11institutetext: Institut de Mathématiques de Toulouse, Université Paul
Sabatier, 118 route de Narbonne, 31062 Toulouse, France; School of
Mathematical Sciences, Fudan University, 220 Handan Road, 200433 Shanghai,
China<EMAIL_ADDRESS>
# Dyson processes on the octonion algebra

Songzi Li 11

###### Abstract

We consider Brownian motion on symmetric matrices of octonions, and study the
law of the spectrum. Due to the fact that the octonion algebra is
nonassociative, the dimension of the matrices plays a special role. We provide
two specific models on octonions, which give some indication of the relation
between the multiplicity of eigenvalues and the exponent in the law of the
spectrum.

##  1 Introduction

The study of the laws of the spectrum is one of the most important topics in
random matrix theory. One may consider stochastic diffusion processes on
specific set of matrices, for example symmetric or Hermitian matrices. Usually
one considers the empirical measure of the spectrum, which is often again a
stochastic diffusion process, called a Dyson process, see the works of Wigner
Wig58 , Mehta Meh , Dyson dyson , Anderson-Guionnet-Zeitouni AndGuioZeit ,
Erdös and co erdosandco3 ; erdosandco2 ; erdosandco1  , Forrester Forrester
and references therein.

Let us recall some classical results on this topic. Consider specific matrices
with independent Gaussian elements : real symmetric ($\beta=1$), Hermitian
($\beta=2$) and real quaternionic ($\beta=4$). Then the law of their
eigenvalues $(\lambda_{i})_{1\leq i\leq n}$, ordered as
$\lambda_{1}\leq...\leq\lambda_{n}$, has a density with respect to the
Lebesgue measure $d\lambda_{1}...d\lambda_{n}$ which is

$C_{\beta,n}{\rm
exp}(-\frac{\beta}{2}\sum^{n}_{j=1}\lambda^{2}_{j})\prod_{1\leq j\leq k\leq
n}|\lambda_{k}-\lambda_{j}|^{\beta},$ (1.1)

where $C_{\beta,n}$ are constants depending on $\beta$, $n$.

On the other hand, one can also consider random matrices with stochastic
process as entries. In his paper dyson , Dyson derived the stochastic
equations of the eigenvalues of Hermitian matrices whose elements are
independent complex Brownian motions (see also in this direction Anderson-
Guionnet-Zeitouni AndGuioZeit , Mehta Meh , Li-Li-Xie LLX ). The stochastic
process on the spectrum provides a dynamic way to study the law of the
eigenvalues of the matrix with Gaussian entries. In fact there are two ways:
one is through the law of the eigenvalues of matrices with Brownian motions as
entries, considered at time $t=1$; the other one is through the matrix whose
elements are Ornstein-Uhlenbeck process, since when $t\rightarrow\infty$ the
law of the matrix converges to a matrix with Gaussian entries, and the law of
its spectrum is invariant through O-U process: in this case, the law of the
spectrum may be seen as the invariant (in fact reversible, see definition 4)
law of the process. This is in general a much easier way to identify the law,
since reversible measures are easy to identify through the knowledge of the
generator.

Meanwhile, if we consider real symmetric matrix, Hermitian matrix and real
quaternionics matrix as real ones of dimension respectively $n\times n$,
$2n\times 2n$, $4n\times 4n$, the multiplicity of the eigenvalues is again 1,
2 and 4. This fact leads us to wonder whether this exponent factor in the
density reflects the multiplicity of the eigenvalues. However, this is not
true.

In a recent paper, Bakry and Zani BakZ , the authors considered real symmetric
matrices whose elements are independent Brownian motions depending on some
associative algebra structure of the Clifford type. Their computation of the
law of the spectrum shows that, even though there is still the term
$\prod_{1\leq j\leq k\leq n}|\lambda_{k}-\lambda_{j}|^{\beta}$ with
$\beta=1,2,4$, the factor $\beta$ here reflects the structure of the algebra,
known as Bott periodicity, rather than the dimension of the eigenspaces, which
in this situation may be as large as we want.

The previous study on Dyson Brownian motion, including the work of Bakry and
Zani BakZ  on Clifford algebra, mainly concentrated on the case where the
underlying algebra is associative. It is therefore worth understanding how
important this property is in the study of the related Dyson processes. The
octonion algebra, which is nonassociative but only alternative, provides a
good example for us to start with. Its structure differs from the Clifford
one, although Clifford algebras with 1 or 2 generators coincide with complex
numbers and quaternions, the Clifford algebra with 3 generators does not
coincide with octonions, even if the algebras have the same real dimension 8.
In his book Forrester , section $1.3.5$, Forrester mentions that the
distribution (1.1) with $\beta=8$ can be realized by $2\times 2$ matrices on
octonions, with Gaussian entries. It is therefore worth to look at the
associated Dyson process, which could also provide this result through the
study of its reversible measure.

There are only four normed division algebras: ${\mathbb{R}}$, ${\mathbb{C}}$,
${\mathbb{H}}$ and ${\mathbb{O}}$. We are familiar with ${\mathbb{R}}$,
${\mathbb{C}}$, and while the quaternion algebra ${\mathbb{H}}$ is
noncommutative but associative,the octonion algebra ${\mathbb{O}}$ is
nonassociative, but only alternative. Even though their properties are not so
nice, octonions have some important connections to different fields of
mathematics, such as geometry, topology and algebra. One interesting example
is its role in the classification of simple Lie algebra. There are 3 infinite
families of simple Lie algebras, coming from the isometry groups of the
projective spaces $\mathbb{RP}^{n}$, $\mathbb{CP}^{n}$ and $\mathbb{HP}^{n}$.
The remaining 5 simple Lie algebras were later discovered to be in connection
with octonions: they come from the isometry groups of the projective planes
over ${\mathbb{O}}$, ${\mathbb{O}}\otimes{\mathbb{C}}$,
${\mathbb{O}}\otimes{\mathbb{H}}$, ${\mathbb{O}}\otimes{\mathbb{O}}$ and the
automorphism group of octonions. It is also worth to mention that, according
to the independent work by Kervaire Kerv  and Bott-Milnor BottMilnor  in 1958,
there are only 4 parallelizable spheres: ${\mathbb{S}}^{0}$,
${\mathbb{S}}^{1}$, ${\mathbb{S}}^{3}$ and ${\mathbb{S}}^{7}$, which
correspond precisely to elements of unit norm in the normed division algebras
of the real numbers, complex numbers, quaternions, and octonions. See more
examples in the paper by J. Baez BaezJ .

For the eigenvalue problem of matrices on octonions, Y.G.Tian proved in his
paper tian  that $2\times 2$ Hermitian matrix on octonions has 2 eigenvalues,
each of them has multiplicity 8. For $3\times 3$ Hermitian octonionic matrix,
Dray-Manogue drayM  and Okubo oku  showed that it has 6 eigenvalues with
multiplicity 4. For $4\times 4$ and $5\times 5$ Hermitian octonionic matrices,
there are only numerical results, indicating that the eigenvalues have
multiplicity 2 tian . It is still unknown for matrices in higher dimension.
Following the analysis of Bakry and Zani BakZ , one may expect that the study
of probabilistic models on matrices of octonions could give new insights in
these directions.

In this paper, we consider Brownian motions on symmetric matrices of
octonions. Due to the fact that octonions are nonassociative, and in contrast
with the Clifford case, the dimension of the matrices plays a specific role.
In fact, contrary to the real, complex and quaternionic cases, octonions do
not give rise to infinite series of Lie groups but only specific ones, which
are closely related to dimension 2. Thus the study of Dyson processes is
mainly pertinent in this dimension, although we introduce another
probabilistic model related to the octonion algebra, but with a special
structure, see Section 4.2. To study the law of the spectrum of the matrices,
we consider the processes on the characteristic polynomials $P(X)$, as
introduced in the paper by Bakry and Zani BakZ . Because of the specific
structure of octonions, the traditional way to compute the law of the spectrum
turns out to be quite hard, while computation on the process of $P(X)$
provides a simpler and more efficient method to see things clearly.

The paper is organized as follows. Section 2 gives an introduction to the
basics of the octonion algebra; Section 3 explains briefly the language and
tools of symmetric diffusion process; Section 4 states our main results, two
specific models on octonions, and then we explain what is so special about
dimension 2; Section 5 is devoted to the demonstration of the connection
between the algebra structure and the Euclidean structure associated with the
associated symmetric matrices, and the fact that the the two exponents,
multiplicity of eigenvalues and exponent in the law of eigenvalues, are not
correlated.

##  2 The octonion algebra

In this section, we recall some facts about the octonion algebra, and we refer
to BaezJ  for more details. We start with a few definitions.

###### Definition 1

An algebra A is a division algebra if for any $a,b\in A$, with $ab=0$, then
either $a=0$ or $b=0$. A normed division algebra is a division algebra that is
also a normed vector space with $\|ab\|=\|a\|\|b\|$.

###### Definition 2

An algebra $A$ is alternative if the subalgebra generated by any two elements
is associative. By a theorem of Artin scha , this is equivalent to the fact
that for any $a,b\in A$, $(aa)b=a(ab),\ \ \ (ba)a=b(aa)$.

As mentioned earlier, there are only four normed division algebras,
${\mathbb{R}}$, ${\mathbb{C}}$, ${\mathbb{H}}$, ${\mathbb{O}}$. There is a
nice way called "Cayley-Dickson construction" to produce this sequence of
algebras: the complex number $a+ib$ can be seen as a pair of real numbers
$(a,b)$; the quaternions can be defined as a pair of complex number; and
similarly the octonions is a pair of quaternions. As the construction
proceeds, the property of the algebra becomes worse and worse: the quaternions
are noncommutative but associative, while the octonions are only alternative
but not associative.

Since octonions and Clifford algebra are both the algebra with dimension
$2^{n}$ (in this case $n=3$), which share some special property, we can use
the presentation provided in Bakry and Zani BakZ  to describe the algebra
structure on a basis of octonions, in order to simplify the computations. This
presentation is not classical, and we shall therefore use the table below.

Define $E=\\{1,2,3\\}$, and let $\mathcal{P}(E)$ denote the set of the subsets
of $E$. For every set $A\in\mathcal{P}(E)$,we associate a basis element
$\omega_{A}$ in the octonion algebra, with $\omega_{\emptyset}={\rm{Id}}$, the
identity element. Then an element $x\in{\mathbb{O}}$ can be written in the
form

$x=\sum_{A}x_{A}\omega_{A},~{}x_{A}\in{\mathbb{R}},$

and the product of two elements $x$ and $y$ is given by

$xy=\sum_{A,B}x_{A}y_{B}\omega_{A}\omega_{B}.$

It remains to define $\omega_{A}\omega_{B}$ for given $A,B\in\mathcal{P}(E)$,
which is given through the following rule: denote by $A.B$ the symmetric
difference $A\cup B\setminus(A\cap B)$, then
$\omega_{A}\omega_{B}=(A|B)\omega_{A.B}$, where $(A|B)$ takes value in
$\\{-1,1\\}$. Then, the multiplication rule in the octonion algebra is defined
by a sign table, which is as follows :

| $\emptyset$ | $\\{1\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{1,2\\}$ | $\\{1,3\\}$ | $\\{2,3\\}$ | $\\{1,2,3\\}$
---|---|---|---|---|---|---|---|---
$\emptyset$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
$\\{1\\}$ | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1
$\\{2\\}$ | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1
$\\{3\\}$ | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1
$\\{1,2\\}$ | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1
$\\{1,3\\}$ | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1
$\\{2,3\\}$ | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1
$\\{1,2,3\\}$ | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1

In this table, the element $(i,j)$ is the sign $(A_{i}|A_{j})$, where $A_{i}$
is the $i$th element in the first column, $A_{j}$ is the $j$th element in the
first row.

From the facts that for $A,B\neq\emptyset$, $\omega^{2}_{A}=-1$ and
$\omega_{A}\omega_{B}=-\omega_{B}\omega_{A}$, it is easy to get the following
rules:

$\displaystyle(A|A)$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-1,&\hbox{$A\neq\emptyset$,}\\\
1,&\hbox{$A=\emptyset$, }\end{array}\right.$ $\displaystyle(A|B)$
$\displaystyle=$ $\displaystyle-(B|A),\ \ \ for\ \ \ B\neq
A,A,B\neq\emptyset.$

It can be seen from the above table that ${\mathbb{O}}$ is an algebra, non-
associative but alternative. Moreover ${\mathbb{O}}$ can be equipped with the
Euclidean structure obtained by identifying ${\mathbb{O}}$ as a 8 dimensional
(real) vector space via

$x=\sum_{A}x_{A}\omega_{A}\mapsto(x_{\emptyset},x_{\\{1\\}},x_{\\{2\\}},x_{\\{3\\}},x_{\\{1,2\\}},x_{\\{1,3\\}},x_{\\{2,3\\}},x_{\\{1,2,3\\}})\,,$

so that the inner product and the norm are respectively :

$\langle x,y\rangle=\sum_{A}x_{A}y_{A}\ \ ,\ \
\|x\|=(\sum_{A}x_{A}^{2})^{1/2}\,,$

so that $\\{\omega_{A},~{}A\in{\mathcal{P}}(E)\\}$ form a real orthonormal
basis for the algebra ${\mathbb{O}}$.

Let us recall that to prove that ${\mathbb{O}}$ is a division algebra, it is
usual to introduce the conjugate

$x=\sum_{A}x_{A}\omega_{A}\mapsto x^{*}=\sum_{A}x_{A}\omega_{A}(A|A),$

and observe that $(xy)^{*}=y^{*}x^{*}$ , $xx^{*}=x^{*}x$ and
$\|x\|^{2}=xx^{*}$, so that
$\|xy\|^{2}=(xy)(xy)^{*}=(xy)(y^{*}x^{*})=x(yy^{*})x^{*}=\|x\|^{2}\|y\|^{2}$.

Altough the previous table does not provide an associative algebra, the
octonion algebra satisfies however some useful identities. In what follows, we
shall make a strong use of Moufang identities, which are stated as follows :
for elements $x,y,z$ belongs to ${\mathbb{O}}$, we have

$\displaystyle z(x(zy))$ $\displaystyle=$ $\displaystyle(zxz)y,$
$\displaystyle((xz)y)z$ $\displaystyle=$ $\displaystyle x(zyz),$
$\displaystyle(zx)(yz)$ $\displaystyle=$ $\displaystyle(z(xy))z,$
$\displaystyle(zx)(yz)$ $\displaystyle=$ $\displaystyle z((xy)z).$

We shall mainly use this for the elements $\omega_{A}\in{\mathbb{O}}$,
although Moufang identities provide more information than this.

According to the alternativity property, we can get some basic formulae about
the sign table $\\{(A|B)\\}$ for octonions,

###### Lemma 1

For $A,B,C,D\in\mathcal{P}(E)$, we have

  1. 1.

$(A.B|B)=(A|B)(B|B)$.

  2. 2.

$(A.B|A)(A.B|B)=(A.B|A.B)$.

  3. 3.

If $A.B\neq\emptyset$, $(A.C|A)(B.C|B)=-(A.C|B)(B.C|A)$.

  4. 4.

If $A.B.C.D=\emptyset$,

$(B.C|C)(C.D|D)(D.A|A)(A.B|B)=(B.D|B.D).$

###### Proof

The first one is just the result of alternativity:

$(\omega_{A}\omega_{B})\omega_{B}=(A|B)\omega_{A.B}\omega_{B}=(A.B|B)\omega_{A},$

while

$(\omega_{A}\omega_{B})\omega_{B}=\omega_{A}(\omega_{B}\omega_{B})=(B|B)\omega_{A},$

Hence

$(A.B|B)=(A|B)(B|B).$

The second one can be easily proved by the first statement.

For the third statement, we first remark that, for any vector $x=\sum
x_{C}\omega_{C}$, $x\omega_{A}$ is always orthogonal to $x\omega_{B}$ if
$A\neq B$. Indeed, to see this, we may reduce to the case where $\|x\|=1$, and
then observe that the fact that the algebra is a division algebra shows that
for any $y\in{\mathbb{O}}$, $y\mapsto xy$ is an orthogonal transformation. For
$A,B\in{\mathcal{P}}(E)$, $A.B\neq\emptyset$, and any $C\in{\mathcal{P}}(E)$,
choose $D=A.B.C$ and set $x=\omega_{C}+\omega_{D}$. Then

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\langle
x\omega_{A},x\omega_{B}\rangle$ $\displaystyle=$
$\displaystyle\langle(C|A)\omega_{C.A}+(D|A)\omega_{D.A},(C|B)\omega_{C.B}+(D|B)\omega_{D.B}\rangle$
$\displaystyle=$
$\displaystyle\langle(C|A)\omega_{C.A},(D|B)\omega_{D.B}\rangle+\langle(D|A)\omega_{D.A},(C|B)\omega_{C.B}\rangle$
$\displaystyle=$ $\displaystyle(C|A)(D|B)+(D|A)(C|B).$

Since $D=A.B.C$, the above formula indicates that for
$A,B,C\in{\mathcal{P}}(E)$, $A.B\neq\emptyset$,

$(C|A)(A.B.C|B)+(A.B.C|A)(C|B)=0.$

By changing $C$ into $A.C$, we get

$(A.C|A)(B.C|B)+(B.C|A)(A.C|B)=0,$

then the third statement is proved.

For the last statement, denote

$\Theta:=(B.C|C)(C.D|D)(D.A|A)(A.B|B).$ (2.3)

Then, from Moufang identities,

$\displaystyle((\omega_{B.C}\omega_{C})(\omega_{D.A}\omega_{A}))((\omega_{C.D}\omega_{D})(\omega_{A.B}\omega_{B}))$
$\displaystyle=$
$\displaystyle(B.C|C)(C.D|D)(D.A|A)(A.B|B)(\omega_{B}\omega_{D})(\omega_{C}\omega_{A})$
$\displaystyle=$ $\displaystyle\Theta(B|D)(C|A)\omega_{B.D}\omega_{C.A}$
$\displaystyle=$ $\displaystyle\Theta(B|D)(C|A)\omega_{B.D}\omega_{B.D}$
$\displaystyle=$ $\displaystyle\Theta(B|D)(C|A)(B.D|B.D).$

On the other hand,

$\displaystyle\omega_{A}\omega_{B}=(A.B|A.B)(A|A)(B|B)\omega_{B}\omega_{A}.$
$\displaystyle((\omega_{B.C}\omega_{C})(\omega_{D.A}\omega_{A}))((\omega_{C.D}\omega_{D})(\omega_{A.B}\omega_{B}))$
$\displaystyle=$
$\displaystyle(D|D)(B.C|B.C)(A|A)(A|A)(B|B)(A.B|A.B)((\omega_{B.C}\omega_{C})(\omega_{A}\omega_{B.C}))((\omega_{A.B}\omega_{D})(\omega_{B}\omega_{A.B}))$
$\displaystyle=$
$\displaystyle(D|D)(B.C|B.C)(B|B)(A.B|A.B)(\omega_{B.C}(\omega_{C}\omega_{A})\omega_{B.C})(\omega_{A.B}(\omega_{D}\omega_{B})\omega_{A.B})$
$\displaystyle=$
$\displaystyle(D|D)(B.C|B.C)(B|B)(A.B|A.B)(C|A)(D|B)(\omega_{B.C}\omega_{C.A}\omega_{B.C})(\omega_{A.B}\omega_{D.B}\omega_{A.B})$
$\displaystyle=$ $\displaystyle(D|D)(B|B)(C|A)(D|B)\omega_{C.A}\omega_{D.B}$
$\displaystyle=$ $\displaystyle(D|D)(B|B)(C|A)(D|B)(B.D|B.D).$

Hence,

$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle(B|D)(C|A)(B.D|B.D)(D|D)(B|B)(C|A)(D|B)(B.D|B.D)$
$\displaystyle=$ $\displaystyle(B|D)(D|D)(B|B)(D|B)$ $\displaystyle=$
$\displaystyle(B.D|B.D),$

which ends the proof of the lemma.

For a $n\times n$ matrix on octonions, write it as
$\mathcal{M}=\sum_{A}M^{A}\omega_{A}$, where $\\{M^{A}\\}$ are real $n\times
n$ matrices. For an $n$ dimensional vector $\sum_{B}X^{B}\omega_{B}$,

$\displaystyle(\sum_{A}M^{A}\omega_{A})(\sum
X^{B}\omega_{B})=\sum_{A,B}M^{A}X^{B}(A|B)\omega_{A.B}=\sum_{A,B}(A.B|B)M^{A.B}X^{B}\omega_{A}.$

Therefore, $\mathcal{M}$ can be expressed by the real $8n\times 8n$ block
matrix $\\{M^{A,B}_{ij}\\}$, where $M^{A,B}_{ij}=(A.B|B)M_{ij}^{A.B}$.

This leads to the following definition:

###### Definition 3

A $(2^{3}\times n)\times(2^{3}\times n)$ block matrix $M^{A,B}$ (where
$A,B\subset\\{1,2,3\\}$) is a real octonionic if $M^{A,B}=(A.B|B)M^{A.B}$,
where $M^{A}=M^{A,\emptyset}$ is a family of 8 $n\times n$ square matrices. It
is the real form of a matrix with octonionic entries. We shall denote it as
${\mathcal{M}}=\sum_{A}M^{A}\omega_{A}$.

Then, we shall say that an octonionic matrix is symmetric if its real form is
symmetric. This corresponds to the fact that, for any $A\in{\mathcal{P}}(E)$,
$(M^{A})^{t}=(A|A)M^{A}$.

That is to say, $(M^{A,B})^{t}=(A.B|B)(M^{A.B})^{t}=M^{B,A}=(B.A|A)M^{A.B}$.
Due to property 2 of Lemma 2.3, this leads to the fact that for any
$A\in{\mathcal{P}}(E)$, $(M^{A})^{t}=(A|A)M^{A}$, i.e. $M^{\emptyset}$ is
symmetric while $M^{A}$ is antisymmetric for any $A\neq\emptyset$.

It is worth to point out that since the octonion algebra is not associative,
there is no matrix presentation of the algebra structure for the octonions,
and therefore the matrix multiplication of the real octonionic matrices does
not corresponds to the octonionic multiplication of the associated matrices
with octonion entries. Even the product of octonionic matrices is not
octonionic in general.

The inverse of an octonionic matrix is in general not octonionic, and its
exact structure is not easy to decipher; the octonionic property may not be
preserved. The following lemma gives a condition for this last property to
hold, and will play an important role in the rest of this paper.

###### Lemma 2

Let $M=\sum M^{A}\omega_{A}$ be an octonionic matrix such that $M^{\emptyset}$
is invertible. Assume moreover that, for any $A,B\in\mathcal{P}(E)$

$M^{A}(M^{\emptyset})^{-1}M^{B}=M^{B}(M^{\emptyset})^{-1}M^{A},$ (2.4)

and that $\sum_{C}M^{C}(M^{\emptyset})^{-1}M^{C}$ is invertible. Then, $M$ is
invertible and its inverse $N$ is octonionic, satisfying
$N=\sum_{A}\omega_{A}N^{A}$, with

$N^{A}=-N^{\emptyset}M^{A}(M^{\emptyset})^{-1},\ \ forA\neq\emptyset,\\\ $
(2.5) $N^{\emptyset}=(\sum_{C}M^{C}(M^{\emptyset})^{-1}M^{C})^{-1}.\\\ $ (2.6)

###### Proof

In fact, assume the octonionic matrix
$N^{A,B}=(A.B|B)N^{A.B}=(A.B|A)\widetilde{N}^{A.B}$ is the inverse of $M$,
where

$N^{A.B}=(A.B|A.B)\widetilde{N}^{A.B}=\left\\{\begin{array}[]{ll}-\widetilde{N}^{A.B},&\hbox{$A.B\neq\emptyset$;}\\\
\widetilde{N}^{\emptyset},&\hbox{$A.B=\emptyset$.}\end{array}\right.$

Then

$\sum_{C}(A.C|A)(C.B|B)\widetilde{N}^{A.C}M^{C.B}=0,\ \ forA\neq B,\\\ $ (2.7)
$\sum_{C}(A.C|A)(C.A|A)\widetilde{N}^{A.C}M^{C.A}={\rm{Id}}.\\\ $ (2.8)

Changing $C$ into $A.B.C$ in $(\ref{eq3})$, we get
$\sum_{C}(B.C|A)(A.C|B)\widetilde{N}^{B.C}M^{C.A}=0$, then it is enough to
have

$(A.C|A)(C.B|B)\widetilde{N}^{A.C}M^{C.B}+(B.C|A)(A.C|B)\widetilde{N}^{B.C}M^{C.A}=0.$

According to Lemma 1, it holds as soon as

$\widetilde{N}^{A.C}M^{C.B}=\widetilde{N}^{B.C}M^{C.A}.$ (2.9)

Choosing $C=B$ and then setting $D=A.C$, this leads to

$\widetilde{N}^{D}=N^{\emptyset}M^{D}(M^{\emptyset})^{-1},$ (2.10)

for every $D\in{\mathcal{P}}(E)$. Now choose $C=\emptyset$ in $(\ref{eq1})$
and apply $(\ref{eqN})$ to $N^{A}$ and $N^{B}$, plugging into $(\ref{eq1})$,
this reduces to

$N^{\emptyset}M^{A}(M^{\emptyset})^{-1}M^{B}=N^{\emptyset}M^{B}(M^{\emptyset})^{-1}M^{A}.$

Now $(\ref{eq4})$ is

$\sum_{D}\widetilde{N}^{D}M^{D}={\rm{Id}},\ $

and using $(\ref{eqN})$, this gives

$N^{\emptyset}(\sum M^{C}(M^{\emptyset})^{-1}M^{C})^{-1}={\rm{Id}},$

which means that $N^{\emptyset}$ is invertible, and gives its inverse, such
that $(\ref{eqN2})$ holds true. Then we can use $(\ref{eqN2})$ and
$(\ref{eqN})$ to get $(\ref{eqN1})$.

###### Remark 1

It is worth to observe for later use that if the matrix ${\mathcal{M}}$ on the
octonions satisfies the assumptions of Lemma 2, then it is also the case of
${\mathcal{M}}-X{\rm{Id}}$.

##  3 Symmetric diffusion operators on matrices

We introduce the basics on symmetric diffusion operators, in a simplified
version adapted to our case. For further details see bglbook .

Let $E$ be an open set in ${\mathbb{R}}^{n}$, endowed with a $\sigma$-finite
measure $\mu$ and let ${\mathcal{A}}_{0}$ be the set of smooth compactly
supported functions, or of polynomials functions on $E$. For any linear
operator ${\rm L}:{\mathcal{A}}_{0}\mapsto{\mathcal{A}}_{0}$, we define its
carré du champ operator as

$\Gamma(f,g)=\frac{1}{2}\Big{(}{\rm L}(fg)-f{\rm L}(g)-g{\rm L}(f)\Big{)}\,.$

We have the following

###### Definition 4

A symmetric diffusion operator is a linear operator ${\rm L}$:
${\mathcal{A}}_{0}\oplus 1\mapsto{\mathcal{A}}_{0}$, such that

  1. 1.

${\rm L}(1)=0$,

  2. 2.

$\forall f,g\in{\mathcal{A}}_{0}\oplus 1,~{}\int f{\rm L}(g)\,d\mu=\int g{\rm
L}(f)\,d\mu$,

  3. 3.

$\forall f\in{\mathcal{A}}_{0},\Gamma(f,f)\geq 0$,

  4. 4.

$\forall f=(f_{1},\cdots,f_{n})$, where $f_{i}\in{\mathcal{A}}_{0}$ , $\Phi$
is a smooth function ${\mathbb{R}}^{n}\mapsto{\mathbb{R}}$ and $\Phi(0)=0$,

${\rm L}(\Phi(f))=\sum_{i}\partial_{i}\Phi(f){\rm
L}(f_{i})+\sum_{i,j}\partial^{2}_{ij}\Phi(f)\Gamma(f_{i},f_{j}).$ (3.11)

Consider an open set $\Omega\subset E$, and a given system of coordinates
$(x^{i})$, then we can write

${\rm
L}(f)=\sum_{ij}g^{ij}(x)\partial^{2}_{ij}f+\sum_{i}b^{i}(x)\partial_{i}f,$

where

$g^{ij}(x)=\Gamma(x^{i},x^{j}),~{}b^{i}(x)={\rm L}(x^{i}).$

In this paper, we perform computations on the characteristic polynomial
$P(X)={\rm{det}}(M-XId)$ of a matrix $M$. Assume that we have some diffusion
operator acting on the entries of a matrix $M$, described by the values of
${\rm L}(M_{ij})$ and $\Gamma(M_{ij},M_{kl})$ for any $(i,j,k,l)$. Then, we
have,

$\displaystyle\Gamma(\log P(X),\log P(Y))$ $\displaystyle=$
$\displaystyle\sum_{i,j,k,l}\partial_{M_{ij}}\log(P(X))\partial_{M_{kl}}\log(P(Y))\Gamma(M_{ij},M_{kl}),$
$\displaystyle{\rm L}(\log P(X))$ $\displaystyle=$
$\displaystyle\sum_{i,j}\partial_{M_{ij}}\log(P(X)){\rm
L}(M_{ij})+\sum_{i,j,k,l}\partial_{M_{ij}}\partial_{M_{kl}}\log(P(X))\Gamma(M_{ij},M_{kl}).$

To compute $\partial_{M_{ij}}\log(P(X))$ and
$\partial_{M_{ij}}\partial_{M_{kl}}\log(P(X))$ in the above formulae, we use
the Lemma 6.1 in Bakry and Zani BakZ , which we quote here without proof.

###### Lemma 3

Let $M=(M_{ij})$ be a matrix and $M^{-1}$ be its inverse, on the set
$\\{{\rm{det}}M\neq 0\\}$ we have

$\displaystyle\partial_{M_{ij}}\log{\rm{det}}M$ $\displaystyle=$
$\displaystyle M^{-1}_{ji},$
$\displaystyle\partial_{M_{ij}}\partial_{M_{kl}}\log{\rm{det}}M$
$\displaystyle=$ $\displaystyle-M^{-1}_{jk}M^{-1}_{li}.$

Hence, with $M^{-1}(X)=(M-XId)^{-1}$,

$\displaystyle\Gamma(\log P(X),\log
P(Y))=\sum_{i,j,k,l}M^{-1}(X)_{ji}M^{-1}(Y)_{lk}\Gamma(M_{ij},M_{kl}),$ (3.12)
$\displaystyle{\rm L}(\log P(X))=\sum_{i,j}M^{-1}_{ji}(X){\rm
L}(M_{ij})-\sum_{i,j,k,l}M^{-1}_{jk}(X)M^{-1}_{li}(X)\Gamma(M_{ij},M_{kl}).$
(3.13)

According to Bakry and Zani BakZ , one can get from $\Gamma(P(X),P(Y))$ and
${\rm L}(P(X))$ informations about the multiplicities of the eigenvalues, and
on the invariant measure of the operator ${\rm L}$ acting on $P(X)$:
If for some constants $\alpha_{1},\alpha_{2},\alpha_{3}$,

${\rm L}(P)=\alpha_{1}P^{\prime\prime}+\alpha_{2}\frac{P^{\prime
2}}{P},~{}\Gamma(\log P(X),\log
P(Y))=\frac{\alpha_{3}}{Y-X}\Big{(}\frac{P^{\prime}(X)}{P(X)}-\frac{P^{\prime}(Y)}{P(Y)}\Big{)}.$
(3.14)

and if there exists for some $a\in{\mathbb{R}}$, $a\neq 0$ which satisfies

$a^{2}(\alpha_{1}+\alpha_{2})-a(\alpha_{1}+\alpha_{3})+\alpha_{3}=0,$ (3.15)

Then

  1. 1.

If $a$ is a positive integer, it is the multiplicity of the eigenvalues of
$M$;

  2. 2.

Write $P(X)=\prod^{n}_{i=1}(X-x_{i})^{a}$, the invariant measure for the
operator ${\rm L}$ in the Weyl chamber $\\{x_{1}<...<x_{n}\\}$ is, up to a
multiplicative constant,

$d\mu=(\prod_{i<j}(x_{i}-x_{j})^{2})^{-\frac{a^{2}(\alpha_{1}+\alpha_{2})}{\alpha_{3}}}d\mu_{0},$

where $d\mu_{0}$ is the Lebesgue measure.

##  4 Symmetric matrices on octonions

Our aim is to describe the law of the spectrum of the real form of symmetric
matrices on octonions. The block matrix is
$\mathcal{M}=((A.B|B)M^{A.B})_{A,B\in{\mathcal{P}}(E)}$, satisfying
$(M^{A})^{t}=(A|A)M^{A}$ from the symmetry assumption.

We will focus on cases where the symmetry condition $(\ref{symm})$ of matrix
$\mathcal{M}-X\mathrm{Id}$ is satisfied, i.e. where the matrix

$U(X):=(\mathcal{M}-X\mathrm{Id})^{-1}$

is octonionic (almost surely for the stochastic process under consideration).

Setting

$P(X):={\rm{det}}(\mathcal{M}-X\mathrm{Id}),$

by Lemma 3 we have

$\displaystyle\Gamma(\log P(X),\log P(Y))$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}A,B,C,D\\\
i,j,k,l\end{subarray}}U_{ji}^{B,A}U_{lk}^{D,C}\Gamma(M_{ij}^{A,B},M_{kl}^{C,D})$
(4.16) $\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}A,B,C,D\\\
i,j,k,l\end{subarray}}(A.B|A.B)(C.D|C.D)U_{ji}^{A.B}U_{lk}^{C.D}\Gamma(M_{ij}^{A.B},M_{kl}^{C.D})$

(where we used property 2 of Lemma 2.3), and

${\rm L}(\log P(X))=\sum_{\begin{subarray}{c}A,B\\\
i,j\end{subarray}}U^{A,B}_{ji}(X){\rm
L}(M^{A,B}_{ij})-\sum_{\begin{subarray}{c}A,B,C,D\\\
i,j,k,l\end{subarray}}U^{B,C}_{jk}(X)U^{D,A}_{li}(X)\Gamma(M^{A,B}_{ij},M^{C,D}_{kl}).\\\
$ (4.17)

For further use in both examples, we state (without proof) two preliminary
lemmas. The first lemma collects some elementary facts, consequences of the
definition, the symmetries and property 2 of Lemma 1.

###### Lemma 4

For $F\in{\mathcal{P}}(E)$, we have

$\displaystyle\textmd{trace}\,\\!\ U(X)^{F}$ $\displaystyle=$
$\displaystyle\sum_{i}U(X)^{F}_{ii}\,,$ (4.18)
$\displaystyle\textmd{trace}\,\\!\ U(X)$ $\displaystyle=$ $\displaystyle 8\\!\
\textmd{trace}\,\\!\ U(X)^{\emptyset}\,,$ (4.19)
$\displaystyle\textmd{trace}\,\\!\ [U(X)^{F}U(Y)^{F}]\ $ $\displaystyle=$
$\displaystyle\sum_{ij}U(X)^{F}_{ji}U(Y)^{F}_{ij}\,,$ (4.20)
$\displaystyle(F|F)\textmd{trace}\,\\!\ [U(X)^{F}U(Y)^{F}]$ $\displaystyle=$
$\displaystyle\sum_{ij}U(X)^{F}_{ji}U(Y)^{F}_{ji}\,,$ (4.21)
$\displaystyle\textmd{trace}\,\\!\ [U(X)U(Y)]$ $\displaystyle=$ $\displaystyle
8\sum_{C}(C|C)\textmd{trace}\,\\!\ [U(X)^{C}U(Y)^{C}]\,.$ (4.22)

The second lemma gives expressions of the traces in terms of the
characteristic polynomials. The first two identities are obtained by
derivation from

$\log P(X)=\textmd{trace}\,\\!\
\log(\mathcal{M}-X\hbox{Id})=-\textmd{trace}\,\\!\ \log U(X)\,,$

and the third one is a consequence of the first one and the resolvent
equation.

###### Lemma 5

$\displaystyle\textmd{trace}\,\\!\ U(X)$ $\displaystyle=$
$\displaystyle\frac{P^{\prime}(X)}{P(X)}\,,$ (4.23)
$\displaystyle\textmd{trace}\,\\!\ (U(X)^{2})$ $\displaystyle=$
$\displaystyle\frac{P^{\prime}(X)^{2}}{P(X)^{2}}-\frac{P^{\prime\prime}(X)}{P(X)}\,,$
(4.24) $\displaystyle\textmd{trace}\,\\!\ [U(X)U(Y)]$ $\displaystyle=$
$\displaystyle\frac{1}{Y-X}\left(\frac{P^{\prime}(X)}{P(X)}-\frac{P^{\prime}(Y)}{P(Y)}\right)\,.$
(4.25)

###  4.1 The dimension 2 case

Consider $M=\sum M^{A}\omega_{A}$, where $\\{M^{A}\\}$ are matrices whose
elements are independent Brownian motions. For $A\neq\emptyset$, due to
symmetry of $\mathcal{M}$, $(M^{A})^{t}=(A|A)M^{A}=-M^{A}$. Such matrices
naturally satisfy the symmetry restriction 2.4 in dimension 2, since the
$2\times 2$ antisymmetric matrices are all of the form
$\left(\begin{array}[]{cc}0&-z\\\ z&0\\\ \end{array}\right)$, and they are
therefore all proportional to each other. This property fails in higher
dimensions. Set

$\Gamma(M^{A}_{ij},M^{B}_{kl})=\frac{1}{2}\delta_{A,B}(\delta_{ik}\delta_{jl}+(A|A)\delta_{il}\delta_{jk}),\
\ \ L(M^{A}_{ij})=0,$ (4.26)

which reflects the symmetry of the matrices. Notice that the inverse matrix
$U(X)$ is also symmetric with $(U^{A})^{t}=(A|A)U^{A}$. We have the following
result

###### Proposition 1

For the $2\times 2$ symmetric matrix $M=\sum M^{A}\omega_{A}$,

$\Gamma(\log P(X),\log
P(Y))=\frac{8}{Y-X}\left(\frac{P^{\prime}(X)}{P(X)}-\frac{P^{\prime}(Y)}{P(Y)}\right),$
(4.27) $L(\log
P(X))=3\left(\frac{P^{\prime}(X)^{2}}{P(X)^{2}}-\frac{P^{\prime\prime}(X)}{P(X)}\right)-\frac{1}{2}\frac{P^{\prime}(X)^{2}}{P(X)^{2}}.\\\
$ (4.28)

###### Proof

Proof of (4.27) : From (4.16) and (4.26) we have

$\displaystyle\Gamma(\log P(X),\log P(Y))$ $\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}A,B,C,D\\\
AB=CD\end{subarray}}(A.B|A.B)(C.D|C.D)U(X)^{B.A}_{ji}U(Y)^{D.C}_{lk}\frac{1}{2}(\delta_{ik}\delta_{jl}+(A.B|A.B)\delta_{il}\delta_{jk})$
$\displaystyle=$ $\displaystyle 8\sum_{F}{\rm tr}(U(X)^{F}U(Y)^{F})(F|F)$
$\displaystyle=$ $\displaystyle 8{\rm
tr}(U(X)U(Y))=\frac{8}{Y-X}\left(\frac{P^{\prime}(X)}{P(X)}-\frac{P^{\prime}(Y)}{P(Y)}\right),$

where we applied (4.21), (4.20), (4.22) and formula 3 in Lemma 5. This ends
the proof of (4.27).

Proof of (4.28) : In the following, we write $U$ is for $U(X)$. From (4.17)
and (4.26), we have

$\displaystyle{\rm L}(\log P(X))$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{B,C,D}(B.D|B.D)\sum_{ij}(U_{ji}^{B.C})^{2}$
$\displaystyle-\frac{1}{2}\sum_{\begin{subarray}{c}A,B,C,D\\\
A.B.C.D=\emptyset\end{subarray}}(B.D|B.D)(A.B|A.B)(\sum_{i}U_{ii}^{BC})^{2}.$

On the one hand, in view of (4.21) and (4.22)

$\displaystyle\sum_{B,C,D}(B.D|B.D)\sum_{ij}\left(U_{ji}^{B.C}\right)^{2}=-6\\!\
\textmd{trace}\,\\!\ (U^{2})\,.$

On the other hand, since for $F\not=\emptyset$, $U^{F}$ is antisymmetric, so
that $\hbox{tr}\\!\ U^{F}=0$,

$\displaystyle\displaystyle\sum_{\begin{subarray}{c}A,B,C,D\\\
A.B.C.D=\emptyset\end{subarray}}(B.D|B.D)(A.B|A.B)(\sum_{i}U_{ii}^{B.C})^{2}=8^{2}\left(\textmd{trace}\,\\!\
U^{\emptyset}\right)^{2}.$

Combined as in (4.1), these sums give :

$L(\log P(X))=3\hbox{tr}\\!\ U^{2}-4\cdot 8\left(\hbox{tr}\\!\
U^{\emptyset}\right)^{2}=3\hbox{tr}\\!\ U^{2}-\frac{1}{2}\left(\hbox{tr}\\!\
U\right)^{2},$ (4.29)

which, in view of (4.23) and (4.24) ends the proof of (4.28) and then the
proof of the Proposition.

###### Remark 2

s By the results of above proposition and formula $(\ref{formulaL})$, it is
easy to get

$\displaystyle
L(P)=(11-\frac{1}{2})\frac{P^{\prime}(X)^{2}}{P(X)}-11P^{\prime\prime}(X).$

Now chose $\alpha_{1}=-11$, $\alpha_{2}=11-\frac{1}{2}$ and $\alpha_{3}=8$ in
formula $(\ref{eq.gal.P})$ : the resulting value for $a$ is $a=8$. This shows
that the multiplicity of the eigenvalues is 8.
Assume $\rho$ is the density of the invariant measure of $L$ of the
coordinates $\\{x_{i}\\}$ in the Weyl chamber, then according to our
discussion in the previous section, we have

$\rho=C\prod_{i<j}(x_{i}-x_{j})^{8}.$

###  4.2 Another model in any dimension

We now provide another set of random octonionic matrices for which the
symmetry condition $(\ref{symm})$ is automatically satisfied.

Let $M^{\emptyset}$ be a symmetric matrix with independent Brownian motions as
its entries. For all $A,B\neq\emptyset$, let $M^{A}=M^{B}={\mathcal{A}}$ be a
random antisymmetric matrix with independent Brownian motion as its off
diagonal entries. Then consider
$M=M^{\emptyset}\omega_{\emptyset}+{\mathcal{A}}\sum_{C\neq\emptyset}\omega_{C}$.
This model is similar to the Hermitian case considered in Bakry and Zani BakZ
(see Remark 4). Similarly to the Hermitian case, we set

$\Gamma(M^{\emptyset}_{ij},M^{\emptyset}_{kl})=\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$
(4.30)
$\Gamma({\mathcal{A}}_{ij},{\mathcal{A}}_{kl})=\frac{1}{14}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}),$
(4.31) $\Gamma(M^{\emptyset}_{ij},{\mathcal{A}}_{kl})=0,\ \ \ {\rm
L}(M^{A}_{ij})=0.$ (4.32)

Due to Lemma 1, for the inverse matrix $U(X)=(M-XI)^{-1}$, we have for every
$C\neq\emptyset$

$U^{C}=-U^{\emptyset}M^{C}(M^{\emptyset}-XI)^{-1}=-U^{\emptyset}{\mathcal{A}}(M^{\emptyset}-XI)^{-1},$

which means for all $C\neq\emptyset$, $U^{C}$ is the same, and we denote it by
$U_{a}$.

###### Proposition 2

For the matrix
$M=M^{\emptyset}\omega_{\emptyset}+\sum_{C\neq\emptyset}{\mathcal{A}}\omega_{C}$
on the octonions,

$\displaystyle\Gamma(\log P(X),\log P(Y))$ $\displaystyle=$
$\displaystyle\frac{8}{Y-X}\left(\frac{P^{\prime}(X)}{P(X)}-\frac{P^{\prime}(Y)}{P(Y)}\right),$
$\displaystyle L(\log P)$ $\displaystyle=$
$\displaystyle-\frac{1}{8}\frac{P^{\prime}(X)^{2}}{P(X)^{2}}.$

###### Proof

On the one hand, from (4.16) and (4.32)

$\Gamma(\log P(X),\log P(Y))=\frac{1}{2}S_{1}+\frac{1}{14}S_{2},$ (4.33)

where

$\displaystyle S_{1}$ $\displaystyle:=$
$\displaystyle\sum_{A=B,C=D}\sum_{i,j,k,l}U(X)^{\emptyset}_{ji}U(Y)^{\emptyset}_{lk}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$
$\displaystyle S_{2}$ $\displaystyle:=$
$\displaystyle\sum_{A\not=B,C\not=D}\sum_{i,j,k,l}U(X)^{A.B}_{ji}U(Y)^{C.D}_{lk}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})\,.$

A careful computation, using the fact that $U(Y)^{\emptyset}$ is symmetric and
$U(Y)_{a}$ is antisymmetric gives

$S_{1}=2\cdot 8^{2}\\!\ \textmd{trace}\,\\!\
\left[U(X)^{\emptyset}U(Y)^{\emptyset}\right]\ ,\ S_{2}=-2\cdot 7^{2}\cdot
8^{2}\\!\ \textmd{trace}\,\\!\ \left[U(X)_{a}U(Y)_{a}\right]\,,$ (4.34)

and then, using (4.33)

$\Gamma(\log P(X),\log P(Y))=8^{2}\\!\ \textmd{trace}\,\\!\
\left[U(X)^{\emptyset}U(Y)^{\emptyset}\right]-7\cdot 8^{2}\\!\
\textmd{trace}\,\\!\ \left[U(X)_{a}U(Y)_{a}\right].$

Going back to (4.22) we see that

$\displaystyle\hbox{tr}\\!\ \left[U(X)U(Y)\right]$ $\displaystyle=$
$\displaystyle 8\\!\ \textmd{trace}\,\\!\
\left[U(X)^{\emptyset}U(Y)^{\emptyset}\right]-7\cdot 8\\!\
\textmd{trace}\,\\!\ \left[U(X)_{a}U(Y)_{a}\right],$ (4.35)

so that

$\Gamma(\log P(X),\log P(Y))=8\\!\ \hbox{tr}\\!\
\left[U(X)U(Y)\right]=\frac{8}{Y-X}\left(\frac{P^{\prime}(X)}{P(X)}-\frac{P^{\prime}(Y)}{P(Y)}\right)\,.$

On the other hand, with $U$ for $U(X)$

$\displaystyle{\rm L}(\log P)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\displaystyle\sum_{\begin{subarray}{c}A=B,C=D\\\
i,j,k,l\end{subarray}}(B.C|C)(D.A|A)(A.B|B)(C.D|D)U(X)^{B.C}_{jk}U(X)^{D.A}_{li}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$
$\displaystyle-\frac{1}{14}\displaystyle\sum_{\begin{subarray}{c}A\not=B,C\not=D\\\
i,j,k,l\end{subarray}}(B.C|C)(D.A|A)(A.B|B)(C.D|D)U(X)^{B.C}_{jk}U(X)^{D.A}_{li}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})$
$\displaystyle=:$
$\displaystyle-\frac{1}{2}S^{\prime}_{1}-\frac{1}{14}S^{\prime}_{2}\,.$

Let us first remark that

$\sum_{ijkl}U_{jk}^{F}U_{li}^{G}(\delta_{ik}\delta_{jl}\pm\delta_{il}\delta_{jk})=(F|F)\textmd{trace}\,\\!\
(U^{F}U^{G})\pm(\textmd{trace}\,\\!\ U^{F})(\textmd{trace}\,\\!\ U^{G}).$
(4.36)

For the first part, we have

$\displaystyle S^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\sum_{B,C}\textmd{trace}\,\\!\
(U^{B.C})^{2}+\sum_{B,C}(B.C|B.C)[\textmd{trace}\,\\!\ U^{B.C}]^{2}$
$\displaystyle=$ $\displaystyle 8\left[\textmd{trace}\,\\!\
(U^{\emptyset})^{2}+7\textmd{trace}\,\\!\
(U_{a})^{2}\right]+8[\textmd{trace}\,\\!\ U^{\emptyset}]^{2}.$

For the second part, going back to the notation (2.3) for $\Theta$ and
applying (4.36) we have

$\displaystyle
S^{\prime}_{2}=\sum_{A\not=B,C\not=D}\Theta\times\left((B.C|B.C)\textmd{trace}\,\\!\
[U^{B.C}U^{A.D}]-\textmd{trace}\,\\!\ U^{B.C}\textmd{trace}\,\\!\
U^{A.D}\right).$ (4.37)

Let us split the sum into four parts according to $B=C$ or not, and $A=D$ or
not.

i) When $A\not=D,B=C$, the sum vanishes. Indeed, in this case,

$(B.C|B.C)\textmd{trace}\,\\!\ [U^{B.C}U^{A.D}]-(\textmd{trace}\,\\!\
U^{B.C}\textmd{trace}\,\\!\ U^{A.D})=\textmd{trace}\,[U^{\emptyset}U_{a}],$
(4.38)

and $\Theta=(A.D|A)(A.B|B)(B.D|D)$ which is antisymmetric in $A,B$.

ii) The same occurs when $A=D$ and $B\not=C$.

iii) When $A=D$ and $B=C$,

$(B.C|B.C)\textmd{trace}\,\\!\ [U^{B.C}U^{A.D}]-(\textmd{trace}\,\\!\
U^{B.C}\textmd{trace}\,\\!\ U^{A.D})=\textmd{trace}\,\\!\
[U^{\emptyset}]^{2}-(\textmd{trace}\,\\!\ U^{\emptyset})^{2}\,,$

and $\Theta=(B.D|B.D)$ in view of property 4 of Lemma 2.3, so that the
contribution is

$\displaystyle\sum_{A\not=B,C\not=D,B=C,A=D}(B.D|B.D)\left((B.C|B.C)\textmd{trace}\,\\!\
[U^{B.C}U^{A.D}]-\textmd{trace}\,\\!\ U^{B.C}\textmd{trace}\,\\!\
U^{A.D}\right)$ $\displaystyle=$ $\displaystyle-7\cdot
8\left(\textmd{trace}\,\\!\ [U^{\emptyset}]^{2}-(\textmd{trace}\,\\!\
U^{\emptyset})^{2}\right)\,.$

iv) When $A\not=D,B\not=C$,

$(B.C|B.C)\textmd{trace}\,\\!\ [U^{B.C}U^{A.D}]-(\textmd{trace}\,\\!\
U^{B.C}\textmd{trace}\,\\!\ U^{A.D})=-\textmd{trace}\,\\!\ (U_{a})^{2}\,.$

With the help of some computer algebra, we get

$\sum_{A\not=B,C\not=D,B\not=C,A\not=D}\Theta=2^{3}\cdot 7^{2}\,,$

and finally, all the contributions in (4.37) give so that

$\displaystyle S^{\prime}_{2}=-7\cdot 8\left(\textmd{trace}\,\\!\
[U^{\emptyset}]^{2}-(\textmd{trace}\,\\!\ U^{\emptyset})^{2}\right)-7^{2}\cdot
8\\!\ \textmd{trace}\,\\!\ (U_{a})^{2}\,.$

Going back to (4.2) and (4.2), we conclude, using again (4.32)

${\rm L}(\log P)=-8(\textmd{trace}\,\\!\ U^{\emptyset})^{2}\
=-\frac{1}{8}(\textmd{trace}\,\\!\ U)^{2}\,,$

which ends the proof of the proposition.

###### Remark 3

Similarly we have

${\rm
L}(P(X))=(8-\frac{1}{8})\frac{P^{\prime}(X)^{2}}{P}-8P^{\prime\prime}(X).\\\ $

Chose $\alpha_{1}=-8$, $\alpha_{2}=8-\frac{1}{8}$, $\alpha_{3}=8$, we still
obtain the multiplicity $a=8$, while the density of the invariant measure of
$L$ is then

$C\prod_{i<j}|x_{i}-x_{j}|^{2}.$

###### Remark 4

Recall that in Bakry and Zani BakZ , section 7.1, for a Hermitian matrix
$H=M+iA$ with independent Brownian motions as its entries (where $M$ is
symmetric, $A$ is anti-symmetric), we have

$\displaystyle\Gamma(M_{ij},M_{kl})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$
$\displaystyle\Gamma(A_{ij},A_{kl})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}).$

In our model
$M=M^{\emptyset}\omega_{\emptyset}+{\mathcal{A}}\sum_{C\neq\emptyset}\omega_{C}$,
denote $e$ the specific element in the octonion algebra,
$e=\sum_{C\neq\emptyset}\omega_{C}$. Notice that

$e^{2}=-7,$

which indicates that $e$ works like $i$ in the Hermitian matrices, just with a
different variance. Therefore, this example is indeed similar to the case of
Hermitian matrices.

###### Remark 5

In this remark we would like to discuss why the dimension 2 is so special.
Consider a more general model: let

$M=M^{\emptyset}\omega_{0}+\sum_{C\neq\emptyset}M^{C}\omega_{C},$

with $M^{C}=x_{C}A_{0}$, where $M^{\emptyset}$ is a Brownian motion on
symmetric matrices, $\\{x_{C}\\}$ a series of Brownian motions on
$\mathbb{R}$, and $A_{0}=\\{a_{ij}\\}$ a fixed anti-symmetric matrix.
Obviously this model satisfies the symmetry condition 2.4. When $M$ is a
$2\times 2$ matrix, it can be considered as a special example of the first
case. Let $e=\sum_{C\neq\emptyset}x_{C}\omega_{C}$. Different from the
previous model, in this case $e$ can be considered as a Brownian motion on the
basis of octonions satisfying
$e^{2}=-\sum_{C\neq\emptyset}|x_{C}|^{2}=-|e|^{2}$.

Therefore,

$\displaystyle\Gamma(M^{\emptyset}_{ij},M^{\emptyset}_{kl})$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$
$\displaystyle\Gamma(M^{A}_{ij},M^{B}_{kl})$ $\displaystyle=$
$\displaystyle\delta_{A=B}a_{ij}a_{kl},A,B\neq\emptyset,$
$\displaystyle\Gamma(M^{\emptyset}_{ij},M^{A}_{kl})$ $\displaystyle=$
$\displaystyle 0,A\neq\emptyset.$

Similar computations yield

$\displaystyle\Gamma(\log P(X),\log P(Y))$ $\displaystyle=$ $\displaystyle
8\sum_{A.B=\emptyset}{\rm
tr}(U(X)^{\emptyset}U(Y)^{\emptyset})+8\sum_{A.B\neq\emptyset}({\rm
tr}(U(X)^{B.A}A_{0}){\rm tr}(U(Y)^{B.A}A_{0})),$ $\displaystyle L(\log P)$
$\displaystyle=$ $\displaystyle-\frac{1}{2}\sum_{A.C\neq\emptyset}{\rm
tr}(U(X)^{A.C})^{2}-4{\rm tr}(U(X)^{\emptyset})^{2}-4({\rm
tr}U(X)^{\emptyset})^{2}$ $\displaystyle+5\sum_{B.C\neq\emptyset}{\rm
tr}(U(X)^{B.C}A_{0}U(X)^{B.C}A_{0})+56{\rm
tr}(U(X)^{\emptyset}A_{0}U(X)^{\emptyset}A_{0}),$

which are hard to describe in terms of $P$. When the matrix is $2\times 2$,
the following equalities hold: for $C\neq\emptyset$,

$\displaystyle{\rm tr}(U(X)^{C}A_{0}){\rm tr}(U(Y)^{C}A_{0})$ $\displaystyle=$
$\displaystyle-{\rm tr}(U(X)^{C}U(Y)^{C}),$ $\displaystyle{\rm
tr}(U(X)^{C}A_{0}U(X)^{C}A_{0})$ $\displaystyle=$ $\displaystyle-{\rm
tr}(U(X)^{C})^{2},$ $\displaystyle{\rm
tr}(U(X)^{\emptyset}A_{0}U(X)^{\emptyset}A_{0})$ $\displaystyle=$
$\displaystyle{\rm tr}(U^{\emptyset})^{2}-({\rm tr}U^{\emptyset})^{2}.$

which give rise to the results in the first model.

However, in higher dimensions, the above conditions are hard to satisfy. In
fact when $n=2$, it is enough to take $A_{0}=\left(\begin{array}[]{cc}0&-1\\\
1&0\\\ \end{array}\right)$. Set $x_{\\{1\\}}=z$. By the formula

$\displaystyle U^{\emptyset}$ $\displaystyle=$
$\displaystyle(M^{\emptyset}+z^{2}A_{0}(M^{\emptyset}){-1}A_{0})^{-1},$
$\displaystyle U^{\\{1\\}}$ $\displaystyle=$ $\displaystyle-
zU^{\emptyset}A_{0}(M^{\emptyset}){-1}.$

It is easily seen that $U^{\\{1\\}}$ is a $2\times 2$ antisymmetric matrix and
can be written as

$U^{\\{1\\}}=\lambda A_{0},$

where $\lambda=\frac{z}{z^{2}-{\rm{det}}(M^{\emptyset})}$. Compare it with the
expressions of $U^{\emptyset}$ and $U^{\\{1\\}}$, we have
$A_{0}=\frac{\lambda}{\lambda z^{2}-z}M^{\emptyset}A_{0}M^{\emptyset}$. Since
$A_{0}^{2}=-I$, this leads to $(M^{\emptyset}A_{0})^{2}=\frac{z-\lambda
z^{2}}{\lambda}I=-{\rm{det}}(M^{\emptyset})I$, which is impossible to hold for
any symmetric matrix $M^{\emptyset}$ in higher dimensions. This restriction
insures the first two conditions, and the third one is proved by this and the
fact that ${\rm tr}(M^{2})-({\rm tr}M)^{2}=-2{\rm{det}}(M)$ holds in dimension
2.

##  5 Some remarks

Our two models provide examples where the multiplicity of eigenvalues and the
exponent $\beta$ in the law are not related, which is in accordance with the
conclusion in Bakry and Zani BakZ , that the exponent reflects the structure
of the algebra while the multiplicity of the eigenvalues is decided by the
dimension of the eigenspaces.

As we have seen, the octonionic structure of the matrix plays an important
role. For higher dimension, the problem may be studied by our method if we
know the structure of the inverse matrix, which is not necessarily octonionic.
The main obstacle is still the non-associativity, which prevents any matrix
presentation for octonionic multiplication. Let us recall that the $3\times 3$
matrices on octonions have been studied by Dray and Manogue drayM  and Okubo
oku  using algebraic method, showing that there are 6 eigenvalues with
multiplicity 4. It it still an open problem to provide a probabilistic model
in this case which would lead to this conclusion.

##  6 Acknowledgement

I would like to thank my supervisor, Prof. Dominique Bakry, for his precious
suggestions and helpful discussions. I am also indebted to my other
supervisors, Prof. Jiangang Ying and Prof. Xiangdong Li, for their support and
help. I would also like to express my gratitude to the referee for careful
reading and suggestions.

This research was supported by China Scholarship Council.

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