\|\widetilde{\textsf{V}}_{n}\right\|\leq
A$

From the definitions in part $(2)$

(4.14)
$\displaystyle\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle=\langle
CX,X\rangle$

Next, use equation (3.16) from part $(2)$ proof, combine with the definition
of $\widetilde{\textsf{V}}_{n}$, see above, write

$\displaystyle\widetilde{\textsf{V}}_{n}=Q_{1}U_{n}Q_{1}^{t}+S_{n-1}^{t}\times
U_{n-1}^{-1}\times S_{n-1}=B+S_{n-1}^{t}\times U_{n-1}^{-1}\times S_{n-1}=$
$\displaystyle\left(b_{i,j}\right)_{1\leq i,j\leq
W}+\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle&\lambda_{1}\xi_{1,1}\tilde{Y}_{1}+\left(\check{T}^{t}\check{C}\tilde{X}_{1}\right)^{t}\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{T}^{t}\check{C}\tilde{X}_{1}&\lambda_{1}\tilde{Y}^{t}_{1}\times\tilde{Y}_{1}+\check{T}^{t}\check{C}\check{T}\end{bmatrix}$

In particular,

(4.15)
$\displaystyle\widetilde{\textsf{V}}_{n}e_{1}=\left(b_{i,1}\right)_{1\leq
i,j\leq W}+\begin{bmatrix}\langle CX,X\rangle\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{T}^{t}C\tilde{X}_{1}\end{bmatrix}$

Use the equations in part $(0)$, combine with (4.12),(4.14), (4.15), get

$\displaystyle
b_{1,1}=\langle\widetilde{\textsf{V}}_{n}e_{1},e_{1}\rangle-\langle
CX,X\rangle,$ $\displaystyle\gamma\left(\textsf{x}\right)r^{2}=b_{1,1}\langle
C\textsf{x},\textsf{x}\rangle r^{2}=b_{1,1}\langle
CX,X\rangle=\langle\widetilde{\textsf{V}}_{n}e_{1},e_{1}\rangle\langle
CX,X\rangle-\langle CX,X\rangle^{2}=$
$\displaystyle\sqrt{2}\langle\widetilde{\textsf{V}}_{n}e_{1},e_{1}\rangle\textsf{a}r^{2}-2\textsf{a}^{2}r^{4},$
$\displaystyle\big{|}\gamma\left(\textsf{x}\right)r^{2}+2\textsf{a}^{2}\left(\textsf{x}\right)r^{4}\big{|}=\sqrt{2}\big{|}\langle\widetilde{\textsf{V}}_{n}e_{1},e_{1}\rangle\big{|}\textsf{a}\left(\textsf{x}\right)r^{2}\leq\sqrt{2}A\textsf{a}\left(\textsf{x}\right)r^{2},$
$\displaystyle\left(\langle\widetilde{\textsf{V}}_{n}e_{1},e_{1}\rangle\right)_{2\leq
i\leq
W}=\tilde{B}_{1}+\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{T}^{t}C\tilde{X}_{1},$
$\displaystyle
r^{2}\beta^{2}\left(\textsf{x}\right)=r^{2}\left\|\lambda_{1}\textsf{x}_{1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{\textsf{x}}\right\|^{2}=\left\|\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\right\|^{2},$
$\displaystyle\textsf{c}\left(\textsf{x}\right)r=2r\langle\lambda_{1}\textsf{x}_{1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{\textsf{x}},\tilde{B}_{1}\rangle=2\langle\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1},\tilde{B}_{1}\rangle=$
$\displaystyle
2\langle\left(\langle\widetilde{\textsf{V}}_{n}e_{1},e_{i}\rangle\right)_{2\leq
i\leq
W},\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\rangle-2\left\|\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\right\|^{2}=$
$\displaystyle
2\langle\left(\langle\widetilde{\textsf{V}}_{n}e_{1},e_{i}\rangle\right)_{2\leq
i\leq
W},\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\rangle-2\beta^{2}\left(\textsf{x}\right)r^{2},$
$\displaystyle\left|\textsf{c}\left(\textsf{x}\right)r+2\beta^{2}\left(\textsf{x}\right)r^{2}\right|=2\left|\langle\left(\langle\widetilde{\textsf{V}}_{n}e_{1},e_{i}\rangle\right)_{2\leq
i\leq
W},\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\rangle\right|\leq$
$\displaystyle
2\left\|\widetilde{\textsf{V}}_{n}\right\|\times\left\|\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\right\|\leq
2A\beta\left(\textsf{x}\right)r$

The verification of the equations

$\displaystyle\textsf{a}r^{2}<A+\left\|B_{1}\right\|,$
$\displaystyle|\gamma|r^{2}\leq\left\|B_{1}\right\|\left(A+\left\|B_{1}\right\|\right),$
$\displaystyle\beta^{2}r^{2}\leq\left(A+\left\|B_{1}\right\|\right)^{2},$
$\displaystyle|b|r^{2}\leq
A^{2}+\left(A+\left\|B_{1}\right\|\right)^{2}+\left\|B_{1}\right\|\left(A+\left\|B_{1}\right\|\right),$
$\displaystyle|\textsf{c}|r\leq
2\left\|B_{1}\right\|\left(A+\left\|B_{1}\right\|\right)$

is similar and actually a bit shorter. We skip it. That finishes the proof of
part $(2)$.

$(3)$ Use part $(2)$, combine with the standard disintegration argument and
with Chebyshev inequality, get the statements in $(3)$.

$(4)$ Use the definitions, get

$\displaystyle\textsf{a}^{2}={1\over 2}\langle
C\textsf{x},\textsf{x}\rangle^{2}\leq{1\over
2}\left\|C\textsf{x}\right\|^{2}={1\over
2}\left\|X\right\|^{-2}\left\|CX\right\|^{2}={1\over 2}\zeta^{2}$

Next, use

$\displaystyle
S^{t}CX=\begin{bmatrix}\lambda_{1}\xi^{2}_{1,1}+\langle\check{C}\tilde{X}_{1},\tilde{X}_{1}\rangle\\\
\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\end{bmatrix},$

see Proposition 3.1, part $(3)$. Use the definitions, combine with the
equation (4.12), write

$\displaystyle\beta^{2}r^{2}=\left\|\lambda_{1}\xi_{1,1}\tilde{Y}^{t}_{1}+\check{S}^{t}\check{C}\tilde{X}_{1}\right\|^{2}\leq\left\|S^{t}CX\right\|^{2}\leq
A^{2}\left\|CX\right\|^{2}=A^{2}\left\|\textsf{Z}\right\|^{2}=A^{2}\zeta^{2}r^{2},$
$\displaystyle\beta^{2}\leq A^{2}\zeta^{2}$

as claimed. Finally, take
$\textsf{x}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-1,n,A}$,
$r\notin{\mathcal{B}}_{\texttt{r},n-1,n,A}$ as in $(3)$, write

$\displaystyle\big{|}\gamma r^{2}\big{|}\leq\big{|}\gamma
r^{2}+2\textsf{a}^{2}r^{4}\big{|}+2\textsf{a}^{2}r^{4}\leq\sqrt{2}A\textsf{a}r^{2}+2\textsf{a}^{2}r^{4}\leq
A\zeta r^{2}+\zeta^{2}r^{4},$ $\displaystyle\left|\textsf{c}\right|r\leq
2\beta^{2}r^{2}+\left|\textsf{c}r+2\beta^{2}r^{2}\right|\leq
2A^{2}\zeta^{2}r^{2}+2A\zeta r$

Use that $r\leq A$ if $r\notin{\mathcal{B}}_{\texttt{r},n-1,n,A}$, conclude

$\displaystyle\big{|}\gamma\big{|}\leq A\zeta+\zeta^{2}r^{2}\leq
A\zeta+A^{2}\zeta^{2},$ $\displaystyle\left|\textsf{b}\right|\leq
1+\beta^{2}+\big{|}\gamma\big{|}\leq
1+A^{2}\zeta^{2}+A\zeta+A^{2}\zeta^{2}=1+A\zeta+2A^{2}\zeta^{2},$
$\displaystyle\left|\textsf{c}\right|\leq 2A^{3}\zeta^{2}+2A\zeta,$

as claimed. Since $r$ does not enter the estimates in the last display the
estimates hold without condition
$r\notin{\mathcal{B}}_{\texttt{r},\textsf{x},n-1,n,A}$.

$(5)$ Rewrite the recurrence equation (3.12) from Proposition 3.1 part $(4)$
as follows

$\displaystyle\left\|\textsf{B}_{n-2}\left(\textsf{y}\right)\right\|=\left\|X_{1,n-1}\right\|\left\|\textsf{Z}_{n-1}\right\|^{-1}=\zeta_{n-1}^{-1}\left(\textsf{x}\right)$

Take

$\displaystyle\textsf{y}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-2,n-1,A},\quad\rho\notin{\mathcal{B}}_{\texttt{r},\textsf{x},n-2,n-1,A}$

Use the part $(2)$ last display estimates with $(n-2)$ in the role of $(n-1)$.
Use $\rho$ for the notation for the sperical radius and y for the spherical
variable. Take

$\displaystyle\textsf{y}\notin\widehat{{\mathcal{B}}}_{\texttt{x},n-2,n-1,A},\quad\rho\notin{\mathcal{B}}_{\texttt{r},\textsf{x},n-2,n-1,A},$

write

$\displaystyle\textsf{a}_{n-2}\left(\textsf{y}\right)\rho^{2}<A+\left\|\textsf{B}_{n-2}\left(\textsf{y}\right)\right\|,$
$\displaystyle\beta_{n-2}^{2}\left(\textsf{y}\right)\rho\leq\left(A+\left\|\textsf{B}_{n-2}\left(\textsf{y}\right)\right\|\right)^{2},$
$\displaystyle|\textsf{c}_{n-2}\left(\textsf{y}\right)|\rho\leq
2\left\|\textsf{B}_{n-2}\left(\textsf{y}\right)\right\|\left(A+\left\|\textsf{B}_{n-2}\left(\textsf{y}\right)\right\|\right)$

If

$\zeta_{n-1}\left(\textsf{x}\right)>\textsf{C},$

then

$\displaystyle\left\|\textsf{B}_{n-2}\left(\textsf{y}\right)\right\|<\textsf{C}^{-1},$
$\displaystyle\textsf{a}_{n-2}\left(\textsf{y}\right)\rho^{2}<A+\textsf{C}^{-1},$
$\displaystyle|\gamma_{n-2}\left(\textsf{y}\right)|\rho^{2}\leq\textsf{C}^{-1}\left(A+\textsf{C}^{-1}\right),$
$\displaystyle\beta^{2}_{n-2}\left(\textsf{y}\right)\rho^{2}\leq\left(A+\textsf{C}^{-1}\right)^{2},$
$\displaystyle\textsf{b}_{n-2}\left(\textsf{y}\right)\rho^{2}\leq
A^{2}+2\left(A+\textsf{C}^{-1}\right)^{2},$
$\displaystyle|\textsf{c}_{n-2}\left(\textsf{y}\right)|\rho\leq
2\textsf{C}^{-1}\left(A+\textsf{C}^{-1}\right)$

_regardless how large is C_. Now we got to verify that dichotomy for
$\textsf{b}:=\textsf{b}_{n-2}\left(\textsf{y}\right)$. Apply the current
proposition part $(0)$ and Proposition 2.2 part $(2)$ with $(n-2)$ in the role
$(n-1)$. It is convenient though to consider $(n-1)$ instead of $(n-2)$ That
way we just use the same equations without any comments. Use

$\displaystyle\textsf{a}^{2}\left(\textsf{x}\right)={1\over 2}\langle
C\textsf{x},\textsf{x}\rangle^{2},$
$\displaystyle\textsf{b}=\textsf{b}\left(\textsf{x}\right)=1+\beta^{2}+\gamma,$
$\displaystyle\gamma\left(\textsf{x}\right)=b_{1,1}\langle
C\textsf{x},\textsf{x}\rangle,$
$\displaystyle|\gamma\left(\textsf{x}\right)|=|b_{1,1}|\left|\langle
C\textsf{x},\textsf{x}\rangle\right|<\textsf{C}^{-1}\left|\langle
C\textsf{x},\textsf{x}\rangle\right|=\textsf{C}^{-1}\sqrt{2}|\textsf{a}\left(\textsf{x}\right)|$

see part $(0)$ in the current proposition. If
$\textsf{b}\left(\textsf{x}\right)>0$, then we are done. Assume
$\textsf{b}\left(\textsf{x}\right)\leq 0$. Then the last dispay implies

$\displaystyle
1+\beta^{2}<|\gamma|<\textsf{C}^{-1}\sqrt{2}|\textsf{a}\left(\textsf{x}\right)|,$
$\displaystyle|\textsf{b}\left(\textsf{x}\right)|\leq
1+\beta^{2}+|\gamma\left(\textsf{x}\right)|<2|\gamma\left(\textsf{x}\right)|<2\textsf{C}^{-1}\sqrt{2}|\textsf{a}\left(\textsf{x}\right)|,$

as claimed.

∎

##  5\. Concentration Estimates and Log-Variance of Spherical Radius Against
Special Super-Exponential Densitities.

The proof of Theorem B is split into few proposition. Each of these
propositions targets a specific technological problem which enter our method.
The first target consists of concentration estimates and log-variance of
spherical radius against super-exponential densitities from Proposition 4.1,
part 4,

(5.1) $\displaystyle d\mathbb{P}(r)=\mu
r^{W-1}e^{-W\left(\textsf{a}^{2}r^{4}+\textsf{b}r^{2}+\textsf{c}r\right)}dr:=\mu\phi(r)dr,\quad
r\geq 0$

where a, b, c are constant coefficients, $\textsf{a}^{2}>0$,
$\mu=\mu\left(\textsf{a},\textsf{b},\textsf{c}\right)$ is the normalizing
factor. Actually, in Proposition 4.1 the coefficients a, b , c _depend on the
spherical variable $\textsf{x}\in\mathbb{S}^{W-1}$_. The coefficients depend
also on some other variables too. On the other hand Proposition 4.1 says that
off a set of very small probability $($ the actual estimate of the set in
question will play an important role in Section 6, but not in the curent
section$)$, the coefficient obey some very important estimates. In the current
section we list some of these estimates as the main conditions under which the
result of the section hold.

Here is the list of conditions in question.

$(I)$ Crude upper bound estimates:

$\displaystyle\max\left(\textsf{a}^{2},\left|\textsf{b}\right|,\left|\textsf{c}\right|\right)\leq
2A^{2}W^{2D},$

with absolute constants $A,D$

$(II)$ The distribution ”essential” support size estimate

(5.2) $\displaystyle\mathbb{P}\left\\{r\geq A\right\\}\leq
C_{0}e^{-c_{0}A^{2}W},$

with absolute constants $c_{0},C_{0}$. Here $\mathbb{P}$ stands for the
probability distribution in(5.1).

$(III)$ The dichotomy conditions. Either

(5.3)
$\displaystyle\max\left(\textsf{a}^{2},\left|\textsf{b}\right|,\left|\textsf{c}\right|\right)\leq\textsf{A}$

with absolute constant A, or there exists a set ${\mathcal{B}}_{\texttt{r}}$

$\displaystyle\mathbb{P}{\mathcal{B}}_{\texttt{r}}\leq C_{0}e^{-c_{0}A^{2}W},$

such that for any $r\notin{\mathcal{B}}_{\texttt{r}}$ holds

(5.4)
$\displaystyle\max\left(\textsf{a}^{2}r^{4},\textsf{b}r^{2},\left|\textsf{c}\right|r\right)\leq\textsf{A}$

Furthermore, in this case, either

(5.5) $\displaystyle\textsf{b}>0,$

or

(5.6) $\displaystyle|\textsf{b}|<\textsf{C}^{-1}|\textsf{a}|$

The main result of this section is the following proposition

######  Proposition 5.1.

Assume that conditions $(I)-(III)$ hold. Then

(5.7) $\displaystyle{\mathrm{Var}}\log r\geq cW^{-1}$

with absolute constant $c>0$. Furthemore, there exists a set
${\mathcal{B}}_{\phi}\subset(0,+\infty)$ such that

$\displaystyle\mathbb{P}{\mathcal{B}}_{\phi}\leq
Ce^{-cW}\qquad\qquad\qquad\qquad$ (5.8) $\displaystyle r\geq
c_{1}W^{-\textsf{d}},\quad\text{for any
$r\in(0,+\infty)\setminus{\mathcal{B}}_{\phi}$},$

where $c,c_{1},C,\textsf{d}$ are absolute constants.

The proof of Proposition 5.1 is a simple corollary of the Proposition 5.2
below. The latter proposition establishes a detailed analysis of the
concentration estimates addressing several different narrow intervals of size
$\sim W^{-{1\over 2}}$ of the random variable $r$ values. The final result of
the analysis itself is a combination of a very long list of elementary
calculus estimates, which address integrals of the density $\phi(r)$ over
specific intervals where $\phi$ is monotonic.

######  Proposition 5.2.

Assume that conditions $(I)-(III)$ hold. Denote

$\displaystyle
f\left(r\right)=W^{-1}\log\phi\left(r\right)=\qquad\qquad\qquad\qquad$ (5.9)
$\displaystyle W^{-1}\log\mu+\left(1-W^{-1}\right)\log
r-\textsf{a}^{2}r^{4}-\textsf{b}r^{2}-\textsf{c}r,\quad r\geq 0$

$(i)$ There exist $0<r_{1}<2A$ and $\delta_{1}>0$, such that
$f^{\prime}\left(r\right)\geq 0$ for $0\leq r\leq r_{1}$, and
$f^{\prime}\left(r\right)<0$ for $r_{1}<r\leq r_{1}+\delta_{1}$. In
particular, $f^{\prime}\left(r_{1}\right)=0$,
$f^{\prime\prime}\left(r_{1}\right)<0$. The estimates

$\displaystyle r_{1}\geq c_{0}\min\left(\textsf{a}^{-{1\over
2}},|\textsf{b}|^{-{1\over 2}},|\textsf{c}|^{-1}\right)\geq cW^{-\textsf{d}}$

hold with absolute constants $c,\textsf{d}$.

$(ii)$ Assume that $\textsf{b}>0$. Then $r_{1}$ is the only critical point of
$f$ on the interval $[0,2A]$.

$(iii)$ There might be at most one point $0<r_{1}<r_{3}$ with
$f^{\prime}\left(r_{3}\right)=0$, $f^{\prime\prime}\left(r_{3}\right)\leq 0$.
If $r_{3}$ exists, then $(iiia)$ $f^{\prime\prime}\left(r_{3}\right)<0$,
$(iiib)$ $f(r)$ has a local minimum at $r_{1}<r_{2}<r_{3}$ and no other
critical points on $(0,+\infty)$, $(iiic)$ $f^{\prime\prime}(r)$ has exactly
two zeros, both sitting in $[r_{1},r_{3}]$ and no other positive zeros.

$(iv)$ Take $r_{j}$, $j\in\\{1,3\\}$ and $\eta>-1$. The following equation
holds

$\displaystyle
f\left(r_{j}\left(1+\eta\right)\right)-f\left(r_{j}\right)=\qquad\qquad\qquad\qquad$
(5.10)
$\displaystyle\left(1-\nu\right)\log\left(1+\eta\right)-\left(1-\nu\right)\eta-\left(6\textsf{a}^{2}r_{j}^{4}+\textsf{b}r_{1}^{2}\right)\eta^{2}-4\textsf{a}^{2}r_{j}^{4}\eta^{3}-\textsf{a}^{2}r_{j}^{4}\eta^{4}$

In particular, for $|\eta|\leq{1\over 4}$

$\displaystyle\exp W\left[-{2\over
3}\left(1-\nu\right)\eta^{2}-6\textsf{a}^{2}r_{j}^{4}\left(1+{3\eta\over
2}\eta+{\eta^{2}\over
6}\right)\eta^{2}-\textsf{b}r_{j}^{2}\eta^{2}\right]\times$ (5.11)
$\displaystyle\phi\left(r_{j}\right)\leq\phi\left(r_{j}\left(1+\eta\right)\right)\leq\phi\left(r_{j}\right)\times\qquad\qquad$
$\displaystyle\exp W\left[-{2\over
5}\left(1-\nu\right)\eta^{2}-6\textsf{a}^{2}r_{j}^{4}\left(1+{3\eta\over
2}+{\eta^{2}\over 6}\right)\eta^{2}-\textsf{b}r_{j}^{2}\eta^{2}\right]\quad$

$(v)$ If $\textsf{b}\geq 0$, then the unction $f(r)$ decreases monotonically
on each of the intervals $({3\over 4}r_{1},r_{1})$ and $(r_{1},{5\over
4}r_{1})$. If $\textsf{b}<0$, then for $|\eta|\leq{1\over 4}$

(5.12)
$\displaystyle\qquad\exp\left[-W\textsf{A}_{1}\eta^{2}\right]\times\phi\left(r_{j}\right)\leq\phi\left(r_{j}\left(1+\eta\right)\right)\leq\phi\left(r_{j}\right)\times\exp\left[W\textsf{A}_{1}\eta^{2}\right]\quad$

with absolute constant $\textsf{A}_{1}>0$.

$(vi)$ Assume $\textsf{b}\geq 0$. Then for $|\eta|\leq{1\over 8}$ holds

$\displaystyle{\phi\left(r_{1}\left(1+2\eta\right)\right)\over\phi\left(r_{1}\left(1+\eta\right)\right)}<\exp
W\left[-{3\over
5}\left(1-\nu\right)\eta^{2}-\textsf{a}^{2}r_{1}^{4}\eta^{2}-\textsf{b}r_{1}^{2}\eta^{2}\right],$

$(vii)$ Assume $\textsf{b}\geq 0$. There exists unique
$0<\phi_{\textsf{m}}<\phi(r_{1})$, such that

$\displaystyle\int_{\phi(r)<\phi_{\textsf{m}}}\phi\left(r\right)dr=\int_{\phi(r)>\phi_{\textsf{m}}}\phi\left(r\right)dr$

The set $I_{\textsf{m}}=\\{\phi(r)>\phi_{\textsf{m}}\\}$ is an interval,
$I_{\textsf{m}}:=(r_{1,<},r_{1,>})$.

$(viii)$ Assume $\textsf{b}\geq 0$. Then

(5.13)
$\displaystyle\max\left[\textsf{a}^{2}r_{1}^{4},\textsf{b}r_{1}^{2}\right]<8A^{2}$

$(ix)$ Use the notations in $(vii)$. Assume that

$\displaystyle\int_{r_{1}<r<r_{1,>}}\phi\left(r\right)dr\geq\int_{r_{1,<}<r<r_{1}}\phi\left(r\right)dr$

Given $0<\tau<1$ there exists unique $r_{1}<r_{>|\tau}<r_{1,>}$, such that

$\displaystyle\int_{r_{1}<r<r_{>|\tau}}\phi\left(r\right)dr=\tau\int_{r_{1}<r<r_{1,>}}\phi\left(r\right)dr$

Furthermore,

$\displaystyle r_{>|{j+1\over 5}}>r_{>|{j\over
5}}\left(1+{c_{j}\over\sqrt{W}}\right),\quad j=0,1,2,3$

where $c_{j}>0$ are absolute constants.

A completely similar statement holds in case

$\displaystyle\int_{r_{1,<}<r<r_{1}}\phi\left(r\right)dr\geq\int_{r_{1}<r<r_{1,>}}\phi\left(r\right)dr$

with the interval $(r_{1,<},r_{1})$ in the role of $(r_{1},r_{1,>})$ and
$-\eta$ in the role of $\eta$. We denote the respective points as
$r_{<|{j+1\over 5}}<r_{<|{j\over 5}}$, $j=0,1,2,3$ and the main part of the
statement is that

$\displaystyle r_{>|{j+1\over 5}}<r_{>|{j\over
5}}\left(1-{c_{j}\over\sqrt{W}}\right),\quad j=0,1,2,3$

In particular, the statement implies that $r_{1}-r_{1,<}\geq
r_{1}\left(1-{c\over\sqrt{W}}\right)$ with an absolute constant $c>0$.

$(x)$ Assume $\textsf{b}<0$. Assume also there is one local maximum
$r_{1}<2A$. Assume that

$\displaystyle\int_{0<r<r_{1}}\phi\left(r\right)dr\leq\int_{r_{1}<r<+\infty}\phi\left(r\right)dr$

Given $0<\tau<1$ there exists unique $r_{1}<r_{>|\tau}$, such that

$\displaystyle\int_{r_{1}<r<r_{>|\tau}}\phi\left(r\right)dr=\tau\int_{r_{1}<r<+\infty}\phi\left(r\right)dr$

Furthermore,

$\displaystyle r_{>|{j+1\over 5}}>r_{>|{j\over
5}}\left(1+{c_{j}\over\sqrt{W}}\right),\quad j=0,1,2,3$

where $c_{j}>0$ are absolute constants.

A completely similar statement holds in case

$\displaystyle\int_{0<r<r_{1}}\phi\left(r\right)dr\leq\int_{r_{1}<r<+\infty}\phi\left(r\right)dr$

$(xi)$ Assume $\textsf{b}<0$. Assume there are two local maxima $r_{1}<r_{3}$
as in part $(3)$. Assume

$\displaystyle\int_{0<r<r_{2}}\phi\left(r\right)dr\geq\int_{r_{2}<r<\infty}\phi\left(r\right)dr,$

where $r_{1}<r_{2}<r_{3}$ is the only local minimum of $f(r)$, see part
$(iii)$. Then all the statements in part $(x)$ hold. The same is true in the
complementary case

$\displaystyle\int_{0<r<r_{2}}\phi\left(r\right)dr<\int_{r_{2}<r<\infty}\phi\left(r\right)dr,$

with $r_{3}$ in the role of $r_{1}$.

###### Proof.

$(i)$ Write

$\displaystyle f\left(0\right)=-\infty=f\left(+\infty\right),$ $\displaystyle
f^{\prime}={1-W^{-1}\over
r}-\left(4\textsf{a}^{2}r^{3}+2\textsf{b}r+\textsf{c}\right)=-{4\textsf{a}^{2}r^{4}+2\textsf{b}r^{2}+\textsf{c}r-\left(1-\nu\right)\over
r},$ $\displaystyle\nu=W^{-1}\ll 1,\quad
f^{\prime}\left(0\right)=\infty=-f^{\prime}\left(+\infty\right)$

Assume $f^{\prime}\left(r\right)\geq 0$ for $0\leq r\leq 2A$. Then
$f^{\prime}\left(r\right)>0$ for $0\leq r\leq 2A$, and therefore
$\phi\left(r\right)$ increases for $0\leq r\leq 2A$. In particular,

$\displaystyle\mathbb{P}\left\\{r>A\right\\}=\int^{\infty}_{A}\phi\left(r\right)dr\geq\int^{2A}_{A}\phi\left(r\right)dr\geq\int^{A}_{0}\phi\left(r\right)dr=\mathbb{P}\left\\{r\leq
A\right\\}=$ $\displaystyle 1-\mathbb{P}\left\\{r>A\right\\},$

i.e. $\mathbb{P}\left\\{r>A\right\\}\geq{1/2}$, contrary to one of the
conditions assumed. That implies the statement. The estimate $r_{1}\geq{1\over
6\sqrt{2}}\min\left(\textsf{a}^{-{1\over 2}},|\textsf{b}|^{-{1\over
2}},|\textsf{c}|^{-1}\right)$ follows from the equation

$\displaystyle
4\textsf{a}^{2}r^{4}+2\textsf{b}r^{2}+\textsf{c}r-\left(1-\nu\right)=0$

$(ii)$ Use $\textsf{b}>0$, compute, get

$\displaystyle f^{\prime\prime}=-{\left(1-\nu\right)\over
r^{2}}-12\textsf{a}^{2}r^{2}-2\textsf{b}<0,\quad\text{ for $r>0$}$

By Rolle theorem $f^{\prime}$ can not have more than one root in
$(0,+\infty)$, as claimed.

$(iii)$ Write the equation

$\displaystyle f^{\prime\prime}=-{\left(1-\nu\right)\over
r^{2}}-12\textsf{a}^{2}r^{2}-2\textsf{b}=0$

The equation has two positive roots at most, for it is bi-quadratic. By Rolle
theorem $f^{\prime}$ can not have more than three positive roots counted with
their multiplicities. Assume there exists $0<r_{1}<r_{3}$ with
$f^{\prime}\left(r_{3}\right)=0$, $f^{\prime\prime}\left(r_{3}\right)\leq 0$.
Denote $0<r_{1}\leq r_{2}\leq r_{3}$ the point at which $f$ assumes its
minimum on $[r_{1},r_{3}]$. Then $f^{\prime}\left(r_{2}\right)=0$,
$f^{\prime\prime}\left(r_{2}\right)\geq 0$. Part $(1)$ implies, in particular,
$r_{1}<r_{2}$, for $f^{\prime\prime}\left(r_{1}\right)<0$. Assume
$f^{\prime\prime}\left(r_{2}\right)=0$. Then
$f^{\prime\prime\prime}\left(r_{2}\right)=0$, for $r_{2}$ is a local minimum.
Hence, $r_{2}$ is a root of $f^{\prime}$ of order 3 at least. Since
$f^{\prime}\left(r_{1}\right)=0$ and $r_{1}<r_{2}$, $f^{\prime}$ has at least
four roots when conted with their multiplicities. This is contrary to
conclusion we made before. Thus, $f^{\prime\prime}\left(r_{2}\right)>0$. In
particular $r_{2}<r_{3}$ and $f^{\prime}$ has at least three roots
$0<r_{1}<r_{2}<r_{3}$. Since $f$ can not have four roots this verifies
$(iiia)$, $(iiib)$ along with the $r_{3}$–uniqueness statement in $(iii)$.
Combine Rolle theorem with $(iiib)$, get $(iiic)$.

$(iv)$ Equation $f^{\prime}\left(r_{j}\right)=0$ reads

$\displaystyle
4\textsf{a}^{2}r_{j}^{4}+2\textsf{b}r_{j}^{2}+\textsf{c}r_{1}-1+\nu=0$

Take $|\eta|<1$, use (5.9), combine with the equation in the display, write

$\displaystyle f\left(r_{j}\left(1+\eta\right)\right)-f\left(r_{j}\right)=$
$\displaystyle\left(1-\nu\right)\log\left(1+\eta\right)-\textsf{a}^{2}r_{1}^{4}\left(1+\eta\right)^{4}-\textsf{b}r_{1}^{2}\left(1+\eta\right)^{2}-\textsf{c}r_{j}\left(1+\eta\right)+\textsf{a}^{2}r_{1}^{4}+\textsf{b}r_{j}^{2}+\textsf{c}r_{j}=$
$\displaystyle\left(1-\nu\right)\log\left(1+\eta\right)-\left(4\textsf{a}^{2}r_{j}^{4}+2\textsf{b}r_{j}^{2}+\textsf{c}r_{j}\right)\eta-\left(6\textsf{a}^{2}r_{j}^{4}+\textsf{b}r_{j}^{2}\right)\eta^{2}-4\textsf{a}^{2}r_{j}^{4}\eta^{3}-\textsf{a}^{2}r_{j}^{4}\eta^{4}=$
$\displaystyle\left(1-\nu\right)\log\left(1+\eta\right)-\left(1-\nu\right)\eta-\left(6\textsf{a}^{2}r_{j}^{4}+\textsf{b}r_{j}^{2}\right)\eta^{2}-4\textsf{a}^{2}r_{j}^{4}\eta^{3}-\textsf{a}^{2}r_{j}^{4}\eta^{4},$

as claimed. Combine (5.10) with the standard logarithm power series expansion,
get (5.11).

$(v)$ The statement for $\textsf{b}\geq 0$ follows from part $(ii)$. Assume
$\textsf{b}<0$. Then it is the first case in the the dichotomy conditions in
$(III)$, i.e.

(5.14)
$\displaystyle\max\left(\textsf{a}^{2},\left|\textsf{b}\right|,\left|\textsf{c}\right|\right)\leq\textsf{A}$

Combine (5.11) with (5.14), get the second statement in $(v)$.

$(vi)$ Use (5.11) with $|\eta|\leq{1\over 8}$, combine with $\textsf{b}\geq
0$, get

$\displaystyle{\phi\left(r_{1}\left(1+2\eta\right)\right)\over\phi\left(r_{1}\left(1+\eta\right)\right)}\leq\exp
W\left[-{8\over
5}\left(1-\nu\right)\eta^{2}-16\textsf{a}^{2}r_{j}^{4}\eta^{2}-\textsf{b}r_{j}^{2}\eta^{2}\right]\times$
$\displaystyle\exp W\left[{2\over
3}\left(1-\nu\right)\eta^{2}+6\textsf{a}^{2}r_{1}^{4}\eta^{2}+\textsf{b}r_{1}^{2}\eta^{2}\right]<$
$\displaystyle\exp W\left[-{3\over
5}\left(1-\nu\right)\eta^{2}-\textsf{a}^{2}r_{1}^{4}\eta^{2}-\textsf{b}r_{1}^{2}\eta^{2}\right],$

as claimed.

$(vii)$ In this part statement we assume $\textsf{b}\geq 0$. Therefore $r_{1}$
is the only critical point in the interval $(0,2A)$. The function

$\displaystyle\Pi(t)=\int_{\phi(r)<t}\phi\left(r\right)dr,\quad
0<t<\phi(r_{1})$

increases monotonically and continuos. There exists unique
$0<t:=\phi_{\textsf{m}}<\phi(r_{1})$, such that
$\Pi\left(\phi_{\textsf{m}}\right)={1\over 2}\Pi\left(\phi(r_{1})\right)$.
Since the function $\phi(r)$ has unique maximum at $r=r_{1}$, the set
$I_{\textsf{m}}=\\{\phi(r)>\phi_{\textsf{m}}\\}$ is an interval,
$I_{\textsf{m}}=(r_{1,<},r_{1,>})$, $r_{1,<}<r_{1}<r_{1,>}$. Assume
$\left|I_{\textsf{m}}\right|\geq{r_{1}\over 4}$. Then
$\max\left(r_{1}-r_{1,<},r_{1,>}-r_{1}\right)\geq{r_{1}\over 8}$. Assume first
that $r_{1}-r_{1,<}\leq r_{1,>}-r_{1}$. Denote $\eta={1\over 16}>0$,
$r_{1,k\eta}=r_{1}(1+k\eta)$, $k=1,2$. Then $r_{1,2\eta}\leq r_{1,>}$. Use
part $(v)$, write

$\displaystyle\phi\left(r_{1,>}\right)\leq\phi\left(r_{1,2\eta}\right)\leq\phi\left(r_{1,\eta}\right)\times\exp
W\left[-{3\over
5}\left(1-\nu\right)\eta^{2}-\textsf{a}^{2}r_{1}^{4}\eta^{2}-\textsf{b}r_{1}^{2}\eta^{2}\right]$

Next,

$\displaystyle\int_{r_{1}<r<r_{1,\eta}}\phi\left(r\right)dr<\int_{I_{\textsf{m}}}\phi\left(r\right)dr:=\textsf{M}_{c}=\int_{(0,+\infty)\setminus
I_{\textsf{m}}}\phi\left(r\right)dr=$
$\displaystyle\int_{0<r<r_{1,<}}\phi\left(r\right)dr+\int_{r_{1,>}<r<+\infty}\phi\left(r\right)dr:=\textsf{M}_{<}+\textsf{M}_{>}$

Assume first that $r_{1,>}<A$. Truncate the variable $r$ to the level $r=A$ in
the integral $\textsf{M}_{>}$. For that use (5.2), write

$\displaystyle 1\gg\mathbb{P}\left[{\mathcal{B}}_{\texttt{r}}\right]\geq
p_{A}:=\mathbb{P}\left\\{A<r<+\infty\right\\}={\textsf{M}_{A}\over\textsf{M}_{<}+\textsf{M}_{c}+\textsf{M}_{>}}={\textsf{M}_{A}\over
2\textsf{M}_{c}},$
$\displaystyle\textsf{M}_{A}:=\int_{A<r<+\infty}\phi\left(r\right)dr,$
$\displaystyle\textsf{M}_{c}=\textsf{M}_{<}+\textsf{M}_{>}=\textsf{M}_{<}+\textsf{M}_{>,A}+\textsf{M}_{A}=\textsf{M}_{<}+\textsf{M}_{>,A}+2p_{A}\textsf{M}_{c},$
$\displaystyle\textsf{M}_{>,A}:=\int_{r_{1,>}<r<A}\phi\left(r\right)dr,$
$\displaystyle\textsf{M}_{c}=\left(1-2p_{A}\right)^{-1}\left[\textsf{M}_{<}+\textsf{M}_{>,A}\right]<2\left[\textsf{M}_{<}+\textsf{M}_{>,A}\right],$

Combine, get

$\displaystyle\int_{r_{1}<r<r_{1,\eta}}\phi\left(r\right)dr<\left(1-2p_{A}\right)^{-1}\left[\textsf{M}_{<}+\textsf{M}_{>,A}\right]<2\left[\textsf{M}_{<}+\textsf{M}_{>,A}\right]$

On the other hand, compare

$\displaystyle\phi\left(r\right)\leq\phi\left(r_{1,>}\right)\leq\phi\left(r_{1,2\eta}\right),\quad\text{for
$r_{1,>}<r<+\infty$},$
$\displaystyle\phi\left(r\right)\leq\phi\left(r_{1,<}\right)=\phi\left(r_{1,>}\right),\quad\text{for
$0<r<r_{1,<}$},$ $\displaystyle\textsf{M}_{<}+\textsf{M}_{>,A}\leq
A\phi\left(r_{1,>}\right)\leq A\phi\left(r_{1,\eta}\right)\times\exp
W\left[-{3\over
5}\left(1-\nu\right)\eta^{2}-\textsf{a}^{2}r_{1}^{4}\eta^{2}-\textsf{b}r_{1}^{2}\eta^{2}\right]<$
$\displaystyle Ar_{1}^{-1}\exp W\left[-{3\over
5}\left(1-\nu\right)\eta^{2}-\textsf{a}^{2}r_{1}^{4}\eta^{2}-\textsf{b}r_{1}^{2}\eta^{2}\right]\int_{r_{1}<r<r_{1,\eta}}\phi\left(r\right)dr$

Combine, use the last estimate in part $(i)$, get

$\displaystyle\int_{r_{1}<r<r_{1,\eta}}\phi\left(r\right)dr<2Ar_{1}^{-1}\exp
W\left[-{3\over
8}\left(1-\nu\right)\eta^{2}-4\textsf{a}^{2}r_{1}^{4}\eta^{2}-\textsf{b}r_{1}^{2}\eta^{2}\right]\int_{r_{1}<r<r_{1,\eta}}\phi\left(r\right)dr,$
$\displaystyle\exp W\left[{3\over
5}\left(1-\nu\right)\eta^{2}+\textsf{a}^{2}r_{1}^{4}\eta^{2}+\textsf{b}r_{1}^{2}\eta^{2}\right]<2Ar_{1}^{-1}<2AW^{\textsf{d}},$
$\displaystyle W\left[{3\over
5}\left(1-\nu\right)\eta^{2}+\textsf{a}^{2}r_{1}^{4}\eta^{2}+\textsf{b}r_{1}^{2}\eta^{2}\right]<\log
2+\log A+\textsf{d}\log W,$

where d is an absolute constant. Each term in the last line is non-negative.
In particular

$\displaystyle{3\over 8}\left(1-\nu\right)\eta^{2}<W^{-1}\log 2+W^{-1}\log
A+\textsf{d}+W^{-1}\log W$

Put here $\eta={1\over 16}$, conclude that this is in contradiction with the
$W\gg 1$, $\textsf{d}\sim 1$ setup. That verifies the statement for the case
it was argued. To finish the proof consider the complementary cases. Assume
$r_{1,>}\geq A$. In this case write

$\displaystyle 1\gg\mathbb{P}\left[{\mathcal{B}}\right]\geq
p_{A}:=\mathbb{P}\left\\{A<r<+\infty\right\\}\geq\mathbb{P}\left\\{r_{1,>}<r<+\infty\right\\}={\textsf{M}_{>}\over
2\textsf{M}_{c}},$
$\displaystyle\textsf{M}_{c}=\textsf{M}_{<}+\textsf{M}_{>}<\textsf{M}_{<}+2p_{A}\textsf{M}_{c},$
$\displaystyle\textsf{M}_{c}=\left(1-2p_{A}\right)^{-1}\textsf{M}_{<}<2\textsf{M}_{<}$

From this point the argument goes completely similar to the case $r_{1,>}<A$.
Finally, consider the last remaining case $r_{1}-r_{1,<}>r_{1,>}-r_{1}$. If
$r_{1,<}\geq A$, then $r_{1,>}>A$ and the argument goes completely similar to
the case $r_{1,>}<A$. If $r_{1,<}<A$, the argument goes similar to the case
$r_{1,>}<A$, but it is shorter for there is no need to truncate the variable
$r$. That finishes $(vii)$.

$(viii)$ Use the notations from the part $(vii)$ proof, write

$\displaystyle\mathbb{P}\left\\{r_{1,<}<r<r_{1,>}\right\\}={\textsf{M}_{c}\over\textsf{M}_{<}+\textsf{M}_{c}+\textsf{M}_{>}}={\textsf{M}_{c}\over
2\textsf{M}_{c}}={1\over 2}$

Use (5.4), conclude there exists $\check{r}\notin{\mathcal{B}}_{\texttt{r}}$,
$r_{1,<}<\check{r}<r_{1,>}$, which implies

$\displaystyle\textsf{a}^{2}\check{r}^{4},\textsf{b}\check{r}^{2}\leq
4A^{2}\qquad\qquad$

Combine this with the assumption $\left|I_{\textsf{m}}\right|<{r_{1}\over 4}$,
get

$\displaystyle r_{1}\leq\check{r}+{r_{1}\over 9},\quad r_{1}\leq{9\over
8}\check{r},$
$\displaystyle\max\left[\textsf{a}^{2}r_{1}^{4},|\textsf{b}|r_{1}^{2}\right]<2\max\left[\textsf{a}^{2}\check{r}^{4},|\textsf{b}|\check{r}^{2}\right]\leq
8A^{2},$

as claimed.

$(ix)$ The argument for the first statement in $(ix)$ is completely similar to
the one for the first statement in $(vii)$. To prove the second statement use
(5.10), combine with (5.13), get

$\displaystyle
f\left(r_{1}\left(1+\eta\right)\right)-f\left(r_{1}\left(1+\eta+{\kappa\over\sqrt{W}}\right)\right)\leq$
$\displaystyle\left(1-\nu\right)\log\left(1+\eta\right)-\left(1-\nu\right)\log\left(1+\eta+{\kappa\over\sqrt{W}}\right)+CA^{2}\times{\kappa\over\sqrt{W}}\leq$
$\displaystyle 2CA^{2}\times{\kappa\over\sqrt{W}},$

where $C$ is an absolute constant. As in $(ix)$ statement, denote for
convenience $r_{>;\eta^{\prime}}=r_{1}\left(1+\eta^{\prime}\right)$. Take
arbitrary integer $k\geq 1$, set in the display
$\kappa=\kappa_{k}=C^{-1}A^{-2}k^{-2}$, $\eta=\eta_{j}=j\kappa_{k}$,
$j=0,...,k$, write

$\displaystyle\textsf{M}_{>;j}:=\int_{r_{>;\eta_{j}}<r<r_{>;\eta_{j+1}}}\phi\left(r\right)dr\leq
e^{{1\over
k^{2}}}\int_{r_{>;\eta_{j+1}}<r<r_{>;\eta_{j+2}}}\phi\left(r\right)dr:=\textsf{M}_{>;j+1},$
$\displaystyle\textsf{M}_{>;0}\leq e^{{j\over k^{2}}}\textsf{M}_{>;j},\quad
j=0,...,k-1,$
$\displaystyle\int_{0<r<+\infty}\phi\left(r\right)dr\geq\int_{0<r<r_{>;\eta_{k}}}\phi\left(r\right)dr=\sum_{0\leq
j\leq k-1}\textsf{M}_{>;j}\geq$
$\displaystyle\textsf{M}_{>;0}\times\sum_{0\leq j\leq k-1}e^{-{j\over
k^{2}}}\geq\textsf{M}_{>;0}\times\sum_{0\leq j\leq k-1}1-{j\over
k^{2}}>(k-1)\times\textsf{M}_{>;0}$

Recall that we assume in $(ix)$ that

$\displaystyle\textsf{M}_{>,c}:=\int_{r_{1}<r<r_{1,>}}\phi\left(r\right)dr\geq\int_{r_{1,<}<r<r_{1}}\phi\left(r\right)dr:=\textsf{M}_{<,c}$

Use the proof of part $(viii)$, combine with the last display, get

$\displaystyle\textsf{M}_{c}+\textsf{M}_{<}+\textsf{M}_{>}=\int_{0<r<+\infty}\phi\left(r\right)dr,$
$\displaystyle\textsf{M}_{<}+\textsf{M}_{>}=\int_{0<r<r_{1,<}}\phi\left(r\right)dr+\int_{r_{1,>}<r<+\infty}\phi\left(r\right)dr,\textsf{M}_{c}=\textsf{M}_{<}+\textsf{M}_{>},$
$\displaystyle\textsf{M}_{>,c}\geq{1\over
4}\int_{0<r<+\infty}\phi\left(r\right)dr$

Combine, get

$\displaystyle\textsf{M}_{>,c}\geq{1\over
4}\int_{0<r<+\infty}\phi\left(r\right)dr>{k-1\over 4}\times\textsf{M}_{>;0}$

Take here $k=21$, write

$\displaystyle\int_{r_{1}<r<r_{>;\eta_{1}}}\phi\left(r\right)dr=\textsf{M}_{>;0}<{1\over
5}\textsf{M}_{>,c}={1\over 5}\int_{r_{1}<r<r_{1,>}}\phi\left(r\right)dr$

That implies

$\displaystyle r_{>|{1\over
5}}>r_{>;\eta_{1}}=r_{1}\left(1+{c_{1}\over\sqrt{W}}\right),$

where $c_{1}:=\kappa_{21}$ is an absolute constant. Note that

$\displaystyle\int_{r_{>|{1\over 5}}<r<r_{1,>}}\phi\left(r\right)dr={4\over
5}\textsf{M}_{>,c}$

That allows to use the above arguments with $r_{>|{1\over 5}}$ in the role of
$r_{1}$. That leads to

$\displaystyle\int_{r_{>|{1\over 5}}<r<r_{1,>}}\phi\left(r\right)dr\geq{1\over
5}\int_{0<r<+\infty}\phi\left(r\right)dr>{k-1\over 5}\times\int_{r_{>|{1\over
5}}<r<r_{>|{1\over 5}}\left(1+\kappa_{k}\right)}\phi\left(r\right)dr$

One can see that $k=21$ is again a right choice. Indeed, write

$\displaystyle\int_{r_{>|{1\over 5}}<r<r_{>|{1\over
5}}\left(1+\kappa_{k}\right)}\phi\left(r\right)dr<{1\over 4}\int_{r_{>|{1\over
5}}<r<r_{1,>}}\phi\left(r\right)dr\leq$ $\displaystyle{1\over
5}\int_{r_{1}<r<r_{1,>}}\phi\left(r\right)dr$

That implies

$\displaystyle r_{>|{2\over 5}}>r_{>|{1\over
5}}\left(1+{c_{1}\over\sqrt{W}}\right),$

where $c_{2}$ is an absolute constant. The rest of the proof for $(ix)$ is
similar.

$(x)$ The proof in case $\textsf{b}<0$ is similar to the one for b and
actually shorter. The reason for the shortcut is that in this case equation
(5.14) holds. In particular,

(5.15)
$\displaystyle\max\left(\textsf{a}^{2}r_{1}^{4},\left|\textsf{b}\right|r_{1}^{2}\right)\leq
A_{1}^{2}$

with another absolute constant $A_{1}$. This estiame is the only one where we
used the function $\phi(r)$ decay rate on its two monotonicity intervals from
part $(iv)$. With ${eq:dichotomy121}$ in hand the proof goes completely
similar to the one in $(ix)$ and we skip it.

$(xi)$ The argument is completely similar to the one in $(x)$ and we skip it.
∎

Now we prove Proposition 5.1.

###### Proof of Proposition 5.1.

Assume that conditions $(I)-(III)$ hold. Use parts $(ix)$, $(x)$, $(xi)$ from
Proposition 5.2, conclude that in any event there exist five points, which we
denote as $r_{|j}<r_{<|j+1}$, $j=0,1,2,3$, such that the following equations
hold

$\displaystyle\int_{r_{|0}<r<r_{|4}}\phi\left(r\right)dr\geq{1\over
8}\int_{0<r<+\infty}\phi\left(r\right)dr,$
$\displaystyle\int_{r_{|j}<r<r_{|j+1}}\phi\left(r\right)dr\geq{1\over
5}\int_{r_{|0}<r<r_{|4}}\phi\left(r\right)dr,\quad j=0,1,2,3$ $\displaystyle
r_{|j+1}>r_{|j}\left(1+{c\over\sqrt{W}}\right),$

where $c>0$ is an absolute constant. Combine first two equations, get

$\displaystyle\mathbb{P}\left\\{r_{|j}<r<r_{|j+1}\right\\}\geq{1\over
40},\quad j=0,1,2,3$

Take arbitrary constant $\bar{r}\geq 0$. Find smallest $j$ such that
$\bar{r}\leq r_{|j}$, evaluate

$\displaystyle\mathbb{E}\left[\log r-\log\bar{r}\right]^{2}\geq$
$\displaystyle{1\over 40}\begin{cases}\left[\log r_{|2}-\log
r_{|1}\right]^{2}\geq\left[\log\left(1+{c\over\sqrt{W}}\right)\right]^{2}\geq{c^{2}\over
2W},\quad\text{if $j=0$},\\\ \left[\log r_{|3}-\log
r_{|2}\right]^{2}\geq\left[\log\left(1+{c\over\sqrt{W}}\right)\right]^{2}\geq{c^{2}\over
2W},\quad\text{if $j=1$},\\\ \left[\log r_{|4}-\log
r_{|3}\right]^{2}\geq\left[\log\left(1+{c\over\sqrt{W}}\right)\right]^{2}\geq{c^{2}\over
2W},\quad\text{if $j=2$},\\\ \left[\log r_{|1}-\log
r_{|0}\right]^{2}\geq\left[\log\left(1+{c\over\sqrt{W}}\right)\right]^{2}\geq{c^{2}\over
2W},\quad\text{if $j=3$},\\\ \end{cases},$

That implies the first statement.

To verify the second statement use Proposition 5.2, $(i)$, write

(5.16) $\displaystyle r_{1}\geq c_{0}\min\left(\textsf{a}^{-{1\over
2}},|\textsf{b}|^{-{1\over 2}},|\textsf{c}|^{-1}\right)\geq cW^{-\textsf{d}}$

with absolute constants $c,\textsf{d}$. Next we need to use the function
$\phi(r)$ decay rate on its monotonicity interval $(0,r_{1})$. For that reason
we again use the dichotomy conditions in $(III)$ and consider separately the
cases $\textsf{b}\geq 0$ and $\textsf{b}<0$.

Assume first that $\textsf{b}\geq 0$. In this case we use (5.11) from
Proposition 5.2, $(iv)$ and write

$\displaystyle\phi\left(r_{1,-\eta}\right)\leq\phi\left(r_{1}\right)\times\exp\left[-W{3\over
5}\left(1-\nu\right)\eta^{2}\right],\quad 0<\eta<{1\over 8},$
$\displaystyle\phi\left(r\right)\leq\phi\left(r_{1,-\eta}\right)\quad
0<r<\eta<{1\over 8}$

On the other hand in this case

$\displaystyle\max\left(\textsf{a}^{2}r_{1}^{4},\left|\textsf{b}\right|r_{1}^{2}\right)\leq
8A$

Use (5.11) left inequality, get

$\displaystyle\phi\left(r_{1}\left(1+{\eta\over\sqrt{W}}\right)\right)\geq$
$\displaystyle\exp\left[-{2\over
3}\left(1-\nu\right)\eta^{2}-6\textsf{a}^{2}r_{j}^{4}\left(1+{3\eta\over
2}\eta+{\eta^{2}\over
6}\right)\eta^{2}-\textsf{b}r_{j}^{2}\eta^{2}\right]\times\phi\left(r_{1}\right)\geq$
$\displaystyle e^{-A_{1}\eta^{2}}\times\phi\left(r_{1}\right),\quad-1<\eta<1$

with absolute constant $A_{1}$. Combine, get

$\displaystyle\mathbb{P}\left\\{0<r<r_{1,-{1\over
8}}\right\\}={\textsf{M}_{(0,r_{1,-{1\over
8}})}\over\textsf{M}_{(0,+\infty)}}\leq{\textsf{M}_{(0,r_{1,-{1\over
8}})}\over\textsf{M}_{(r_{1,-{1\over\sqrt{W}}},r_{1,{1\over\sqrt{W}}})}},$
$\displaystyle\textsf{M}_{(0,r_{1,-{1\over 8}})}\leq
r_{1}\times\phi\left(r_{1}\right)\times\exp\left[-W{3\over 5\times
64}\left(1-\nu\right)\right]\leq$ $\displaystyle
2A\times\phi\left(r_{1}\right)\times\exp\left[-W{3\over 5\times
64}\left(1-\nu\right)\right],$
$\displaystyle\textsf{M}_{(r_{1,-{1\over\sqrt{W}}},r_{1,{1\over\sqrt{W}}})}\geq{2\over\sqrt{W}}\times
e^{-A_{1}}\times\phi\left(r_{1}\right),$
$\displaystyle\mathbb{P}\left\\{0<r<r_{1,-{1\over 8}}\right\\}\leq
A\sqrt{W}\times\exp\left[-W{3\over 5\times
64}\left(1-\nu\right)+A_{1}\right]\leq e^{-cW},$

where $c$ is absolute constant. Combine with (5.16), get the statement in case
$\textsf{b}\geq 0$.

Assume $\textsf{b}<0$. Then it is the first case in the the dichotomy
conditions in $(III)$. In particular

(5.17) $\displaystyle\textsf{a}^{2},\left|\textsf{b}\right|\leq\textsf{A}$

Write

$\displaystyle f^{\prime\prime}(r)=-{\left(1-\nu\right)\over
r^{2}}-12\textsf{a}^{2}r^{2}-2\textsf{b}<-{\left(1-\nu\right)\over
r^{2}}+\textsf{A}<$ $\displaystyle-{\left(1-\nu\right)\over
2r^{2}},\quad\text{ for $0<r<c$},$ $\displaystyle
f^{\prime}(r)<-c_{1},\quad\text{ for $0<r<c$},$ $\displaystyle
f(r)<f(r_{1})-c_{1}(r_{1}-r),\quad\text{ for $0<r<\min\left(r_{1},c\right)$}$

with absolute constants $c,c_{1}$. With (5.17) and the last inequality in hand
the argument goes completely similar to the one in case $\textsf{b}<0$ and we
skip it.

That finishes the proof of Proposition 5.1. ∎

##  6\. Proof of Theorem B.

The proof of Theorem B basically consists of application of Propositions
3.1,4.1,5.1 to each site $n$ factor in the Green function edge–to–edge matrix
vector action

(6.1) $\displaystyle G_{[1,N]}(1;N)g=(-1)^{N-1}U_{1}^{-1}\times
T_{1}\times...\times U^{-1}_{N-1}\times T_{N-1}\times U_{N}^{-1}g$

It is convenient to split the proof of Theorem B into two proposition and a
closing argument.The main point of such presentation is that each proposition
statement introduces notations and desribes in a transparent way what is the
target. The proofs are easy because we use detailed statements from of
Propositions 3.1,4.1,5.1. The first proposition below describes repeated
application of Proposition 3.1 to the factors in (6.1).

######  Proposition 6.1.

$(0)$ Fix arbitrary $U_{1},U_{2},...,U_{N}$. In (6.1) redenote for convenience
$g:=g_{N}$. Apply Propositions 3.1 with $n=N$, denote $Q_{1,N}$ the orthogonal
matrix $Q_{1}$ in the equation (3.6), $Q_{2,N-1}$ the orthogonal matrix
$Q_{2}$ defined in the part $(2)$ of the proposition, $X_{1,N-1|T_{N-1}}$ the
vector in the equation (3.10), $g_{N-1|T_{N-1}}$ the vector $h$, defined in
the equation (3.9). Apply Propositions 3.1 with $n=N-1$, $g_{N-1|T_{N-1}}$ in
the role of $g$, denote $X_{1,N-2|T_{N-2},T_{N-1}}$ the vector from (3.10),
$g_{N-1|T_{N-2},T_{N-1}}$ the vector from (3.9). Repeat the application, get
the vectors $X_{1,n-1|T_{n-1},...,T_{N-1}}$, $g_{1,n-1|T_{n-1},...,T_{N-1}}$.

The following equation holds

$\displaystyle U_{1}^{-1}\times T_{1}\times...\times U^{-1}_{N-1}\times
T_{N-1}\times U_{N}^{-1}g=\qquad\qquad\qquad\qquad$ (6.2)
$\displaystyle\prod_{2\leq n\leq
N}\left\|U_{n}^{-1}g_{1,n-1|T_{n-1},...,T_{N-1}}\right\|^{-1}\times\left\|X_{1,n-1|T_{n-1},...,T_{N-1}}\right\|^{-1}\times
U_{1}^{-1}g_{1,1|T_{1},...,T_{N-1}}$

$(1)$ The random variables $\left\|X_{1,n-1|T_{n-1},...,T_{N-1}}\right\|$,
$n=2,...,N$ are independent when conditioned on the random matrices
$U_{2},...,U_{N}$ and on the random vectors $g_{1,n-1|T_{n-1},...,T_{N-1}}$,
$n=2,...,N$.

###### Proof.

Rewrite the equations (3.3), (3.4) (3.9) using the above notations :

$\displaystyle\textsf{v}_{g}\left(T_{n-1}\big{|}\right):=T_{n-1}\times
U_{n}^{-1}g_{1,n-1|T_{n-1},...,T_{N-1}}$
$\displaystyle\left\|\textsf{v}_{n}\left(T_{n-1}\big{|}\right)\right\|^{-1}\textsf{v}_{n}\left(T_{n-1}\big{|}\right)=g_{1,n-2|T_{n-2},...,T_{N-1}}=$
$\displaystyle\quad\left\|X_{1,n-1|T_{n-1},...,T_{N-1}}\right\|^{-1}Q_{2,n-1}X_{1,n-1|T_{n-1},...,T_{N-1}}$

Put this in (6.1), get the equation in part $(0)$.

To verify part $(1)$, recall that he conditional distribution of the random
vector $X_{1,n-1|T_{n-1},...,T_{N-1}}$ is as follows

$\displaystyle
d\mathbb{P}\left(X_{1,n-1|T_{n-1},...,T_{N-1}}\big{|}\right)=\qquad\qquad$
(6.3) $\displaystyle\mu e^{-{W\over
2}\phi\left(X_{1,n-1|T_{n-1},...,T_{N-1}}\big{|}\right)}\times
dX_{1,n-1|T_{n-1},...,T_{N-1}},$

where $\mu$ is the normalizing factor and the function
$\phi\left(X_{1,n-1|T_{n-1},...,T_{N-1}}\big{|}\right)$ depends on
$U_{n-1},U_{n}$ and the vector $g_{1,n|T_{n},...,T_{N-1}}$ only, see the
statement in part $(3)$ of Propositions 3.1 for more details. That implies
$(1)$. ∎

Next we apply Propositions 4.1 to each factor
$\left\|X_{1,n-1|T_{n-1},...,T_{N-1}}\right\|^{-1}$ in (6.2).

######  Proposition 6.2.

Use the notations in Proposition 6.1.

$(0)$ Use the spherical variables, write

$\displaystyle X_{1,n-1|T_{n-1},...,T_{N-1}}=r_{|n-1}\textsf{x}_{|n-1},$
$\displaystyle
r_{|n-1}=\left\|X_{1,n-1|T_{n-1},...,T_{N-1}}\right\|,\quad\textsf{x}_{|n-1}\in\mathbb{S}^{W-1}$

Then

(6.4) $\displaystyle
g_{1,n-2|T_{n-2},...,T_{N-1}}=Q_{2,n-1}\textsf{x}_{|n-1},$ (6.5)
$\displaystyle\log\left\|G_{[1,N]}(1;N)g\right\|=L+\sum_{2\leq n\leq N}\log
r_{|n-1},$

where $L$ does not depend on the variables $r_{|n-1}$. The variables
$r_{|n-1}$, $n=2,...,N$ are independent when conditioned on the random
matrices $U_{2},...,U_{N}$ and on the random vectors $\textsf{x}_{|n-1}$,
$n=2,...,N$.

$(1)$ The conditional distribution $d\mathbb{P}\left(r_{|n-1}\big{|}\right)$
is as follows

$\displaystyle
d\mathbb{P}\left(r_{|n-1}\big{|}\right)=\mu\Big{(}U_{n-1},U_{n},\textsf{x}_{|n-1},\textsf{x}_{|n}\Big{)}\times$
$\displaystyle r_{|n-1}^{W-1}e^{-{W\over
2}\left(\textsf{a}_{|n-1}^{2}r_{|n-1}^{4}+\textsf{b}_{|n-1}r_{|n-1}^{2}+\textsf{c}_{|n-1}r_{|n-1}\right)}dr_{|n-1},$

where

$\displaystyle\text{$\textsf{a}_{|n-1}^{2},\textsf{b}_{|n-1},\textsf{c}_{|n-1}$
are constants depending $U_{n-1},U_{n},\textsf{x}_{|n-1},\textsf{x}_{|n-1}$},$
$\mu\Big{(}U_{n-1},U_{n},\textsf{x}_{|n-1},,\textsf{x}_{|n}\Big{)}$ is the
normalizing factor, $n=2,...,N$

$(2)$ There exists a set
$\widehat{{\mathcal{B}}}\in\mathcal{F}\left(U_{2},...,U_{N}\right)$, such that

(6.6) $\displaystyle\mathbb{P}_{N}\left[\widehat{{\mathcal{B}}}\right]\leq
CNW^{D-1},$

where $D\gg 1$ is an absolute constant, and for each
$\left(U_{2},...,U_{N}\right)\notin\widehat{{\mathcal{B}}}$ there exist sets
${\mathcal{B}}_{|n-1,A}\in\mathcal{F}\left(T_{n-1},\textsf{x}_{|n}\right)$,
depending on $\left(U_{2},...,U_{N}\right)$, with

(6.7) $\displaystyle\mathbb{P}_{N}{\mathcal{B}}_{|,n-1,A}\leq
C_{1}e^{-c_{1}A^{2}W}$

and for each
$\left(T_{n-1},\textsf{x}_{|n-1}\right)\notin{\mathcal{B}}_{|,n-1,A}$ there
exist sets
${\mathcal{B}}_{|\texttt{r},n-1,A}\in\mathcal{F}\left(r_{|n-1}\right)$,
depending on $\left(T_{n-1},\textsf{x}_{|n}\right)$, with

(6.8) $\displaystyle\mathbb{P}_{N}{\mathcal{B}}_{|\texttt{r},n-1,A}\leq
C_{1}e^{-c_{1}A^{2}W}$

where $A,C_{1},c_{1}$ are absolute constants, $n=2,...,N$, such that for any
$n$ and

$\displaystyle\text{for any
$\left(U_{2},...,U_{N}\right)\notin\widehat{{\mathcal{B}}}$},$
$\displaystyle\text{and any
$\left(T_{n-1},\textsf{x}_{|n}\right)\notin{\mathcal{B}}_{|n-1,A}$,
$\left(T_{n-2},\textsf{x}_{|n-1}\right)\notin{\mathcal{B}}_{|n-2,A}$},$ and
any $r_{|n-1}\notin{\mathcal{B}}_{|\texttt{r},n-1,A}$,
$r_{|n-2}\notin{\mathcal{B}}_{|\texttt{r},n-2,A}$

conditions $(I)-(II)$ in Proposition 5.1 hold for
$d\mathbb{P}\left(r_{|n-1}\big{|}\right)$,$d\mathbb{P}\left(r_{|n-2}\big{|}\right)$.
Moreover, either $d\mathbb{P}\left(r_{|n-1}\big{|}\right)$ obeys Proposition
5.1 condition $(III)$ dichotomy first case, or
$d\mathbb{P}\left(r_{|n-2}\big{|}\right)$ obeys Proposition 5.1 condition
$(III)$ dichotomy second case. Finally, the variables
$\mathcal{I}_{{\mathbb{R}}^{W}\times\mathbb{S}^{W-1}\setminus{\mathcal{B}}_{|n-1,A}}\left(T_{n-1}\right)\times
r_{|n-1}$, $n=2,...,N$ are independent, when conditioned on the random
matrices $U_{2},...,U_{N}$. Here $\mathcal{I}_{{\mathcal{A}}}$ stands for the
indicator of the set ${\mathcal{A}}$.

###### Proof.

Part $(0)$ follows straightforward from Proposition 6.1. Part $(1)$ follows
straightforward from part $(0)$ in Proposition 4.1. To verify $(2)$ recall
that part $(1)$ in Proposition 4.1 reads in particular:

there exists a set
$\widehat{{\mathcal{B}}}_{U,n-2,n-1,D}\in\mathcal{F}\left(U_{n-2},U_{n-1}\right)$,
such that

$\displaystyle\mathbb{P}_{N}\left[\widehat{{\mathcal{B}}}_{U,n-1,n,D,A}\right]\leq
CW^{D-1},$

and a set ${\mathcal{B}}_{S,n-1,A}\in\mathcal{F}\left(S\right)$ with

$\displaystyle\mathbb{P}_{N}{\mathcal{B}}_{S,n-1,A}\leq C_{1}e^{-c_{1}A^{2}W}$

such that for any
$\left(U_{n-1},U_{n}\right)\notin\widehat{{\mathcal{B}}}_{U,n-1,n,D,A}$,
$S\notin{\mathcal{B}}_{S,n-1,A}$ and any x holds

$\displaystyle\max\left(\textsf{a}^{2}\left(\textsf{x}\right),\left|\textsf{b}\left(\textsf{x}\right)\right|,\left|\textsf{c}\left(\textsf{x}\right)\right|\right)\leq
2A^{2}W^{2D}$

Recall also that from its definition $S=Q_{2,n-1}T_{n-1}Q_{1,n-1}$. Denote

$\displaystyle\widehat{{\mathcal{B}}}=\bigcup_{1\leq n\leq
N}\widehat{{\mathcal{B}}}_{U,n-1,n,D,A},$
$\displaystyle{\mathcal{B}}_{|\texttt{T},n-1,A}=\left\\{T_{n-1}:Q_{2,n-1}T_{n-1}Q_{1,n-1}\in{\mathcal{B}}_{S,n-1,A}\right\\},$
$\displaystyle{\mathcal{B}}_{|n-1,A}={\mathcal{B}}_{|\texttt{T},n-1,A}\cup\left\\{\textsf{x}_{|n-1}:\textsf{x}_{|n-1}\in{\mathcal{B}}_{\texttt{x},n-1,n,A}\right\\},$
$\displaystyle{\mathcal{B}}_{|n-1,A}={\mathcal{B}}_{|\texttt{T},n-1,A}=\left\\{r_{|n-1}:r_{|n-1}\in{\mathcal{B}}_{\texttt{r},\textsf{x},n-1,n,A}\right\\},$

where
${\mathcal{B}}_{\texttt{x},n-1,n,A}\in\mathcal{F}\left(\textsf{x}\right)$,
${\mathcal{B}}_{\texttt{r},\textsf{x},n-1,n,A}\in\mathcal{F}\left(r\right)$
are from the statement in part $(3)$ of Proposition 4.1. Equation (6.6) holds
by subadditivity. holds becasue of Gaussian random matrix distribution
orthogonal invariance. Equations (6.7), (6.8) follow from part $(3)$ in
Proposition 4.1. The statement addressing conditions $(I)-(II)$ in Proposition
5.1 follow from parts $(2)$, $(3)$ in Proposition 4.1. The statement
addressing the dichotomy follows from the last statement in part $(5)$ of
Proposition 4.1.

Finally, the variables $T_{n}$, $n=1,...,N-1$ are independent, when
conditioned on $U_{n}$, $n=1,...$. That implies the last statemnt. ∎

The proof of Theorem B consists of a short argument based on the following
corollary, which we use also to prove Theorem C. To state the corollary we
need to introduce yet another collection of sets. Here is the discription. In
the proof of the corollary we use both statements in Proposition 5.1, applied
to the super-exponential density from part $(1)$ in Proposition 6.2, i.e. to

$a_{|n-1}^{2},\textsf{b}_{|n-1},\textsf{c}_{|n-1}$ are constants depending
$U_{n-1},U_{n},\textsf{x}_{|n-1},\textsf{x}_{|n-1}$

in the role of $\textsf{a},\textsf{b},\textsf{c}$, provided conditions in
Proposition 6.2 part $(2)$ hold. Given $\textsf{a},\textsf{b},\textsf{c}$, the
second statement in Proposition 5.1 defines a set ${\mathcal{B}}_{\phi}$, such
that

$\displaystyle\mathbb{P}{\mathcal{B}}_{\phi}\leq
Ce^{-cW}\qquad\qquad\qquad\qquad$ (6.9) $\displaystyle r\geq
c_{1}W^{-\textsf{d}},\quad\text{for any
$r\in(0,+\infty)\setminus{\mathcal{B}}_{\phi}$},$

see (5.8). Given $U_{n-1},U_{n},\textsf{x}_{|n-1},\textsf{x}_{|n-1}$, which
obey conditions in Proposition 6.2 part $(2)$, redenote ${\mathcal{B}}_{\phi}$
as ${\mathcal{B}}_{|\texttt{r},n-1}$. These are new sets we use in sets we use
in the corollary below.

######  Corollary 6.3.

Use the notations in Proposition 6.2 part $(2)$. Take

$\displaystyle\left(U_{2},...,U_{N}\right)\notin\widehat{{\mathcal{B}}}$

Denote

$\displaystyle\eta_{k,|}=\mathcal{I}_{{\mathbb{R}}^{W}\times\mathbb{S}^{W-1}\setminus{\mathcal{B}}_{|N-2k,A}}\left(T_{n-2k},\textsf{x}_{|n-2k}\right)\times\qquad\qquad$
$\displaystyle\mathcal{I}_{{\mathbb{R}}^{W}\times\mathbb{S}^{W-1}\setminus{\mathcal{B}}_{|n-2k+1,A}}\left(T_{n-2k+1},\textsf{x}_{|n-1}\right)\times\qquad\qquad$
(6.10)
$\displaystyle\quad\mathcal{I}_{(0,+\infty)\setminus{\mathcal{B}}_{|\texttt{r},n-2k,A}}\left(r_{n-2k}\right)\times\mathcal{I}_{(0,+\infty)\setminus{\mathcal{B}}_{|\texttt{r},n-2k+1,A}}\left(r_{n-2k+1}\right)\times$
$\displaystyle\mathcal{I}_{(0,+\infty)\setminus{\mathcal{B}}_{|\texttt{r},n-2k}}\left(r_{n-2k}\right)\times\mathcal{I}_{(0,+\infty)\setminus{\mathcal{B}}_{|\texttt{r},n-2k+1}}\left(r_{n-2k+1}\right)\times$
$\displaystyle\left[\log r_{|N-2k}+\log r_{|N-2k+1}\right],\quad
k=1,...\qquad\qquad$

The variables $\eta_{k,|}$ are independent,

$\displaystyle{\mathrm{Var}}\left[\eta_{k,|}\right]\geq{c\over W}\qquad\qquad$
(6.11) $\displaystyle\left|\eta_{k,|}\right|\leq C\log W,\qquad\qquad$

where $c,C$ are absolute constants.

###### Proof.

Combine Proposition 6.2 with Proposition 5.1, get both estimates. ∎

###### Proof of Theorem B.

Use (6.5) from Proposition 6.2, write

$\displaystyle\textsf{L}_{N,|}:=\log\left\|G_{[1,N]}(1;N)g\right\|=L+\sum_{2\leq
n\leq N-1}\log r_{|n-1}=$ (6.12) $\displaystyle\hat{L}+\sum_{2\leq 2k\leq
N-1}\eta_{k,|}\qquad\qquad\qquad\qquad$

Here $\textsf{L}_{N,|}$ stands for the random variable

$\log\left\|G_{[1,N]}(1;N)g\right\|,\quad\text{conditioned on
$\left(U_{2},...,U_{N},\textsf{x}_{1,|},...,\textsf{x}_{N-1,|}\right)$},$

and also on $r_{1,|}$ if $N$ is even, $\hat{L}$ does not depend on the
variables $\eta_{|k}$, see Corollary 6.3. Use Corollary 6.3, get

(6.13) $\displaystyle{\mathrm{Var}}\left[\textsf{L}_{N,|}\right]=\sum_{2\leq
2k\leq N}{\mathrm{Var}}\left[\eta_{k,|}\right]\geq c_{1}NW^{-1},$

where $c_{1}$ is an absolute constant. It follows from Proposition 6.2 that
the variables $\textsf{L}_{N,|}$ is conditioned on, run in the set with
$\mathbb{P}_{N}\geq 1-N\times W^{-3D}$ where $D\gg 1$ is an absolute constant.
Take $N\sim W^{cD}$, with an absolute constant $0<c\ll 1$. Then the
probability estimate results $\mathbb{P}_{N}\geq 1-W^{-2D}$. It follows also
from Proposition 6.2 that each variable $r_{n,|}$ in the setting of Corollary
6.3, runs in the set with $\mathbb{P}_{N}\geq 1-Ce^{-cW}$ with absolute
constants $c,C$. Combine this with (6.13) and (6.12), conclude

(6.14)
$\displaystyle{\mathrm{Var}}_{\mathcal{G}}\left[\log\left\|G_{[1,N]}(1;N)g\right\|\right]\geq
c_{1}NW^{-1},$

where $\mathbb{P}_{N}{\mathcal{G}}\geq 1-W^{-D}$ and
${\mathrm{Var}}_{\mathcal{G}}$ stands for the conditional variance against the
set ${\mathcal{G}}$. Note that it is exactly the scales setting in theorem B.
That implies the statement in Theorem B. ∎

##  7\. Proofs of Theorem C.

We derive Theorem C from the following lemma. The proof is straightforward,
but discuss it for completeness.

######  Lemma 7.1.

Take a random variable $\eta$. Assume

(7.1)
$\displaystyle\left|\mathbb{E}\left[\eta-\mathfrak{a}\right]^{n}\right|\leq
n!\sigma^{2}H^{n-2},\quad\text{for any $n=2,...$},$

where we denote for convenience $\mathbb{E}[\eta]=\mathfrak{a}$,
${\mathrm{Var}}\xi=\sigma^{2}$ and $H>0$ is a constant. Then for $|t|\leq
cH^{-1}$, with absolute constant $0<c\ll 1$ holds

$\displaystyle\Big{[}\mathbb{E}\big{[}e^{t\eta}\big{]}\Big{]}^{2}\leq\mathbb{E}\big{[}e^{2t\eta}\big{]}\times\Big{(}1-{t^{2}\sigma^{2}\over
4}\Big{)}$

###### Proof.

Write the series expansion, combine with equation (7.1), get

$\displaystyle\mathbb{E}\big{[}e^{t(\eta-\mathfrak{a})}\big{]}=\sum_{p=0}^{\infty}{t^{p}\over
p!}\mathbb{E}\big{[}(\eta-\mathfrak{a})^{p}\big{]}\leq 1+{t^{2}\sigma^{2}\over
2}+\sum_{p=3}^{\infty}t^{p}\sigma^{2}H^{p-2}$ $\displaystyle\leq
1+{t^{2}\sigma^{2}\over 2}+2Ht^{3}\sigma^{2}\leq 1+(1+c){t^{2}\sigma^{2}\over
2}$

provided $|t|\leq c\min\big{(}H^{-1},\sigma^{-1}\big{)}$. Similarly,

$\displaystyle\mathbb{E}\big{[}e^{2t(\eta-\mathfrak{a})}\big{]}\geq
1+2(1-c)t^{2}\sigma^{2}$

Combine, get

$\displaystyle\Big{[}\mathbb{E}\big{[}e^{t(\eta-\mathfrak{a})}\big{]}\Big{]}^{2}\leq\mathbb{E}\big{[}e^{2t(\eta-\mathfrak{a})}\big{]}\times{(1+(1+c){t^{2}\sigma^{2}\over
2})^{2}\over(1+2(1-c)t^{2}\sigma^{2})}\leq$
$\displaystyle\mathbb{E}\big{[}e^{2t(\eta-\mathfrak{a})}\big{]}\times{(1+(1+c){t^{2}\sigma^{2}\over
2})^{2}\over(1+(1+c)t^{2}\sigma^{2})}\leq\mathbb{E}\big{[}e^{2t(\eta-\mathfrak{a})}\big{]}\times\big{(}1-{t^{2}\sigma^{2}\over
4}\big{)},$

Combine with

$\displaystyle\Big{[}\mathbb{E}\big{[}e^{t(\eta-\mathfrak{a})}\big{]}\Big{]}^{2}=\Big{[}\mathbb{E}\big{[}e^{t\eta}\big{]}\Big{]}^{2}\times
e^{-2t\mathfrak{a}}$
$\displaystyle\mathbb{E}\big{[}e^{2t(\eta-\mathfrak{a})}\big{]}=\mathbb{E}\big{[}e^{2t\eta}\big{]}\times
e^{-2t\mathfrak{a}},$

get the statement. ∎

######  Corollary 7.2.

Take independent random variables $\eta_{k}$, $k=1,...,N$. Denote
$\sigma^{2}_{k}=Var\eta_{k}$. Assume that each $\eta_{k}$ obeys condition
(7.1) in Lemma 7.1 with the same $H$ for all $k$ and with $\sigma_{k}$ in the
role of $\sigma$. Denote

$\displaystyle\textsf{L}=\sum_{1\leq k\leq N}\eta_{k}$

Then for $|t|\leq cH^{-1}$

(7.2)
$\displaystyle\Big{[}\mathbb{E}\big{[}e^{t\textsf{L}}\big{]}\Big{]}^{2}\leq\mathbb{E}\big{[}e^{2t\textsf{L}}\big{]}\times\prod_{1\leq
k\leq N}\Big{(}1-{t^{2}\sigma_{k}^{2}\over 4}\Big{)}$

###### Proof.

Use Lemma 7.1, combine with independence, get (7.2). ∎

Finally, to prove Theorem C we need the following well-known form of the
Wegner estimate

(7.3)
$\displaystyle\mathbb{E}\left[\left\|G_{[1,N]}\left(1;N\right)\right\|^{t}\right]\leq
C_{1}W^{C_{2}},\quad\text{ for any $0<t<{1\over 2}$}$

with absolute constants $C_{1},C_{2}$. For reference see for instance [Sch09],
Lemma W and equation $(1.7)$.

###### Proof of Theorem C.

Fix

$\displaystyle\left(U_{2},...,U_{N}\right)\notin\widehat{{\mathcal{B}}}$

To simplify the notations us the $\mathbb{P}$ and $\mathbb{E}$ for the
_conditional probability and conditional expectation, at
$\left(U_{2},...,U_{N}\right)$_. Use (6.12) from the proof of Theorem B, write

(7.4)
$\displaystyle\mathbb{E}\left[e^{t\textsf{L}_{N,|}}\big{|}\right]=e^{t\hat{L}}\mathbb{E}\left[e^{t\textsf{L}}\big{|}\right],\quad\textsf{L}=\sum_{2\leq
2k\leq N}\eta_{k,|}$

Here $\mathbb{E}\left[\cdot|\right]$ stands for the conditional expectation
against the variables

$\left(U_{2},...,U_{N},\textsf{x}_{1,|},...,\textsf{x}_{N-1,|}\right)$

and maybe also on $r_{1,|}$ if $N$ is even.

Equation (6.11) in the statement of Corollary 6.3 says

$\displaystyle{\mathrm{Var}}\left[\eta_{k,|}\right]\geq{c\over W}\qquad\qquad$
(7.5) $\displaystyle\left|\eta_{k,|}\right|\leq C\log W,\qquad\qquad$

That implies, in particular, condition (7.1) in Lemma 7.1 with the same
$H=C\log W$ for all $k$. Corollary 7.2 applies:

(7.6)
$\displaystyle\Big{[}\mathbb{E}\big{[}e^{t\textsf{L}}\big{|}\big{]}\Big{]}^{2}\leq\mathbb{E}\big{[}e^{2t\textsf{L}}\big{|}\big{]}\times\prod_{2\leq
2k\leq N-1}\Big{(}1-{t^{2}\sigma_{|,k}^{2}\over 4}\Big{)},$

for any $|t|\leq cH^{-1}$, where
$\sigma_{|,k}^{2}={\mathrm{Var}}\left[\eta_{k,|}\right]$ stands for the
conditional variance, $0<c\ll 1$ is an absolute constant. Combine with (7.5),
get

(7.7)
$\displaystyle\Big{[}\mathbb{E}\big{[}e^{t\textsf{L}}\big{|}\big{]}\Big{]}^{2}\leq\mathbb{E}\big{[}e^{2t\textsf{L}}\big{|}\big{]}\times
e^{-{c_{1}N\over W\left(\log W\right)^{2}}},$

where $c_{1}>0$ is an absolute constant. Combine (7.4) with (7.7) and with
Cauchy-Schwarz, get

$\displaystyle\left[\mathbb{E}\left[\left\|G_{[1,N]}\left(1;N\right)g\right\|^{t}\right]\right]^{2}=\left[\mathbb{E}\left[e^{t\hat{L}}\mathbb{E}\big{[}e^{t\textsf{L}}\big{|}\big{]}\right]\right]^{2}\leq$
$\displaystyle\left[\mathbb{E}\left[e^{t\hat{L}}\left[\mathbb{E}\big{[}e^{2t\textsf{L}}\big{|}\big{]}\right]^{{1\over
2}}\times e^{-{c_{1}N\over 2W\left(\log
W\right)^{2}}}\right]\right]^{2}=e^{-{c_{1}N\over W\left(\log
W\right)^{2}}}\left[\mathbb{E}\left[e^{t\hat{L}}\left[\mathbb{E}\big{[}e^{2t\textsf{L}}\big{|}\big{]}\right]^{{1\over
2}}\right]\right]^{2}\leq$ $\displaystyle e^{-{c_{1}N\over W\left(\log
W\right)^{2}}}\mathbb{E}\left[e^{2t\hat{L}}\mathbb{E}\big{[}e^{2t\textsf{L}}\big{|}\big{]}\right]=e^{-{c_{1}N\over
W\left(\log
W\right)^{2}}}\mathbb{E}\left[e^{2t\hat{L}}e^{2t\textsf{L}}\right]=$
$\displaystyle e^{-{c_{1}N\over W\left(\log
W\right)^{2}}}\mathbb{E}\left[\left\|G_{[1,N]}\left(1;N\right)g\right\|^{2t}\right]$

Combine with (7.3), get

(7.8)
$\displaystyle\mathbb{E}\left[\left\|G_{[1,N]}\left(1;N\right)g\right\|^{t}\right]\leq
C_{1}W^{C_{2}}\times e^{-{c_{1}N\over 2W\left(\log W\right)^{2}}}\leq
e^{-{cN\over W\left(\log W\right)^{2}}}$

Take here $t={C\over\log W}$, combine with Chebishev inequality, get

$\displaystyle\mathbb{P}\left\\{\left\|G_{[1,N]}\left(1;N\right)g\right\|>M\right\\}=\mathbb{P}\left\\{\left\|G_{[1,N]}\left(1;N\right)g\right\|^{t}>M^{t}\right\\}\leq$
$\displaystyle{1\over
M^{t}}\times\mathbb{E}\left[\left\|G_{[1,N]}\left(1;N\right)g\right\|^{t}\right]\leq{1\over
M^{t}}\times e^{-{cN\over W\left(\log W\right)^{2}}},\quad\text{for any
$M>0$},$
$\displaystyle\mathbb{P}\left\\{\left\|G_{[1,N]}\left(1;N\right)g\right\|>e^{-{cN\over
2W\left(\log W\right)^{3}}}\right\\}\leq e^{-{cN\over 2W\left(\log
W\right)^{2}}}$

Recall that here $\mathbb{P}$ stands not for the original probability
$\mathbb{P}_{N}$, but for $\mathbb{P}_{N}$ conditioned on the set
${\mathcal{G}}_{N}$ which is the complement of $\widehat{{\mathcal{B}}}$,
where

(7.9) $\displaystyle\mathbb{P}_{N}\left[\widehat{{\mathcal{B}}}\right]\leq
CNW^{D-1},$

Recall that in Theorem C we assume that $W^{2}<N\leq W^{D_{0}}$ where
$D_{0}\gg 1$ is a constant. Take $D_{0}=c_{0}D$ with $0<c\ll 1$ being an
absolute constant. Combine, run $g=e_{j}$ where $e_{j}$ is the standard basis
in $R^{W}$, add up, get the statement in Theorem C. ∎

##  Appendix A Matrix Variables Orthogonal Change.

######  Definition A.1.

$(1)$ Denote $\textsf{M}_{\textsf{s}}\left(W\right)$ the linear space of all
real symmetric matrices. Denote $\textsf{I}(V)=\textsf{V}$ the identification

$\displaystyle\textsf{M}_{\textsf{s}}\left(W\right)\ni V=(v_{p,q})_{1\leq
p,q\leq W}\to\textsf{V}:=(v_{p,q})_{1\leq p\leq q\leq W}:=$
$\displaystyle(v_{p,p})_{1\leq p\leq W}\oplus(v_{p,q})_{1\leq p<q\leq
W}\in{\mathbb{R}}^{W}\bigoplus{\mathbb{R}}^{{W(W-1)\over
2}}={\mathbb{R}}^{D},\quad D={W(W-1)\over 2}$

$(2)$ Take an orthogonal matrix $Q$. Consider the maps

$\displaystyle\Phi_{Q}:\textsf{M}_{\textsf{s}}\left(W\right)\ni V\to
QVQ^{t}\in\textsf{M}_{\textsf{s}},$
$\displaystyle\tilde{\Phi}_{Q}:{\mathbb{R}}^{D}\ni\textsf{V}\to\textsf{I}\circ\Phi_{Q}\circ\textsf{I}^{-1}(\textsf{V})\in{\mathbb{R}}^{D}$

######  Lemma A.2.

Using the notations in Definition A.1 the Jacobian $J_{\tilde{\Phi}_{Q}}$ of
the map $\tilde{\Phi}_{Q}$ is $=1$.

###### Proof.

Consider the Frobenius inner product in
$\textsf{M}_{\textsf{s}}\left(W\right)$

$\displaystyle\langle V,U\rangle_{F}=Tr\left(U^{t}V\right)$

Use the identification $\textsf{I}(V)=\textsf{V}$, define the inner product

$\displaystyle\langle\textsf{V},\textsf{U}\rangle_{F}:=\langle\textsf{I}^{-1}\textsf{V},\textsf{I}^{-1}\textsf{U}\rangle_{HS}$

The map $\Phi_{Q}(V)=Q^{t}VQ$, $V\in\textsf{M}_{\textsf{s}}\left(W\right)$ is
linear and preserves the Frobenius inner product in
$\textsf{M}_{\textsf{s}}\left(W\right)$. Conclude $\tilde{\Phi}_{Q}(V)$ is a
linear isometric map in the Euclidean space
${\mathbb{R}}^{D},\langle,\rangle_{F}$. Enumerate the standard basis in
${\mathbb{R}}^{W}$, resp. ${\mathbb{R}}^{{W(W-1)\over 2}}$, as $e_{p,p}$,
$1\leq p\leq W$, resp. $e_{p,q}$, $1\leq p<q\leq W$. Use this enumeration for
the standard basis in
${\mathbb{R}}^{D}={\mathbb{R}}^{W}\bigoplus{\mathbb{R}}^{{W(W-1)\over 2}}$,
denote $v_{p,p}$, $1\leq p\leq W$, resp. $v_{p,q}$, $1\leq p<q\leq W$ the
respective coordinates. Use the definitionof the trace, conclude that

$\displaystyle\langle\textsf{V},\textsf{V}\rangle_{F}=\sum_{1\leq p\leq
W}v^{2}_{p,p}+2\sum_{1\leq p<q\leq W}v^{2}_{p,q}$

The map $\tilde{\Phi}_{Q}(V)$ preserves the norm
$\left\|\textsf{V}\right\|_{F}:=\sqrt{\langle\textsf{V},\textsf{V}\rangle_{F}}$.
In particular, the map preserves the volume $\mathcal{V}_{HS}$ in the
Euclidean space ${\mathbb{R}}^{D},\langle,\rangle_{HS}$. Compare the inner
product $\langle,\rangle_{HS}$ against the standard inner product
$\langle,\rangle$ in ${\mathbb{R}}^{D}$ conclude
$\mathcal{V}_{HS}=C(W)\mathcal{V}$, where $\mathcal{V}$ stands for the
standard volume in ${\mathbb{R}}^{D}$, $C(W)=2^{{W(W-1)\over 4}}$. Conclude
the map $\tilde{\Phi}_{Q}(V)$ preserves the standard volume in
${\mathbb{R}}^{D}$, $\tilde{J}_{\Phi_{Q}}=1$.

∎

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