$\mathrm{X}^{\prime}\subset\mathrm{X}$, and every distributive aggregate
function $\mathrm{\color[rgb]{0,0,0}{F}}$ using
$\mathrm{\color[rgb]{0,0,0}{G}}$ that is applicable to $M$, we have:

(5)
$\mathpzc{Agg}^{*}_{T}(F(M)|\mathrm{\scalebox{0.8}{B}})=\mathpzc{Agg}^{*}_{T^{\prime}}(G(F(M))|\mathrm{\scalebox{0.8}{B}})\mathrm{\;where\;}T^{\prime}=\mathpzc{Agg}^{*}_{T}(\mathrm{\color[rgb]{0,0,0}{F}}(\mathrm{\scalebox{0.8}{M}})|\mathrm{X}^{\prime}\cup\mathrm{\scalebox{0.8}{B}})$

The above definition of summarizability relates to summarizability conditions
as follows. In Definition 4.1, we consider a fact table
$T^{\prime}=\mathpzc{Agg}^{*}_{T}(\mathrm{\color[rgb]{0,0,0}{F}}(\mathrm{\scalebox{0.8}{M}})\mid\mathrm{X})$
resulting from a summarization query over a fact table $T$, where
$\mathrm{\color[rgb]{0,0,0}{F}}$ is a distributive function using
$\mathrm{\color[rgb]{0,0,0}{G}}$. Then we consider that _ M is summarizable
with respect to $\mathrm{X}$ and $\mathrm{\color[rgb]{0,0,0}{F}}$_ when
$\mathpzc{Agg}^{*}_{T}(\mathrm{\color[rgb]{0,0,0}{F}}(\mathrm{\scalebox{0.8}{M}})|\mathrm{Z})=\mathpzc{Agg}^{*}_{T^{\prime}}(\mathrm{\color[rgb]{0,0,0}{G}}(\mathrm{\color[rgb]{0,0,0}{F}}(\mathrm{\scalebox{0.8}{M}}))|\mathrm{Z})$
for any $Z\subset X$. In this case, query
$\mathpzc{Q}\,=\mathpzc{Agg}^{*}_{T^{\prime}}(\mathrm{\color[rgb]{0,0,0}{G}}(\mathrm{\color[rgb]{0,0,0}{F}}(M)|\mathrm{Z})$
is considered to be correct by (Hurtado et al., 2005). In the above
definition, Condition 5 must hold for any grouping set
$\mathrm{X}^{\prime}\subset\mathrm{X}$ in the query defining $T^{\prime}$, but
$\mathrm{Z}$ is restricted to  B . Therefore, the condition to determine if
query $\mathpzc{Q}\,$ is correct, is to enforce that _every attribute B of any
dimension ${D}$ in $\mathrm{Z}$ is summarizable from $\mathrm{X}_{D}$, where
$\mathrm{X}_{D}$ is the set of dimension attributes of ${D}$ in $\mathrm{X}$_
(Equation 5 must hold for all attributes  B of any dimension ${D}$ and any
subset of attributes $\mathrm{X}^{\prime}\subset\mathrm{X}_{D}$).

(Hurtado et al., 2005) show that the summarizability Condition 5 can also be
expressed independently of the fact tables which refer to a given dimension:
attribute  B is summarizable from $\mathrm{X}$ in dimension ${D}$ if and only
if for every bottom level attribute $\mathrm{\scalebox{0.8}{A}}_{\bot}$ of
${D}$, the following equality holds, where $\pi$ denotes the relational
duplicate elimination projection and $\bowtie$ denotes the null-eliminating
join:

(6) $\pi_{A_{\bot},B}(D)=\bigcup_{A_{i}\in
X}(\pi_{A_{\bot},A_{i}}(D)\bowtie\pi_{A_{i},B}(D))$

###### Example 0.

Consider the product dimension ${PROD\\_NEW}$ in Table 33. Using Equation 6,
we can show that attribute  CATEGORY is summarizable from
$\mathrm{X}=\\{\mathrm{\scalebox{0.8}{SUBCATEGORY}}\\}$ because
$\pi_{\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{CATEGORY}}}({PROD\\_NEW})$
is equal to the join
$\pi_{\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{SUBCATEGORY}}}({PROD\\_NEW})\bowtie\allowbreak\pi_{\mathrm{\scalebox{0.8}{SUBCATEGORY}},\mathrm{\scalebox{0.8}{CATEGORY}}}({PROD\\_NEW})$.
However, attribute  CATEGORY is not summarizable from
$\mathrm{X}=\\{\mathrm{\scalebox{0.8}{VIDEO\\_RES}}\\}$ because the natural
join between
$\pi_{\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{VIDEO\\_RES}}}({PROD\\_NEW})$
and
$\pi_{\mathrm{\scalebox{0.8}{VIDEO\\_RES}},\mathrm{\scalebox{0.8}{CATEGORY}}}({PROD\\_NEW})$
eliminates products ’p_03’ and ’p_04’.

Table 33. ${PROD\\_NEW}$ PROD_SKU |  SUBCATEGORY |  CATEGORY |  VIDEO_RES |  BRAND
---|---|---|---|---
p_01 | Video projector | Video | 1920x1080 | Epson
p_02 | TV | Video | 3840x2160 | Philips
p_05 | TV | Video | 3840x2160 | Samsung
p_03 | Radio | Audio | - | Philips
p_04 | CD-player | Audio | - | Samsung

To efficiently check summarizability, (Hurtado et al., 2005) proposes to
transform the summarizability problem into the problem of verifying the
satisfaction of a set of dimension constraints by some dimension ${D}$. Let
$\mathrm{\scalebox{0.8}{A}}_{i}$ be a dimension attribute of a dimension ${D}$
and
$\langle\mathrm{\scalebox{0.8}{A}}_{i},\mathrm{\scalebox{0.8}{A}}_{i+1},...,\mathrm{\scalebox{0.8}{A}}_{i+n}\rangle$
denote a path in ${D}$ such that
$\mathrm{\scalebox{0.8}{A}}_{k}\preccurlyeq\mathrm{\scalebox{0.8}{A}}_{k+1}$,
$i\leq k<i+n$. Then, the following dimension constraints can be defined on
$\mathrm{\scalebox{0.8}{A}}_{k}$ and ${D}$:

  1. (1)

$D\models\langle\mathrm{\scalebox{0.8}{A}}_{i},\mathrm{\scalebox{0.8}{A}}_{i+1},...,\mathrm{\scalebox{0.8}{A}}_{j}\rangle$
means that for every attribute value $v$ of $\mathrm{\scalebox{0.8}{A}}_{i}$,
there exists a tuple $t$ in ${D}$ such that
$t.\mathrm{\scalebox{0.8}{A}}_{i}=v$ and $t.\mathrm{\scalebox{0.8}{A}}_{i+1}$,
…, $t.\mathrm{\scalebox{0.8}{A}}_{j}$ are non-null (all values of
$\mathrm{\scalebox{0.8}{A}}_{i}$ roll-up to a value
$\mathrm{\scalebox{0.8}{A}}_{j}$ through a value of
$\mathrm{\scalebox{0.8}{A}}_{i+1}$ …). We shall say that
$\mathrm{\scalebox{0.8}{A}}_{i}$ _rolls up to_
$\mathrm{\scalebox{0.8}{A}}_{j}$.

  2. (2)

$D\models\langle\mathrm{\scalebox{0.8}{A}}_{i},...,\mathrm{\scalebox{0.8}{A}}_{j}=k\rangle$
means that for every attribute value $v$ of $\mathrm{\scalebox{0.8}{A}}_{i}$,
there exists a tuple $t$ in ${D}$ such that
$t.\mathrm{\scalebox{0.8}{A}}_{i}=v$ and $t.\mathrm{\scalebox{0.8}{A}}_{j}$ is
not null if and only if $t.\mathrm{\scalebox{0.8}{A}}_{j}=k$.

Constraints can then be composed using the usual Boolean logical connectives.
Now, assume that a set of constraints are specified on the schema of ${D}$. To
determine if an attribute  B is summarizable from
$\mathrm{X}=\\{A_{1},...,A_{n}\\}$ in ${D}$, one must determine if, for each
bottom level attribute $\mathrm{\scalebox{0.8}{A}}_{\bot}$ of ${D}$, the
following constraint is satisfied :

(7) $\displaystyle D\models\langle A_{\bot},...,B\rangle\implies(\langle
A_{\bot},...,A_{1},...,B\rangle\oplus...\oplus(\langle
A_{\bot},...,A_{n},...,B\rangle)$

where $\oplus$ denotes an exclusive disjunction (XOR). Intuitively, if all
values of $\mathrm{\scalebox{0.8}{A}}_{\bot}$ roll-up to a value of  B then
all these values either roll-up through values of
$\mathrm{\scalebox{0.8}{A}}_{1}$ or (exclusive) through values of
$\mathrm{\scalebox{0.8}{A}}_{2}$ … or through values of
$\mathrm{\scalebox{0.8}{A}}_{n}$. We use the following examples to illustrate
the use of constraints to determine summarizability.

###### Example 0.

Suppose that in dimension ${PROD\\_NEW}$,  VIDEO_RES has now  SUBCATEGORY as
parent in the hierarchy type (Figure 13). Then the two constraints (a) and (b)
shown in Figure 13 are expressed on ${PROD}$. Note that the disjunction
constraint (a) is more expressive than the context dependency of (Lehner et
al., 1998; Lechtenbörger and Vossen, 2003) because it is not restricted to a
single value.

(a) Hierarchy type of ${PROD\\_NEW}$

Rules
---
(a) |  $\langle\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{VIDEO\\_RES}},\mathrm{\scalebox{0.8}{SUBCATEGORY}}=$ ’TV’ $\rangle$ $\oplus$ $\langle\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{VIDEO\\_RES}},\mathrm{\scalebox{0.8}{SUBCATEGORY}}=$ ’Video projector’ $\rangle$
(b) |  $\langle\mathrm{\scalebox{0.8}{A}}_{i},\mathrm{\scalebox{0.8}{A}}_{j}\rangle$, for all other edges ($\mathrm{\scalebox{0.8}{A}}_{i},\mathrm{\scalebox{0.8}{A}}_{j}$)
Semantics
(a) | A value of  PROD_SKU rolls up to  VIDEO_RES and  SUBCATEGORY
| only for the ’Video Projector’ and ’TV’ values of
|  SUBCATEGORY
(b) | All other attributes directly roll up to their parent attribute
(b) Constraints on ${PROD\\_NEW}$

Figure 13. The dimension schema of ${PROD\\_NEW}$

The attribute  CATEGORY is summarizable from
$\mathrm{X}=\\{\mathrm{\scalebox{0.8}{SUBCATEGORY}}\\}$ because the following
constraint is satisfied (all products roll up to a category through some
subcategory):

(8)
$\displaystyle{PROD}\models\langle\mathrm{\scalebox{0.8}{PROD\\_SKU}},\ldots,\mathrm{\scalebox{0.8}{CATEGORY}}\rangle\Rightarrow\langle\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{SUBCATEGORY}},\mathrm{\scalebox{0.8}{CATEGORY}}\rangle$

Using the constraints in (b), we can compose
$\langle\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{SUBCATEGORY}}\rangle$
and
$\langle\mathrm{\scalebox{0.8}{SUBCATEGORY}},\mathrm{\scalebox{0.8}{CATEGORY}}\rangle$
to yield the final constraint. Thus, if the table
$\mathrm{{PROD\\_NEW\\_SALES}}$ is first aggregated with a grouping set
$\\{\mathrm{\scalebox{0.8}{SUBCATEGORY}}\,\mathrm{\scalebox{0.8}{CATEGORY}}\\}$,
then a query that further aggregates this result with a grouping set
$\\{\mathrm{\scalebox{0.8}{CATEGORY}}\\}$ is correct.

However, attribute  SUBCATEGORY is not summarizable from
$\mathrm{X}=\\{\mathrm{\scalebox{0.8}{VIDEO\\_RES}}\\}$ because the following
constraint cannot be satisfied:

(9)
$\displaystyle{PROD}\models\langle\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{SUBCATEGORY}}\rangle\Rightarrow\langle\mathrm{\scalebox{0.8}{PROD\\_SKU}},\mathrm{\scalebox{0.8}{VIDEO\\_RES}},\mathrm{\scalebox{0.8}{SUBCATEGORY}}\rangle$

Thus, if table $\mathrm{{PROD\\_NEW\\_SALES}}$ is first aggregated with a
grouping set
$\\{\mathrm{\scalebox{0.8}{SUBCATEGORY}},\mathrm{\scalebox{0.8}{VIDEO\\_RES}},\mathrm{\scalebox{0.8}{CATEGORY}}\\}$,
then a query $\mathpzc{Q}\,$ that further aggregates this result with a
grouping set
$\\{\mathrm{\scalebox{0.8}{SUBCATEGORY}},\mathrm{\scalebox{0.8}{CATEGORY}}\\}$
is incorrect. If the tuples of $\mathrm{{PROD\\_NEW\\_SALES}}$ with
SUBCATEGORY attribute values ’TV’ or ’Video projector’ are filtered out then
the summarizability constraint can be satisfied and previous query
$\mathpzc{Q}\,$ will be correct.

It is clear that the data model and constraints proposed by (Hurtado et al.,
2005) subsume the data model with context dependencies of (Lehner et al.,
1998; Lechtenbörger and Vossen, 2003). We already showed that our
summarizability conditions are more expressive than context dependencies by
considering $null$ values as regular values for aggregation. The same
arguments apply to dimension constraints. In addition, (Hurtado et al., 2005)
has the following limitations with respect to our work. First, any non-null
value of a mandatory attribute must map to a single parent value in the
hierarchy. This discards the use of dimension tables like ${SALESORG}$ in
Table 8(b). Second, measure attributes cannot depend on a subset of the
dimensions of a fact table. This discards the use of fact tables resulting
from interactive user queries like the result of the left-merge of
$\mathrm{{T2}}$ with $\mathrm{{DEM^{\prime}}}$in Table 6(b), presented in the
Introduction section. Finally, unlike other works, the notion of applicability
of an aggregation function to an attribute is not covered.

##  6\. Conclusions

In this article, we introduce a new framework for controlling the correctness
of aggregation operations during sessions of interactive analytic queries. Our
framework adopts an _attribute-centric view_ , whereby _aggregable properties_
of attributes are used to describe and control the interaction between
measures, dimensions and aggregation functions. As a first advantage,
aggregable properties enable the designers of analytic tables to describe the
wide variety of semantic properties of measure attributes with respect to
their dimensions defined in previous work (Kimball and Ross, 2013; Horner et
al., 2004; Niemi et al., 2014; Stevens, 1946; Pedersen et al., 2001; Lenz and
Shoshani, 1997). Another advantage of aggregable properties is their ability
to guarantee that aggregate queries over some attributes can only be expressed
if these attributes are summarizable. We provide two definitions of
summarizable attributes. Our first definition covers the case when an
aggregate query is defined over the result of another aggregate query; it
subsumes the definitions of previous work on summarizability (Lenz and
Shoshani, 1997; Pedersen et al., 2001; Lehner et al., 1998; Lechtenbörger and
Vossen, 2003; Hurtado et al., 2005). Our second definition introduces the new
notion of _G-summarizability_ which applies in the case of an aggregate query
defined over the result of an arbitrary analytic queries. The two definitions
are complementary. Our main technical results are the definition of
_propagation_ rules that automatically compute the aggregable properties of
attributes in the result of an analytic query knowing the aggregable
properties of the attributes in the operand tables of the query. We
progressively refine our propagation rules to handle the semantic properties
of measures, and the summarizability and G-summarizability properties of
attributes.

There is a number of perspectives for future work. First, aggregable
properties could be extended to handle other correctness issues of aggregate
queries. Currently, we rely on literal functional dependencies (LFD) for
analyzing the summarizability properties of query results. Simpson’s paradox
(Wagner, 1982) is an example of incorrect causal interpretation of aggregated
attribute values where a statistical observation like ratio or bias on the
measures from several partitions might disappear or be inverted on the
aggregated measures over these partitions. Aggregable properties could be
extended to guard against this kind of statistical errors by exploiting
existing causal dependencies between table attributes (Pearl et al., 2016)
(representing features) in addition to LFD.

Another direction of research is the ”explainability” of aggregated values in
an analytic table resulting from an interactive data analysis session.
Aggregable properties provide explanations for the decision to forbid an
incorrect aggregate query on a table. They also help the user to backtrack in
her session to find an intermediate result over which a desired aggregation
can be expressed. However, as shown in Example 4.19, a priori non-summarizable
aggregate queries could be accepted provided that adequate annotations are
added to the result table so that aggregated values can be properly
interpreted. Generating such minimal annotations and propagating them is still
open.

Finally, although a large part of our data model has already been prototyped
in SAP HANA (Rutian, 2020), a significant effort is needed to integrate our
framework into self-service data preparation tools and analytic database
systems.

Another possible direction is the implementation of our framework as an
independent software component, _e.g_. Python library, which could be part of
data preparation pipelines for Deep Learning applications (Python notebooks).

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# Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices –
unexpected links

Jean-Paul Blaizot  IPTh, CEA-Saclay, CNRS/UMR 3681, F- 91191 Gif-sur-Yvette
cedex, France  Jacek Grela  Marian Smoluchowski Institute of Physics,
Jagiellonian University, ul. Łojasiewicza 11, 30–348 Kraków, Poland  Maciej A.
Nowak∗ Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems
Research Center, ul. Łojasiewicza 11, 30–348 Kraków, Poland  ∗ Contributing
author<EMAIL_ADDRESS>Wojciech Tarnowski  Marian Smoluchowski
Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30–348
Kraków, Poland  Piotr Warchoł  Marian Smoluchowski Institute of Physics,
Jagiellonian University, ul. Łojasiewicza 11, 30–348 Kraków, Poland

###### Abstract

We compare the Ornstein-Uhlenbeck process for the Gaussian Unitary Ensemble to
its non-hermitian counterpart - for the complex Ginibre ensemble. We exploit
the mathematical framework based on the generalized Green’s functions, which
involves a new, hidden complex variable, in comparison to the standard
treatment of the resolvents. This new variable turns out to be crucial to
understand the pattern of the evolution of non-hermitian systems. The new
feature is the emergence of the coupling between the flow of eigenvalues and
that of left/right eigenvectors. We analyze local and global equilibria for
both systems. Finally, we highlight some unexpected links between both
ensembles.

###### pacs:

05.10.Gg, 02.50.-r, 02.50.Ey, 05.70.-a

KEYWORDS: Gaussian Unitary Ensemble, Ginibre Ensemble, Ornstein-Uhlenberck
Process, Burgers equation
SUBJECT AREA: mathematical physics, theory of complex systems, random matrix
theory

##  1 Introduction

In 1962, Dyson suggested an inspiring way to understand the joint probability
distribution function (hereafter jpdf) of the eigenvalues of random matrices.
In order to find it, he introduced an auxiliary dynamics in some fictitious
“time”, which, in the large time limit, lead to a stationary state (Gibbs
state) representing the desired jpdf. As he pointed [1], “after considerable
and fruitless efforts to develop a Newtonian theory of ensembles, we
discovered that the correct procedure is quite different and much simpler. The
$x_{i}$ [eigenvalues] should be interpreted as positions of particles in
Brownian motion”. The resulting stationary distributions (originally for
hermitian or for unitary random matrices) were obtained as a result of
Ornstein-Uhlenbeck diffusion with a drift force coming from electrostatic-like
repulsion of eigenvalues. The success of this description has contributed to
multiple applications of random matrix models in practically all branches of
science. The notion of “time” has evolved as well, so nowadays it can be a
physical parameter, representing either the real time or, e.g., the length of
a mesoscopic wire, the area of a string or an external temperature. The idea
of a noisy walk of eigenvalues recently led also to such concepts as
determinantal processes [2, 3, 4], Loewner diffusion [5], fluctuations of non-
intersecting interfaces in thermal equilibrium [6] and the emergence of pre-
shock spectral waves and universal scaling at the critical points of several
random matrix models.

Three years after Dyson, Ginibre [7] has considered for the first time
strictly non-hermitian random matrix models, whose spectrum does not need to
be confined either to the real line (hermitian operators) or to the unit
circle (unitary operators), but can be located on a two-dimensional support on
the complex plane. The original motivation for the study of complex, random
spectra was purely academic. Today however, non-hermitian random operators
play a role in quantum information processing, in financial engineering (when
lagged correlations are discussed [8]) or in identifying clusters in social or
biological networks using non-backtracking operators [9], to name just a few
recent applications. Additionally, statistical properties of eigenvectors of
non-hermitian operators contribute to understanding scattering problems in
open chaotic cavities [10] and random lasing.

It is surprising that in the last half century, the Dysonian picture of random
walk of eigenvalues was not applied to the complex Ginibre Ensemble (GE). The
Brownian walk problem for the real Ginibre Ensemble was recently studied in
[11]. In this contribution, partially based on our earlier work on this
subject, we show how to fill this logical gap, and we also speculate on the
reasons why the non-hermitian extension of a random walk scenario was far from
obvious.

In Section 2, we start from recalling Dyson’s original construction [1]. Then,
we propose an alternative description, where the fundamental object is the
characteristic polynomial. We show the advantages of such description,
borrowing heavily from the analogies to the simplest model of turbulence,
i.e., the so-called Burgers equation. We also briefly mention, how the seminal
results for the Gaussian Unitary Ensemble (GUE) can be recovered from a
Burgers-like description.

In Section 3, we formulate a mathematical framework, which allows us to
parallel the turbulent picture in the case of the GE. In particular, we
unravel a hidden dynamics associated with a new complex variable, which in
standard descriptions of non-hermitian random matrix models is treated as an
infinitesimal regulator only. We point out, that the non-hermitian character
of the GE binds the dynamics of eigenvalues to the evolution of eigenvectors
in a non-trivial way. Alike in the case of GUE, we demonstrate how the well-
known results of the GE can be easily reclaimed in our formalism. Section 4
contains numerical experiments for both GUE and GE capturing the relevant
diffusive degrees of freedom. In Section 5, we uncover the unexpected links
between the descriptions of the Gaussian Unitary and the Ginibre ensembles.
Section 6 concludes the paper and lists some open problems on noise in matrix
models.

##  2 Diffusion in the Gaussian Unitary Ensemble

According to Dyson, the eigenvalues of a random, $N$ by $N$ hermitian matrix
belonging to the GUE fulfill the following stochastic equation

$\displaystyle{\rm d}\lambda_{i}(\tau)=\frac{1}{\sqrt{N}}{\rm
d}B_{i}(\tau)+\frac{1}{N}\sum_{j=1,j\neq
i}^{N}\frac{1}{\lambda_{i}-\lambda_{j}}{\rm d}\tau-a\lambda_{i}{\rm d}\tau,$
(1)

where $B_{i}$’s are one-dimensional standard Brownian motions, $\lambda_{i}$’s
denote the eigenvalues and $\tau$ is the time variable. The second term
represents a fictitious electric field coming from the logarithmic Coulomb
potential (originating from the Van der Monde determinant) and the last term
represents the drift coming from the confining harmonic potential (Ornstein-
Uhlenbeck process). In the limit when $N$ tends to infinity and
$\tau\rightarrow\infty$, the eigenvalues freeze-out as a result of the
compromise between the repulsion (electric field) and attraction (harmonic
potential). The resulting spectral distribution takes the form of the Wigner
semicircle. Despite the fact that Dyson was primarily interested in the
equilibrium state, and introduced a time in an auxiliary construction, he
pointed out that the transition to the equilibrium is quite subtle. In his own
words [1], the Coulombic term is “measuring the frequency with which two
charges come into coincidence. This term is mainly sensitive to the local
(microscopic) configurations of the gas particles… at the microscopic time
scale … After local equilibrium is established… the gas must adjust itself by
macroscopic motion on the time scale”, which is $N$ times larger compared to
the microscopic one. He also noted that [1]”a rigorous proof that this picture
is accurate would require a much deeper mathematical analysis”. The discussion
in this section gives support to this picture.

Let us introduce an $N\times N$ hermitian matrix $H$ by defining its complex
entries according to:

$\displaystyle H_{ij}=\left\\{\begin{array}[]{l r}x_{ii},&i=j,\\\
x_{ij}+iy_{ij},&i\neq j,\end{array}\right.$ (4)

where $x_{ij}=x_{ji}$ and $y_{ij}=-y_{ji}$, with $x_{ij}$ and $y_{ij}$ real.
Furthermore let $x_{ij}$ and $y_{ij}$ perform white noise driven, independent
random walks, such that

$\displaystyle\left\langle\delta
H_{ij}\right\rangle=-aH_{ij}\delta\tau,\qquad\left\langle\left|\delta
H_{ij}\right|^{2}\right\rangle=\frac{g_{ij}}{N}\delta\tau,$ (5)

with $g_{ij}=1+\delta_{ij}$ and for any $i$ and $j$. Let
$P(x_{ij},\tau)P(y_{ij},\tau)$ be the probability that the off diagonal matrix
entry $H_{ij}$ will change from its initial state to $x_{ij}+iy_{ij}$ after
time $\tau$. Analogically, $P(x_{ii},\tau)$ is the probability of the diagonal
entry $H_{ii}$ becoming equal to $x_{ii}$ at $\tau$. The evolution of these
functions is governed by the following Smoluchowski-Fokker-Planck (SFP)
equations:

$\displaystyle\frac{\partial}{\partial\tau}P(x_{ii},\tau)$
$\displaystyle=\left(\frac{1}{2N}\frac{\partial^{2}}{\partial
x_{ii}^{2}}+a\frac{\partial}{\partial x_{ii}}x_{ii}\right)P(x_{ii},\tau),$
$\displaystyle\frac{\partial}{\partial\tau}P(v_{ij},\tau)$
$\displaystyle=\left(\frac{1}{4N}\frac{\partial^{2}}{\partial
v_{ij}^{2}}+a\frac{\partial}{\partial v_{ij}}v_{ij}\right)P(v_{ij},\tau),\quad
i<j,$ (6)

where the parameter $a$ measures the strength of the harmonic potential
confining the diffusion of the matrix elements and $v_{ij}$ denotes either
$x_{ij}$ or $y_{ij}$. The joint probability density function is thus defined
as

$\displaystyle\qquad
P(x,y,\tau)\equiv\prod_{k=1}^{N}P(x_{kk},\tau)\prod_{i<j=1}^{N}P(x_{ij},\tau)P(y_{ij},\tau)$
(7)

and satisfies the following equation

$\displaystyle\partial_{\tau}P(x,y,\tau)=\mathcal{A}(x,y)P(x,y,\tau),$ (8)

with

$\displaystyle\mathcal{A}(x,y)$
$\displaystyle=\sum_{k=1}^{N}\left(\frac{1}{2N}\frac{\partial^{2}}{\partial
x_{kk}^{2}}+a\frac{\partial}{\partial
x_{kk}}x_{kk}\right)+\frac{1}{4N}\sum_{i<j=1}^{N}\left(\frac{\partial^{2}}{\partial
x_{ij}^{2}}+\frac{\partial^{2}}{\partial y_{ij}^{2}}\right)+$ (9)
$\displaystyle+a\sum_{i<j=1}^{N}\left(\frac{\partial}{\partial
x_{ij}}x_{ij}+\frac{\partial}{\partial y_{ij}}y_{ij}\right).$

A source-like solution of (8) reads:

$\displaystyle
P(x,y,\tau)=C\exp\left(-\frac{Na}{1-e^{-2a\tau}}\mathrm{Tr}(H-H_{0}e^{-a\tau})^{2})\right),$
(10)

with $H(\tau=0)=H_{0}$ and $C$ is a normalization constant. With the setting
thus defined, let us proceed to the derivation of the partial differential
equations obeyed by the averaged characteristic polynomial (hereafter called
ACP) $U(z,\tau)$ associated with the diffusing matrix $H$:

$\displaystyle U(z,\tau)\equiv\langle\det\left(z-H\right)\rangle_{\tau},$ (11)

where the angular brackets denote the averaging over the time dependent
probability density (7). In Appendix A, we show that the ACP satisfies

$\displaystyle\partial_{\tau}U(z,\tau)=-\frac{1}{2N}\partial_{zz}U(z,\tau)+az\partial_{z}U(z,\tau)-aNU(z,\tau).$
(12)

Note that the standard Green’s function (resolvent) associated with $H$ is
related to ACP in the large $N$ limit

$\displaystyle G\equiv G(z,\tau)\equiv\lim_{N\to\infty}\frac{1}{N}\langle{\rm
Tr}\frac{1}{z-H}\rangle=\lim_{N\to\infty}\frac{1}{N}\partial_{z}\ln U.$ (13)

Thus, the spectral density of $H$ is given by

$\displaystyle\rho(\lambda,\tau)=-\frac{1}{\pi}\lim_{\epsilon\to 0_{+}}{\rm
Im}G(z=\lambda+i\epsilon,\tau),$ (14)

through the Sokhotski-Plemelj formula.

Let us define $f_{N}\equiv f_{N}(z,\tau)\equiv\frac{1}{N}\partial_{z}\ln U$
(this is the complex analogue of the Cole-Hopf transform). Due to Eq. (12),
$f_{N}$ satisfies:

$\displaystyle\partial_{\tau}f_{N}+f\partial_{z}f_{N}-a\partial_{z}\left(zf_{N}\right)=-\frac{1}{2N}\partial_{zz}f_{N}.$
(15)

Since $\lim_{N\rightarrow\infty}f_{N}=G$, we see that the Green’s function is
governed by the following complex Burgers-like differential equation:

$\displaystyle\partial_{\tau}G+G\partial_{z}G-a\partial_{z}\left(zG\right)=0.$
(16)

###  2.1 Macroscopic equilibrium

In the $\tau\to\infty$ limit the time derivative in Eq. (16) vanishes and so
this equation is easily solvable since it reduces to:

$\displaystyle\partial_{z}\left(\frac{1}{2}G^{2}-azG\right)=0.$ (17)

Because all moments $\frac{1}{N}\left<TrH^{k}\right>$ are finite for any $k$,
the function $G(z,\tau)$ has to tend to zero as $1/z$ in the
$z\rightarrow\infty$ limit. This observation fixes the integration constant of
Eq. (17) to be equal to $-a$. The resulting solution of the quadratic equation

$\displaystyle\frac{1}{2}G(z)^{2}-azG(z)+a=0$ (18)

reads $G(z)=a(z-\sqrt{z^{2}-2/a})$. Using (14), for the standard value
$a=1/2$, we recover the Wigner semicircle

$\displaystyle\rho(x)=\frac{1}{2\pi}\sqrt{4-x^{2}}.$ (19)

The analogy to the Burgers equation is however deeper, as we pointed out in
[12]. For the real Burgers equation, the solution based on the method of
characteristics breaks down due to the emergence of the pre-shock wave
(singularity). Similar phenomenon takes place on the complex plane for Eq.
(16). The resulting spectral shock waves correspond to the endpoints of the
spectrum. Several other, surprising links between the Burgers equation and
some probabilistic model are discussed in [13].

###  2.2 Microscopic limit

We stress here that the equation (12) for the diffusing characteristic
polynomial is exact for any $N$ and for any initial conditions. We can
therefore use it to retrieve the spectral features at all time scales and at
all points of the spectrum. In order to simplify the analysis, let us start by
performing a useful change of variables belonging to the class of Lamperti
transformations [14, 15, 16, 17]. This change of variables will allow to
establish a connection between the Ornstein-Uhlenbeck process and the case of
free diffusion ($a=0$), where the results are known [12, 18]. To check that
indeed we can recover a free diffusion equation from the Ornstein-Uhlenbeck
process, we explicitly write down the relevant Lamperti transformation:

$\displaystyle
U(z,\tau)=(1+2a\tau^{\prime})^{-N/2}U^{\prime}(z^{\prime},\tau^{\prime}),$
$\displaystyle
z^{\prime}=e^{a\tau}z,\quad\tau^{\prime}=\frac{1}{2a}\left(e^{2a\tau}-1\right).$
(20)

Straightforward but lengthy calculations yield

$\displaystyle\partial_{\tau}U=A^{-2}\left(-\frac{Na}{1+2a\tau^{\prime}}BU^{\prime}+B\partial_{\tau^{\prime}}U^{\prime}\right)+az^{\prime}B\partial_{z^{\prime}}U^{\prime},$
$\displaystyle\partial_{z}U=BA^{-1}\partial_{z^{\prime}}U^{\prime},\qquad\partial_{zz}U=BA^{-2}\partial_{z^{\prime}z^{\prime}}U^{\prime},$

where we define $A=e^{-a\tau}$ and $B=\left(1+2a\tau^{\prime}\right)^{-N/2}$.
Plugging these into (12) produce a free diffusion equation in the variables
$(z^{\prime},\tau^{\prime})$:

$\displaystyle\partial_{\tau^{\prime}}U^{\prime}=-\frac{1}{2N}\partial_{z^{\prime}z^{\prime}}U^{\prime}.$
(21)

In order to avoid unnecessary repetitions, we highlight here only the
consequences of Eq. (21), relegating the details to already published work
[12, 18]. First, it is exactly integrable on the complex plane for any initial
conditions. The corresponding Cole-Hopf transformation maps the diffusion
equation onto the complex viscid Burgers equation

$\displaystyle\partial_{\tau^{{}^{\prime}}}f_{N}^{{}^{\prime}}+f^{{}^{\prime}}\partial_{z^{{}^{\prime}}}f_{N}^{{}^{\prime}}=-\nu_{s}\partial_{z^{{}^{\prime}}z^{{}^{\prime}}}f_{N}^{{}^{\prime}},$
(22)

where the (negative) spectral viscosity reads $\nu_{s}=1/2N$. Second, in the
large $N$ limit (inviscid limit), spectral shock waves form at the endpoints
of the spectra. Third, in the vicinity of the shock waves (endpoints of the
spectra), the above equation captures the microscopic universality of the GUE
ensembles, leading to Airy function oscillations at the edges. Intuitively,
spectral oscillations at the endpoint origin from the negative sign of
viscosity, which causes the “roughening” of the transition instead of
smoothening it, as would be expected in the case of a positive viscosity. Last
but not least, the above picture confirms that Dysonian microscopic
equilibrium is formed already at very short time scales and its character is
determined solely by the global properties of the random matrix, i.e. the
symmetries deciding on the functional form of the Coulombic repulsion.

##  3 Diffusion in the Ginibre Ensemble

Contrary to the case of the hermitian ensembles, the spectrum of non-hermitian
matrices is genuinely complex. Let us define the simplest example, the so-
called complex Ginibre Ensemble where each element of the $N\times N$ matrix
$X$ is drawn from a complex Gaussian distribution. That is, each entry
$X_{ij}=x_{ij}+iy_{ij}$ consists of $x_{ij}$ and $y_{ij}$ drawn from standard
Gaussian distributions. Note that all moments $\left<{\rm Tr}X^{n}\right>$
(and $\left<{\rm Tr}(X^{\dagger})^{n}\right>$) vanish because
$\left<\mathrm{Tr}X^{2}\right>=0$. The only non vanishing moments are the
mixed ones, i.e. $\left<{\rm Tr}(XX^{\dagger})^{n}\right>$. As we will show,
in the large $N$ limit, the eigenvalues condense uniformly on the centered
disc on the complex plane. Therefore, the spectrum exhibits a jump at the rim,
contrary to the hermitian cases, when the real spectrum is continuous at the
endpoints, and only the derivatives of the spectrum are discontinuous.
Moreover, the spectrum is non-analytic inside the disc, which seems to
disqualify all the methods based on analyticity of the complex variable $z$.
This is best visible, when we try to repeat the hermitian construction for the
Green’s function $G(z)=\frac{1}{N}\left<{\rm Tr}\frac{1}{z-X}\right>$. Since
all moments vanish, such a Green’s function is simply equal to
$G(z)=\frac{1}{z}$, and does not reflect correctly the spectral properties of
the ensemble. Similarly, the characteristic determinant is trivial,
$\left<\det(z-X)\right>=z^{N}$.

The way out, based on an electrostatic analogy, was suggested a long time ago
[21]. We define an electrostatic potential

$\displaystyle\Phi\equiv\Phi(z)=\lim_{\epsilon\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{N}\left\langle{\rm
Tr}\ln[|z-X|^{2}+\epsilon^{2}]\right\rangle\ ,$ (23)

where we use a short-hand notation:
$|z-X|^{2}+\epsilon^{2}=(z{\bf{1}}_{N}-X)(\bar{z}{\bf{1}}_{N}-X^{\dagger})+\epsilon^{2}{\bf{1}}_{N}$,
where $\bf{1}_{N}$ is the $N$-dimensional identity matrix. Then, we calculate
the “electric field” as a gradient of the electrostatic potential,

$\displaystyle G(z,\bar{z})=\partial_{z}\Phi=\lim_{\epsilon\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{N}\left<{\rm
Tr}\frac{\bar{z}-X^{\dagger}}{|z-X|^{2}+\epsilon^{2}}\right>.$ (24)

The electric field plays the role of the correct Green’s function. Indeed,
applying the Gauss law, in the next step,

$\displaystyle\rho=\frac{1}{\pi}\partial_{\bar{z}}G=\frac{1}{\pi}\partial_{z\bar{z}}\Phi=\lim_{\epsilon\rightarrow
0}\lim_{N\rightarrow\infty}\frac{1}{\pi N}\left\langle{\rm
Tr}\frac{\epsilon^{2}}{[|z-X|^{2}+\epsilon^{2}]^{2}}\right\rangle,$ (25)

we recover the spectral density
$\rho(z,\bar{z})=\frac{1}{N}\langle\sum_{i=1}^{N}\delta^{(2)}(z-\lambda_{i})\rangle$,
using the known representation of the two-dimensional delta function
$\delta^{(2)}(z)=\lim_{\epsilon\rightarrow
0}\frac{1}{\pi}\frac{\epsilon^{2}}{[|z|^{2}+\epsilon^{2}]^{2}}$. Note that the
Gauss law implies the non-analyticity of $G(z,\bar{z})$. It is crucial that
the limit $N\rightarrow\infty$ is taken first, before taking the infinitesimal
regulator $\epsilon$ to zero, since only such order provides the necessary
coupling between $X$ and $X^{\dagger}$, reflected in non-vanishing mixed
moments. If one took the limits in an opposite order, $X$ and $X^{\dagger}$
would decouple, and we would obtain a trivial result $G(z)=1/z$. The bad news,
however, is that the Green’s function $G(z,\bar{z})$ (24) is given by a very
complicated expression, without any similarity to the standard form of the
resolvent.

One may bypass the difficulty by relying on the the algebraic construction for
the so-called generalized Green’s functions proposed some time ago [19, 20].
First, we notice that

$\displaystyle{\rm
Tr}\ln[|z-X|^{2}+\epsilon^{2}]=\ln\det[|z-X|^{2}+\epsilon^{2}]=\ln\det\left(\begin{array}[]{cc}z-X&i\epsilon\\\
i\epsilon&\bar{z}-X^{\dagger}\end{array}\right),$ (28)

where the argument of the last determinant is a $2N\times 2N$ matrix, built
out of four $N\times N$ blocks. Let us now define a new operation called a
block-trace, defined as ${\rm
bTr}\equiv{\bf{1}}_{2}\otimes\mathrm{Tr}_{N\times N}$, which acts in the
following way:

$\displaystyle{\rm bTr}\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)_{2N\times 2N}\equiv\left(\begin{array}[]{cc}{\rm
Tr}A&{\rm Tr}B\\\ {\rm Tr}C&{\rm Tr}D\end{array}\right)_{2\times 2},$ (33)

converting a $2N\times 2N$ block matrix into a $2\times 2$ matrix built out of
ordinary traces. Additionally, we define another pair of block matrices

$\displaystyle Q=\left(\begin{array}[]{cc}z&-\bar{w}\\\
w&\bar{z}\end{array}\right),\,\,\,\,\,\,\,\,{\cal{X}}=\left(\begin{array}[]{cc}X&0\\\
0&X^{\dagger}\end{array}\right).$ (38)

We are now ready to propose the construction of the generalized resolvent
($2\times 2$ matrix)

$\displaystyle{\cal{G}}(z,w)$ $\displaystyle\equiv$
$\displaystyle\left(\begin{array}[]{cc}{\cal{G}}_{11}&{\cal{G}}_{1\bar{1}}\\\
{\cal{G}}_{\bar{1}1}&{\cal{G}}_{\bar{1}\bar{1}}\end{array}\right)=\frac{1}{N}\left<{\rm
bTr}\frac{1}{Q-\cal{X}}\right>,$ (41)

By construction, $\mathcal{G}_{11}$ is equal to the non-analytic resolvent
$G(z,\bar{z})$ (24), provided we identify $|w|^{2}=\epsilon^{2}$. Note that
the duplication trick allowed us to linearize the problem, since the form of
the generalized resolvent (41) has formally the form of the standard resolvent
for hermitian matrices. One may ask the question, what role is played by the
three remaining elements of the matrix ${\cal G}$? Let us recall, that the
general (non-normal) matrix $X$ is determined in terms of its eigenvalues
($Z$) and a set of left ($\bra{L}$) and right ($\ket{R}$) eigenvectors
($X=\sum_{i}z_{i}\ket{R_{i}}\bra{L_{i}}$), which are bi-orthogonal
$\braket{L_{i}}{R_{j}}=\delta_{ij}$. By applying a transformation $S={\rm
diag}(R,L)$, $S^{-1}={\rm diag}(L^{\dagger},R^{\dagger})$ (where
$L,R,L^{\dagger},R^{\dagger}$ are $N\times N$ matrices built from the
corresponding eigenvectors), we notice that

$\displaystyle\det(Q-{\cal X})=\det[S^{-1}(Q-{\cal
X})S]=\det\left(\begin{array}[]{cc}z-Z&-\bar{w}L^{\dagger}L\\\
wR^{\dagger}R&\bar{z}-Z^{\dagger}\end{array}\right),$ (44)

so the off-diagonal elements of the generalized Green’s functions are related
to the expectation values of the overlaps of eigenvectors. Indeed, the left-
right eigenvector correlator [22] reads:

$\displaystyle
O(z,\tau)\equiv\frac{1}{N^{2}}\left<\sum_{a}O_{aa}\delta^{(2)}(z-z_{a})\right>,$
(45)

where $O_{ij}=\left<L_{i}|L_{j}\right>\left<R_{j}|R_{i}\right>$ is given in
the large $N$ limit by the product of off-diagonal elements of $\mathcal{G}$:

$\displaystyle\lim_{N\to\infty}O(z,\tau)=-\frac{1}{\pi}{\cal
G}_{1\bar{1}}{\cal G}_{\bar{1}1}|_{w=0}.$ (46)

as was proven in [23]. The appearance of this correlator is a genuine feature
of non-hermitian random matrix models, since in the hermitian case left and
right eigenvectors coincide and so $O_{ij}=\delta_{ij}$. Finally, for
completeness we notice that ${\cal G}_{\bar{1}\bar{1}}$ is a complex conjugate
of ${\cal G}_{11}$ and does not bring any new information.

We would now like to comment on the role of the $w$ variable. In the hermitian
case, the method of the resolvent involves the whole complex plane $z$,
despite the fact that the real spectrum comes only as a discontinuity near
$z=\lambda\pm i\epsilon$, corresponding to the imaginary part of the
resolvent. In the non-hermitian case, the spectrum is complex, but one may be
tempted to probe the generalized Green’s function with the complex plane $w$
”orthogonal” to the plane $z$, as schematically depicted on Fig. 1. This
choice of strategy is reinforced by the above observed coupling of the $w$
plane to eigenvector correlators.

Figure 1: Schematic comparison between the domains of hermitian Green’s
function $G(z)$ and the non-hermitian, generalized Green’s function ${\cal
G}(Q)$. Arrows on the left figure signal the discontinuity of the Green’s
function when approaching the cut (solid line), arrows on the right figure
denote an additional variable $w$, which in standard approach is treated as
only an infinitesimal regulator. Shaded disc represents the non-analytic
domain where the eigenvalues condense.

The promotion of the original regulator $i\epsilon$ to a complex variable $w$
has additional advantages. Firstly, from the algebraic point of view, $Q$ is a
quaternion, since $Q=q_{0}+i\sigma_{j}q_{j}$, where $\sigma_{j}$ are Pauli
matrices, so $z=q_{0}+iq_{3}$ and $w=-q_{2}+iq_{1}$. This fact significantly
simplifies the algebraic calculations, since block matrices such as
$\mathcal{X}$ and arguments $Q$ naturally appear in non-hermitian random
matrix models, e.g. in the generalized Green’s function technique [19, 20],in
hermitization methods [24, 25, 26] or in the derivation of the multiplication
law for non-hermitian random matrices [27]. The above construction was also
recently proven rigorously in the mathematical literature [28]. Secondly,
introducing the variable $w$ provides a hint on which object should play the
role of the average characteristic polynomial in the case of the Ginibre
ensemble subjected to Ornstein-Uhlenbeck process, as we now demonstrate.

We define now a determinant expressed in terms of the quaternionic variable
$Q$ by.

$\displaystyle D\equiv D(|w|,z,\bar{z},\tau)\equiv\left\langle{\rm
det}(Q-\cal{X})\right\rangle_{\tau}=\left\langle\det\left(\begin{array}[]{cc}z-X&-\bar{w}\\\
w&\bar{z}-X^{\dagger}\end{array}\right)\right\rangle_{\tau},$ (49)

where the measure over which the averaging is performed reflects the Ornstein-
Uhlenbeck process. More concretely, it is given by the following averaged
increments (compare with the hermitian counterpart (5)):

$\displaystyle\left<\delta
X_{ij}\right>=-aX_{ij}\delta\tau,\qquad\left<\left|\delta
X_{ij}\right|^{2}\right>=\frac{1}{N}\delta\tau,$ (50)

which is also expressible as a Smoluchowski-Fokker-Planck equation for the
jpdf $P(x,y,\tau)=\prod_{i,j=1}^{N}P(x_{ij},\tau)P(y_{ij},\tau)$:

$\displaystyle\partial_{\tau}P(x,y,\tau)=\mathcal{B}(x,y)P(x,y,\tau),$ (51)

with

$\displaystyle\mathcal{B}=\frac{1}{4N}\sum_{i,j=1}^{N}\left(\partial^{2}_{x_{ij}}+\partial^{2}_{y_{ij}}\right)+a\sum_{i,j=1}^{N}\left(\partial_{x_{ij}}x_{ij}+\partial_{y_{ij}}y_{ij}\right).$
(52)

Up to an irrelevant normalization constant $C$, a source-like solution to (51)
reads

$\displaystyle
P(X,\tau)=C\exp\left(-\frac{2Na}{1-e^{-2a\tau}}\mathrm{Tr}|X-X_{0}e^{-a\tau}|^{2}\right).$
(53)

Following similar steps as in the hermitian case, we express the determinant
with the help of the auxiliary Grassmann variables and use the properties of
the diffusion process to arrive (cf. Appendix B for the details) at the exact
(for any matrix size $N$ and for any initial conditions) equation

$\displaystyle\partial_{\tau}D=\frac{1}{N}\partial_{w\bar{w}}D-2NaD+a\textrm{d}D,$
(54)

with the operator
$\textrm{d}=z\partial_{z}+\bar{z}\partial_{\bar{z}}+w\partial_{w}+\bar{w}\partial_{\bar{w}}$.

It is worthy to disentangle the mixed variables present in the last, “drift”
term, by repeating the Lamperti transformation defined by (20) in the
hermitian case (with $N$ replaced by $2N$ and $Q$ instead of $z$). This change
of variables leads to the free diffusion

$\displaystyle\partial_{\tau^{\prime}}D^{\prime}=\frac{1}{N}\partial_{w^{\prime}\bar{w}^{\prime}}D^{\prime}.$
(55)

We contrast this equation to its hermitian counterpart (21). Again, it is
exactly integrable, and the case of free diffusion was considered by us in
[29, 30]. Note that this time the diffusion is two-dimensional, and the
Laplace operator acts in the $w^{\prime}$ space, which, in standard
treatments, is largely ignored by serving merely as a regulator. Alike in the
hermitian case, we may suspect the emergence of the Burgers structure,
provided we apply Cole-Hopf transformation. Since we have at our disposal two
complex variables $z^{\prime}$ and $w^{\prime}$, we may perform two
independent Cole-Hopf transformations

$\displaystyle g^{\prime}\equiv\frac{1}{N}\partial_{z^{\prime}}\ln
D^{\prime},\quad u^{\prime}\equiv\frac{1}{N}\partial_{w^{\prime}}\ln
D^{\prime},$ (56)

which satisfy [29, 30]

$\displaystyle\partial_{\tau^{\prime}}g^{\prime}$
$\displaystyle=\frac{1}{N}\partial_{w^{\prime}\bar{w^{\prime}}}g^{\prime}+\partial_{z^{\prime}}(u^{\prime}\bar{u^{\prime}}),$
(57) $\displaystyle\partial_{\tau^{\prime}}u^{\prime}$
$\displaystyle=\frac{1}{N}\partial_{w^{\prime}\bar{w^{\prime}}}u^{\prime}+\partial_{w^{\prime}}(u^{\prime}\bar{u^{\prime}}).$
(58)

Let us then perform the macroscopic and microscopic limit of above equations.

###  3.1 Macroscopic limit

In the large $N$ limit, the second equation (58) takes the form of an inviscid
Burgers equation in 2+1 dimensions

$\displaystyle\partial_{\tau^{\prime}}u^{\prime}=\partial_{w^{\prime}}|u^{\prime}|^{2}.$
(59)

In our case this equation can be simplified due to the rotational symmetry.
From now on, we follow the solution presented in [29], modulo the primed
variables reflecting the Ornstein-Uhlenbeck process. Introducing
$v^{\prime}=|u^{\prime}|$ and the radial variable $r^{\prime}=|w^{\prime}|$ we
recover the Euler equation known from hydrodynamics. The standard solution,
using the method of characteristics, yields
$v^{\prime}=v_{0}^{\prime}(r^{\prime}+\tau^{\prime}v^{\prime})$, where the
initial condition $X_{0}=0$ corresponds to
$v_{0}^{\prime}(r^{\prime})=r^{\prime}/(|z^{\prime}|^{2}+r^{\prime 2})$. We
identify the $v^{\prime 2}$ with the eigenvector correlator
$O(z^{\prime},\tau^{\prime})$, with an explicit solution

$\displaystyle O(z^{\prime},\tau^{\prime})=\frac{1}{\pi\tau^{\prime
2}}(\tau^{\prime}-|z^{\prime}|^{2}).$ (60)

Having the solution for $v^{\prime 2}$, we can turn back to the first equation
$\partial_{\tau^{\prime}}g^{\prime}=\partial_{z^{\prime}}v^{\prime 2}$.
Elementary integration and initial conditions lead to
$g^{\prime}=\bar{z}^{\prime}/\tau^{\prime}$ which in turn gives the spectral
density

$\displaystyle\rho(z^{\prime},\tau^{\prime})=\frac{1}{\pi\tau^{\prime}}\theta(\tau^{\prime}-|z^{\prime}|^{2}).$
(61)

We can now return to the unprimed variables, using the same Lamperti formulae
(20) and perform the stationary limit $\tau\rightarrow\infty$. In the end, the
eigenvector correlator and the spectral density read simply

$\displaystyle O(z,\bar{z})$ $\displaystyle=$
$\displaystyle\frac{4a^{2}}{\pi}\left(\frac{1}{2a}-|z|^{2}\right),$ (62)
$\displaystyle\rho(z,\bar{z})$ $\displaystyle=$
$\displaystyle\frac{2a}{\pi}\theta\left(\frac{1}{2a}-|z|^{2}\right),$ (63)

which reproduce the known GE results for $a=1/2$.

It is amusing to note, that historically, the first equation for
$O(z,\bar{z})$ in the Ginibre ensemble was delivered by Chalker and Mehlig
[22] more than three decades after the result for the uniform spectral density
$\rho(z,\bar{z})$, originally obtained by Ginibre. In our “turbulent”
formulation, at least in the large $N$ limit, the equation for the eigenvector
correlator is of primary importance, and the solution for the spectral density
follows trivially from the knowledge of the eigenvector correlator. This
observation points at the crucial difference between the hermitian and non-
hermitian random matrix models - whereas in the hermitian case the spectral
properties are dominant and the eigenvectors decouple, in the non-hermitian
case the eigenvectors control the spectral evolution. Technically, this
observation was missed in the literature because the analytic structure driven
by the $w$ variable was overlooked.

###  3.2 Microscopic limit

We refer the detailed discussion to the published work [30]. We mention only,
that since the equation for characteristic determinant is given as exactly
integrable, 2+1 dimensional diffusion equation, valid for any $N$ and any
initial condition, unraveling the universal behavior at the edge of the
spectrum is a consequence of certain limiting procedures of the exact
solution. Interestingly, the pre-shock wave appears in the non-hermitian case
in the eigenvector correlator, contrary to the appearance of the spectral pre-
shock wave in eigenvalue spectrum in the case of the hermitian ensemble.
Looking at the neighborhood of the shock by parameterizing the fluctuations in
the vicinity of the boundary as $|z|-1=\eta N^{-1/2}$, we recover the known
universal result for the microscopic behavior

$\displaystyle\rho(\eta)\approx\frac{1}{2\pi}{\rm Erfc}(\sqrt{2}\eta).$ (64)

The form of the unfolding is expected from the general geometric argument
since the number of eigenvalues on the surface of the disc grows alike $N$,
the scaling on the boundary has to grow alike $\sqrt{N}$. Note that contrary
to the hermitian case, the spectrum does not oscillate wildly at the edge, but
rather smoothly interpolates between the plateau at $1/\pi$ and 0. This
behavior can be linked to the fact, that the viscosity in the non-hermitian
case has a positive sign.

##  4 A qualitative relationship between the dynamics of eigenvectors and
eigenvalues based on numerical experiments

To gain some more insight into the intertwined dynamics of the complex
eigenvalues and eigenvectors of non-hermitian matrices, let us perform some
numerical experiments. First, for comparison, we turn to hermitian matrices.
As was mentioned before, in this case, the diffusion (2) induces the Langevin
equation (1) for the eigenvalues - no coupling to the eigenvectors is present.
It is worth mentioning that the eigenvector dynamics does depend on the
eigenvalues. The associated stochastic equation of motion is given by:

${\rm d}\ket{\psi_{i}(\tau)}=\frac{1}{\sqrt{N}}\sum_{j=1,j\neq
i}^{N}\frac{{\rm
d}B_{ij}(\tau)}{\lambda_{i}-\lambda_{j}}\ket{\psi_{j}}-\frac{1}{2N}\sum_{j=1,j\neq
i}^{N}\frac{{\rm d}\tau}{(\lambda_{i}-\lambda_{j})^{2}}\ket{\psi_{i}},$ (65)

and has been studied extensively in [34] (the $\ket{\psi_{i}}$ is an
eigenvector corresponding to eigenvalue $\lambda_{i}$, ${\rm d}B_{ij}$ is a
multi-dimensional Brownian motion). Nonetheless, the evolution of the
eigenvalues does not depend on the eigenvectors and when interested only in
the dynamics of $\lambda_{i}$, we can ignore the changes of
$\ket{\psi_{i}}$’s. The resulting process is depicted in Fig. 2 where we
present the eigenvalue trajectories of a $N=20$ hermitian matrix initiated
with two distinct eigenvalues $\lambda=-1,1$ with equal multiplicities.
Additionally, we have computed the jump amplitude of a particular eigenvalue
normalized by the simulations time step. Note that there is no distinct
dependence of the jump on how close the eigenvalue is to its neighbours.

Figure 2: Upper inlet, a single numerical realization of stochastic behaviour
of $N=20$ eigenvalues governed by eq. (1). Initially the eigenvalues were put
at $-1,1$ with equal multiplicities. Lower inlet, the jump amplitude of a
given eigenvalue (in bold in the upper plot), normalized by the square root of
the time step used in the simulation.

In the case of non-hermitian dynamics, the Langevin equations are not readily
available - we relegate their derivation and study to future work.
Nonetheless, as was argued in Sec. 3, we now know that the eigenvectors and in
particular $O_{ii}$’s are crucial for the dynamics of complex eigenvalues.

Figure 3: Numerical realization of a stochastic behaviour of $N=2$ eigenvalues
of a non-hermitian matrix diffusing according to (50), with $a=0$. Initial
conditions are $\lambda_{1}=0.3,\lambda_{2}=-0.3$ and the color of paths
encode time evolution. Black edged white dots represent the position of
$\lambda$’s at time $t=0.1$ when the distance is minimized (see Fig. 4).

To show this, we focus on an example of a non-hermitian evolution of a $N=2$
matrix starting from ${\rm diag}(-0.3,0.3)$. In Fig. 2, we observe the
eigenvalues covering the complex plane in a diffusive manner. It is also
expected that they repel each other. To perform a closer inspection (see Fig.
4), we plot three characteristics of their dynamics - the distance between the
eigenvalues $|\lambda_{1}-\lambda_{2}|$, the eigenvector correlator
$O_{11}=\braket{L_{1}}{L_{1}}\braket{R_{1}}{R_{1}}$ and the normalized jump
$\Delta\lambda_{1}/(\Delta t)^{1/2}$ of the first eigenvalue, all as a
function of time. We chose to ignore both $O_{22}=O_{11}$ and
$O_{12}=1-O_{11}$ since they do not offer any additional information. The most
interesting feature of this particular realization occurs around the time
$t_{c}=0.1$ of minimal eigenvalue distance (this precise moment is depicted by
white dots on Fig. 4). We observe that as the distance gets smaller, the
$O_{11}$ blows up in a correlated manner. This is accompanied by an increase
in the jump amplitude of the eigenvalue. W have checked that this effect
prevails when matrix size is larger than two. Note again that it was not
present for eigenvalues of hermitian matrices, for which the distance between
the eigenvalues also drives the evolution. We therefore consider this effect
as a qualitative demonstration of the co-dependence between the evolutions of
eigenvalues and eigenvectors in this scenario.

Figure 4: Time series of eigenvalue distance $|\lambda_{1}-\lambda_{2}|$,
eigenvector $O_{11}$ and eigenvalue diffusion distance
$\delta\lambda_{1}=\Delta\lambda_{1}/(\Delta t)^{1/2}$. Corresponding vertical
axes are out of scale, we identify the time $t_{c}=0.1$ to be of both minimal
distance $|\lambda_{1}-\lambda_{2}|$ and maximal values of $O_{11}$ and
${\delta\lambda_{1}}$.

##  5 Unexpected links

Several unexpected links between the static hermitian and non-hermitian random
matrix models were noted in the past [20]. The spectrum of hermitian matrices
is real, but the main tool relies on introducing the complex valued resolvent
(Green’s function), whose discontinuities allow to infer the spectral
function, using the theory of analytic functions. In the large $N$ limit, a
particular transform, known as the R-transform, related to the Green’s
function by the functional inverse as $R[G(z)]+1/G(z)=z$, plays the role of
the analog of generating function of classical cumulants in the matrix-valued
probability calculus. The R-transform constitutes the cornerstone of the free
probability theory [31] and generates matrix-valued analogues of classical
central limit theorems. In the case of non-hermitian matrices, the spectrum is
complex, but the regulator $\epsilon^{2}$ in the logarithmic potential (23)
behaves as the tip of an iceberg, pointing at a hidden algebraic structure.
Indeed, in order to maintain the analogy to the hermitian case, one has to
embed the structure of the generalized Green’s functions in the algebra of
quaternions. In such a way, a second complex variable $w$, “perpendicular” to
$z$ emerges. In the large $N$ limit, one can adapt the Voiculescu construction
for the R-transform by defining the quaternion valued functional inverse
${\cal{R}}[{\cal{G}}(Q)]+1/{\cal{G}}(Q)=Q$ and thus allowing for non-hermitian
and non-commuting convolution of random matrices [19, 20]. Surprisingly, the
links between the hermitian and non-hermitian random matrix models stretch out
to the area of dynamic processes. In the case of the Gaussian randomness, the
exact diffusion equation for the averaged characteristic polynomial finds its
exact analogue for the averaged characteristic polynomial valued in the
algebra of quaternions. It turns out, that the “hidden” variable $w$, ignored
in standard treatment of non-hermitian random matrix models, plays a crucial
role in determining the two-dimensional pattern of the spectral evolution. In
the large $N$ limit, hermitian and non-hermitian Smoluchowski-Fokker-Planck
equations take the surprisingly similar form of a Burgers-like structure. In
general, the Voiculescu equation $\partial_{\tau}G+R(G)\partial_{z}G=0$ is
replaced by its quaternionic counterpart [32]

$\displaystyle\frac{\partial{\cal{G}}_{ab}}{\partial\tau}+\sum\limits_{c,d=1}^{2}{\cal{R}}[{\cal{G}}]_{cd}\frac{\partial{\cal{G}}_{ab}}{\partial
Q_{cd}}=0,$ (66)

where latin indices label the two-by two quaternionic structure of $Q$, ${\cal
G}$ and ${\cal{R}}$. In both cases, singularities emerge. However, in the
hermitian case, singularities appear in the flow of the eigenvalues, whereas
in the case of non-hermitian ensembles, singularities appear in the flow of a
certain correlator of left and right eigenvectors. In both cases, finite $N$
effects can be taken into account as an appearance of spectral viscosity
proportional to $1/N$. There is however a crucial difference in the sign -
positive spectral viscosity smoothens the edge of the Ginibre spectrum,
yielding universal behavior given by the $\rm{Erfc}$ function, whereas
negative spectral viscosity in the GUE triggers violent oscillations, leading
to the formation of the so-called Airy kernel. Resolving the deep reasons for
these links still remains one of the challenges of random matrix models. We
summarize the unexpected links between the GUE and GE ensembles in Table I.

| GUE | GE
---|---|---
Spectral density | real | complex
Resolvent | complex-valued | quaternion-valued
| $G(z)=\frac{1}{N}\left<{\rm Tr}(z-H)^{-1}\right>$ | ${\cal G}(Q)=\frac{1}{N}\left<{\rm bTr}(Q-{\cal X})^{-1}\right>$
Determinant | $U(z,\tau)=\left<{\rm det}(z-H)\right>$ | $D(Q,\tau)=\left<{\rm det}(Q-{\cal{X}})\right>$
Diffusion eq. | $\partial_{\tau}U=-\frac{1}{2N}\partial_{zz}U$ | $\partial_{\tau}D=+\frac{1}{N}\partial_{w\bar{w}}D$
Viscosity | negative | positive
Universal behavior | oscillatory (Airy kernel) | smooth (Erfc)
R-transform | $R_{GUE}(G)=G$ | ${\cal{R}}_{GG}({\cal{G}})=\left(\begin{array}[]{cc}0&{\cal{G}}_{1\bar{1}}\\\ {\cal{G}}_{\bar{1}1}&0\end{array}\right)$
Voiculescu equation | $\frac{\partial G}{\partial\tau}+R(G)\frac{\partial G}{\partial z}=0$ | $\frac{\partial{\cal{G}}_{ab}}{\partial\tau}+\sum\limits_{c,d=1}^{2}{\cal{R}}[{\cal{G}}]_{cd}\frac{\partial{\cal{G}}_{ab}}{\partial Q_{cd}}=0$
Pre-shock waves | Flow of eigenvalues | Flow of eigenvector correlators
Table 1: Comparison of links between GUE and GE.

##  6 Conclusions

The presented results borrow to a large extent from the conclusions obtained
in the series of papers of the present authors [12, 18, 29, 30], but also
include new solutions. First, we adapted the turbulent scenario to the
Ornstein-Uhlenbeck process for GUE. Technical details are deferred to Appendix
A. Then, by a set of transformations, we provided an exact mapping between the
Ornstein-Uhlenbeck process and free diffusion. This mapping allowed us to
interpolate smoothly between the microscopic limit (Dyson’s local equilibrium)
and the macroscopic limit (Dyson’s global equilibrium). Second, we have
repeated the same scenario of the Ornstein-Uhlenbeck process for the Ginibre
ensemble. Again, we relegate technical details to Appendix B. Last but not
least, we tried to point at rather unexpected analogies and similarities in
both examples. We stressed that such analogies are detectable only when the
quaternion variables are used.

We have proven that a consistent description of non-hermitian ensembles
require the knowledge of the detailed dynamics of co-evolving eigenvalues and
eigenvectors. Moreover, at least in the large $N$ limit, the dynamics of
eigenvectors plays a major role and leads directly to the inference of the
spectral properties. This is a dramatically different scenario comparing to
the standard random matrix models, where the statistical properties of
eigenvalues are of primary importance, and the properties of eigenvectors are
basically trivial due to the their decoupling from the spectra and the fact
that they are Haar distributed on $U(N)$. We conjecture that the hidden
dynamics of eigenvectors observed in the Ginibre ensemble, is a general
feature of all non-hermitian random matrix models.

Our formalism could be exploited to expand the area of application of non-
hermitian random matrix ensembles within problems of growth, charged droplets
in quantum Hall effect and gauge theory/geometry relations in string theory
beyond the subclass of complex matrices represented by normal matrices.

One of the challenges is an explanation, why, despite being so different, the
Smoluchowski-Fokker-Planck equations for hermitian and non-hermitian random
matrix models exhibit structural similarity to simple models of turbulence,
where so-called Burgers equation plays the vital role, establishing the flow
of the spectral density of eigenvalues in the case of the hermitian or unitary
ensembles and the flow of certain eigenvector correlator in the case of non-
hermitian ensembles. Another challenge relies in completing the Langevin-like
equations (1) adapted for non-hermitian cases.

We believe that our findings will contribute to the understanding of several
puzzles of non-hermitian dynamics, alike extreme sensitivity of spectra of
non-hermitian systems to perturbations [22, 33] and the sign problem of
certain Euclidean Dirac operators.

## Acknowledgments

MAN thanks the organizers of the 7th International Conference on Unsolved
Problems on Noise, Barcelona, Spain, for hospitality. We appreciate
collaboration with Zdzisław Burda, Ewa Gudowska-Nowak, Romuald Janik, Andrzej
Jarosz, Jerzy Jurkiewicz, Gabor Papp and Ismail Zahed and discussions with
Neil O’Connell, Yan Fyodorov, Mario Kieburg, Roger Tribe, Dima Savin and Oleg
Zaboronski. This work was supported by the Grant DEC-2011/02/A/ST1/00119 of
the National Centre of Science. WT appreciates partial support from the
Ministry of Science and Higher Education through the Grant 0225/DIA/2015/44.

## Appendix A

In what follows we derive the partial differential equation (12). For
completeness, we again introduce the Ornstein-Uhlenbeck diffusion process in
terms of a Smoluchowski-Fokker-Planck equation (8):

$\displaystyle\partial_{\tau}P(x,y,\tau)=\mathcal{A}(x,y)P(x,y,\tau),$ (67)

with the Laplace operator

$\displaystyle\mathcal{A}(x,y)$
$\displaystyle=\sum_{k=1}^{N}\left(\frac{1}{2N}\frac{\partial^{2}}{\partial
x_{kk}^{2}}+a\frac{\partial}{\partial
x_{kk}}x_{kk}\right)+\frac{1}{4N}\sum_{i<j=1}^{N}\left(\frac{\partial^{2}}{\partial
x_{ij}^{2}}+\frac{\partial^{2}}{\partial y_{ij}^{2}}\right)+$ (68)
$\displaystyle+a\sum_{i<j=1}^{N}\left(\frac{\partial}{\partial
x_{ij}}x_{ij}+\frac{\partial}{\partial y_{ij}}y_{ij}\right).$

As a first step, we write the determinant as a Gaussian integral over a set of
Grassmann variables $\eta_{i},\bar{\eta}_{i}$:

$\displaystyle{\rm det}~{}{A}=C{\int{{\prod_{i,{j}=1}^{N}{{\rm
d}{\eta}_{{i}}}}{{\rm
d}{{{\overline{\eta}}}_{{j}}}}}}\,{\exp{\left({\overline{{\eta}_{{i}}}{A}_{{ij}}{\eta}_{{j}}}\right)}}.$
(69)

with an irrelevant proportionality constant $C$. This allows us to express the
characteristic polynomial $U$ defined by (11) in the following way:

$\displaystyle U(z,t)$
$\displaystyle=C\int\mathcal{D}[\bar{\eta},\eta,x,y]P(x,y,\tau)\exp{\left[\sum_{i,j=1}^{N}\bar{\eta}_{i}\left(z\delta_{ij}-H_{ij}\right)\eta_{j}\right]},$
(70)

where the integration measure is defined by

$\displaystyle\mathcal{D}[\bar{\eta},\eta,x,y]$
$\displaystyle\equiv\prod_{i,j=1}^{N}{\rm d}\eta_{i}{\rm
d}\overline{\eta}_{j}\prod_{k=1}^{N}{\rm d}x_{kk}\prod_{n<m=1}^{N}{\rm
d}x_{nm}{\rm d}{y}_{nm}.$ (71)

The hermiticity condition ($H_{ij}=\bar{H}_{ji}$) is used to write the
argument of the exponent of (70) in an explicit form:

$\displaystyle
T_{g}(\bar{\eta},\eta,x,y,z)\equiv\sum_{i,j=1}^{N}\bar{\eta}_{i}\left(z\delta_{ij}-H_{ij}\right)\eta_{j}=\sum_{r=1}^{N}\bar{\eta}_{r}\left(z-x_{rr}\right)\eta_{r}+$
$\displaystyle-\sum_{n<m=1}^{N}\left[x_{nm}\left(\bar{\eta}_{n}\eta_{m}+\bar{\eta}_{m}\eta_{n}\right)+iy_{nm}\left(\bar{\eta}_{n}\eta_{m}-\bar{\eta}_{m}\eta_{n}\right)\right].$
(72)

Now we make use of the diffusion equation (67) which, after integrating by
parts, gives:

$\displaystyle\partial_{\tau}U=\int\mathcal{D}\partial_{\tau}P\exp\left(T_{g}\right)=\int\mathcal{D}\mathcal{A}P\exp\left(T_{g}\right)=\int\mathcal{D}P\tilde{\mathcal{A}}\exp\left(T_{g}\right),$
(73)

with $\tilde{\mathcal{A}}=\mathcal{A}(a\to-a)$. We calculate the expression:

$\displaystyle\tilde{\mathcal{A}}\exp\left(T_{g}\right)=$
$\displaystyle\left[a\sum_{k=1}^{N}x_{kk}\bar{\eta}_{k}\eta_{k}+\frac{1}{N}\sum_{i<j=1}^{N}\bar{\eta}_{i}\eta_{j}\bar{\eta}_{j}\eta_{i}\right]\exp\left(T_{g}\right)+$
(74)
$\displaystyle+\left[a\sum_{i<j=1}^{N}\left[(x_{ij}+iy_{ij})\bar{\eta}_{i}\eta_{j}+(x_{ij}-iy_{ij})\bar{\eta}_{j}\eta_{i}\right]\right]\exp\left(T_{g}\right).$

by schematically writing down the action of derivatives on $\exp(T_{g})$:

$\displaystyle\partial_{x_{kk}}$
$\displaystyle\quad\rightarrow\quad-\bar{\eta}_{k}\eta_{k},\qquad\partial^{2}_{x_{kk}}\quad\rightarrow\quad
0,$

and for $i\neq j$:

$\displaystyle\partial_{x_{ij}}$
$\displaystyle\quad\rightarrow\quad-(\bar{\eta}_{i}\eta_{j}+\bar{\eta}_{j}\eta_{i}),$
$\displaystyle\partial_{y_{ij}}$
$\displaystyle\quad\rightarrow\quad-i(\bar{\eta}_{i}\eta_{j}-\bar{\eta}_{j}\eta_{i}),$
$\displaystyle\partial^{2}_{x_{ij}}$
$\displaystyle\quad\rightarrow\quad(\bar{\eta}_{i}\eta_{j}+\bar{\eta}_{j}\eta_{i})(\bar{\eta}_{i}\eta_{j}+\bar{\eta}_{j}\eta_{i})=-2\bar{\eta}_{i}\eta_{j}\bar{\eta}_{j}\eta_{i},$
$\displaystyle\partial^{2}_{y_{ij}}$
$\displaystyle\quad\rightarrow\quad-(\bar{\eta}_{i}\eta_{j}-\bar{\eta}_{j}\eta_{i})(\bar{\eta}_{i}\eta_{j}-\bar{\eta}_{j}\eta_{i})=-2\bar{\eta}_{i}\eta_{j}\bar{\eta}_{j}\eta_{i}.$

We rewrite the terms of (74) accordingly:

$\displaystyle\sum_{i=1}^{N}\bar{\eta}_{i}\eta_{i}\exp\left(T_{g}\right)=\partial_{z}\exp\left(T_{g}\right),$
$\displaystyle\sum_{i<j=1}^{N}\bar{\eta}_{i}\eta_{i}\bar{\eta}_{j}\eta_{j}\exp\left(T_{g}\right)=\frac{1}{2}\partial_{zz}\exp\left(T_{g}\right),$
$\displaystyle\sum_{i=1}^{N}\bar{\eta}_{i}\partial_{\bar{\eta}_{i}}\exp\left(T_{g}\right)=\left[z\partial_{z}-\sum_{i=1}^{N}x_{ii}\bar{\eta}_{i}\eta_{i}-\sum_{i<j=1}^{N}\left[(x_{ij}+iy_{ij})\bar{\eta}_{i}\eta_{j}+(x_{ij}-iy_{ij})\bar{\eta}_{j}\eta_{i}\right]\right]\exp\left(T_{g}\right).$

We obtain thus, after recalling (73) and (74):

$\displaystyle\partial_{\tau}U=\int\mathcal{D}P\left(-\frac{1}{2N}\partial_{zz}+az\partial_{z}-a\sum_{i=1}^{N}\bar{\eta}_{i}\partial_{\bar{\eta}_{i}}\right)\exp\left(T_{g}\right),$
(75)

where the last term is explicitly calculable upon integrating by parts

$\displaystyle\int\mathcal{D}P\sum_{i=1}^{N}\bar{\eta}_{i}\partial_{\bar{\eta}_{i}}\exp\left(T_{g}\right)=N\int\mathcal{D}P\exp\left(T_{g}\right),$
(76)

so that we arrive at the equation (12)

$\displaystyle\partial_{\tau}U(z,t)=-\frac{1}{2N}\partial_{zz}U(z,\tau)+az\partial_{z}U(z,\tau)-aNU(z,\tau).$
(77)

## Appendix B

Here we present the derivation for the evolution equation (54). The Ornstein-
Uhlenbeck process in the non-hermitian case is given by Smoluchowski-Fokker-
Planck equation for the jpdf $P(x,y,\tau)$:

$\displaystyle\partial_{\tau}P(x,y,\tau)=\mathcal{B}(x,y)P(x,y,\tau),$ (78)

with the operator

$\displaystyle\mathcal{B}=\frac{1}{4N}\sum_{i,j=1}^{N}\left(\partial^{2}_{x_{ij}}+\partial^{2}_{y_{ij}}\right)+a\sum_{i,j=1}^{N}\left(\partial_{x_{ij}}x_{ij}+\partial_{y_{ij}}y_{ij}\right).$
(79)

To proceed, we open the determinant defined in (49) with the help of Grassmann
variables $\eta_{i},\xi_{i}$:

$\displaystyle
D=C^{\prime}\int\mathcal{D}[...]P(x,y,\tau)\exp\left[\left(\begin{array}[]{cc}\bar{\eta}&\bar{\xi}\end{array}\right)\left(\begin{array}[]{cc}z-X&-\bar{w}\\\
w&\bar{z}-X^{\dagger}\end{array}\right)\left(\begin{array}[]{c}\eta\\\
\xi\end{array}\right)\right],$

with $X_{ij}=x_{ij}+iy_{ij},X^{\dagger}_{ij}=x_{ji}-iy_{ji}$, an irrelevant
constant $C^{\prime}$ and the joint measure

$\displaystyle\qquad\mathcal{D}[\bar{\eta},\eta,\bar{\xi},\xi,x,y]$
$\displaystyle=\prod_{i=1}^{N}\mathrm{d}\eta_{i}\mathrm{d}\bar{\eta}_{i}\mathrm{d}\xi_{i}\mathrm{d}\bar{\xi}_{i}\prod_{i,j=1}^{N}\mathrm{d}x_{ij}\mathrm{d}y_{ij}.$
(81)

The argument of the exponent is equal to

$\displaystyle S_{g}$
$\displaystyle\equiv\sum_{i,j=1}^{N}\left[\bar{\eta}_{i}(z-X)_{ij}\eta_{j}+\bar{\xi}_{i}(\bar{z}-X^{\dagger})_{ij}\xi_{j}+w\bar{\xi}_{i}\eta_{i}-\bar{w}\bar{\eta}_{i}\xi_{i}\right]=$
$\displaystyle=\sum_{i,j=1}^{N}\left[-x_{ij}(\bar{\eta}_{i}\eta_{j}+\bar{\xi}_{j}\xi_{i})-iy_{ij}\left(\bar{\eta}_{i}\eta_{j}-\bar{\xi}_{j}\xi_{i}\right)+z\bar{\eta}_{i}\eta_{i}+\bar{z}\bar{\xi}_{i}\xi_{i}+w\bar{\xi}_{i}\eta_{i}-\bar{w}\bar{\eta}_{i}\xi_{i}\right].$

We make use of the diffusion equation SFP equation and integrate by parts:

$\displaystyle\partial_{\tau}D=\int\mathcal{D}\partial_{\tau}P\exp\left(S_{g}\right)=\int\mathcal{D}\mathcal{B}P\exp\left(S_{g}\right)=\int\mathcal{D}P\tilde{\mathcal{B}}\exp(S_{g}),$
(82)

where $\tilde{\mathcal{B}}=\mathcal{B}(a\to-a)$. The last integrand reads:

$\displaystyle\tilde{\mathcal{B}}\exp(S_{g})=\left[\frac{1}{N}\sum_{i,j=1}^{N}\bar{\eta}_{i}\eta_{j}\bar{\xi}_{j}\xi_{i}+a\sum_{i,j=1}^{N}\left((x_{ij}+iy_{ij})\bar{\eta}_{i}\eta_{j}+(x_{ij}-iy_{ij})\bar{\xi}_{j}\xi_{i}\right)\right]\exp(S_{g}),\qquad)$
(83)

where we used the schematic formulas acting on $\exp(S_{g})$:

$\displaystyle\partial_{x_{ij}}$
$\displaystyle\quad\rightarrow\quad-(\bar{\eta}_{i}\eta_{j}+\bar{\xi}_{j}\xi_{i}),$
$\displaystyle\partial_{y_{ij}}$
$\displaystyle\quad\rightarrow\quad-i(\bar{\eta}_{i}\eta_{j}-\bar{\xi}_{j}\xi_{i}),$
$\displaystyle\partial^{2}_{x_{ij}}$
$\displaystyle\quad\rightarrow\quad(\bar{\eta}_{i}\eta_{j}+\bar{\xi}_{j}\xi_{i})(\bar{\eta}_{i}\eta_{j}+\bar{\xi}_{j}\xi_{i})=2\bar{\eta}_{i}\eta_{j}\bar{\xi}_{j}\xi_{i},$
$\displaystyle\partial^{2}_{y_{ij}}$
$\displaystyle\quad\rightarrow\quad-(\bar{\eta}_{i}\eta_{j}-\bar{\xi}_{j}\xi_{i})(\bar{\eta}_{i}\eta_{j}-\bar{\xi}_{j}\xi_{i})=2\bar{\eta}_{i}\eta_{j}\bar{\xi}_{j}\xi_{i}.$

To continue, we rewrite the terms of (83) proportional to $a$ as follows:

$\displaystyle\sum_{i=1}^{N}\bar{\eta}_{i}\eta_{i}\exp(S_{g})=\partial_{z}\exp(S_{g}),\qquad\sum_{i=1}^{N}\bar{\xi}_{i}\xi_{i}\exp(S_{g})=\partial_{\bar{z}}\exp(S_{g}),$
$\displaystyle\sum_{i=1}^{N}\bar{\eta}_{i}\xi_{i}\exp(S_{g})=-\partial_{\bar{w}}\exp(S_{g}),\qquad\sum_{i=1}^{N}\bar{\xi}_{i}\eta_{i}\exp(S_{g})=\partial_{w}\exp(S_{g}),$
$\displaystyle\sum_{i=1}^{N}\bar{\eta}_{i}\partial_{\bar{\eta}_{i}}\exp(S_{g})=\left(-\sum_{i,j=1}^{N}(x_{ij}+iy_{ij})\bar{\eta}_{i}\eta_{j}+z\sum_{i=1}^{N}\bar{\eta}_{i}\eta_{i}-\bar{w}\sum_{i=1}^{N}\bar{\eta}_{i}\xi_{i}\right)\exp(S_{g}),$
$\displaystyle\sum_{j=1}^{N}\bar{\xi}_{j}\partial_{\bar{\xi}_{j}}\exp(S_{g})=\left(-\sum_{i,j=1}^{N}(x_{ij}-iy_{ij})\bar{\xi}_{j}\xi_{i}+\sum_{j=1}^{N}\bar{z}\bar{\xi}_{j}\xi_{j}+w\sum_{j=1}^{N}\bar{\xi}_{j}\eta_{j}\right)\exp(S_{g}),$

so that

$\displaystyle
a\sum_{i,j=1}^{N}\left((x_{ij}+iy_{ij})\bar{\eta}_{i}\eta_{j}+(x_{ij}-iy_{ij})\bar{\xi}_{j}\xi_{i}\right)\exp(S_{g})=$
$\displaystyle=\left[a\sum_{i=1}^{N}\left(-\bar{\eta}_{i}\partial_{\bar{\eta}_{i}}-\bar{\xi}_{j}\partial_{\bar{\xi}_{j}}\right)+a(z\partial_{z}+\bar{z}\partial_{\bar{z}}+w\partial_{w}+\bar{w}\partial_{\bar{w}})\right]\exp(S_{g}).$
(84)

Plugging the above expressions into (82), gives:

$\displaystyle\partial_{\tau}D=\int\mathcal{D}P$
$\displaystyle\left(\frac{1}{N}\sum_{i,j=1}^{N}\bar{\eta}_{i}\eta_{j}\bar{\xi}_{j}\xi_{i}-a\sum_{i=1}^{N}\left(\bar{\eta}_{i}\partial_{\bar{\eta}_{i}}+\bar{\xi}_{j}\partial_{\bar{\xi}_{j}}\right)+\right.$
(85)
$\displaystyle+a(z\partial_{z}+\bar{z}\partial_{\bar{z}}+w\partial_{w}+\bar{w}\partial_{\bar{w}})\Bigg{)}\exp(S_{g}),$

where the first term is expressible as

$\displaystyle\partial_{w\bar{w}}D$
$\displaystyle=\int\mathcal{D}P\sum_{i,j=1}^{N}\bar{\eta}_{j}\eta_{i}\bar{\xi}_{i}\xi_{j}\exp\left(S_{g}\right),$
(86)

and the second one reads

$\displaystyle\int\mathcal{D}P\sum_{i=1}^{N}\left(\bar{\eta}_{i}\partial_{\bar{\eta}_{i}}+\bar{\xi}_{j}\partial_{\bar{\xi}_{j}}\right)\exp\left(S_{g}\right)=2N\int\mathcal{D}P\exp\left(S_{g}\right).$
(87)

Taking them into account we finally obtain equation (54)

$\displaystyle\partial_{\tau}D=\frac{1}{N}\partial_{w\bar{w}}D-2NaD+a\textrm{d}D,$
(88)

where
$\textrm{d}=z\partial_{z}+\bar{z}\partial_{\bar{z}}+w\partial_{w}+\bar{w}\partial_{\bar{w}}$.

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