Natural Selection and Random Matrix Theory
Mehdi<EMAIL_ADDRESS>
Ferdowsi University of Mashhad
Faculity of Science, Physics Department

We will study the relation between two well-known theories, genetic evolution
and random matrix theory in the context of many-body systems and chaos theory.
We show that the time evolution of certain chaotic quantum mechanical models
is similar to the evolution of a living cell. It is also suggested that
natural selection can be described by a random matrix theory with statistical
distribution in which the genetic evolution acts as a Gross-Witten-Wadia phase
transition.

##  1 Introduction

It is amazing if we look at different branches of science, we would find out
they are originated from the same root. So besides the advanced studies in a
single area, one has to study other branches as well, seeking connections. One
of these branches that can be directly related to quantum physics is the
genetics and the biology of living cells. We aim our focus to the well-known
concept in genetic science called Evolution. Evolution theory was first
proposed by Charles Darwin in 1859 [1] and then it was significantly modified
to the new version by Mendel [2, 3, 4].

It is strongly believed that all species on earth are evaluated from a complex
single-celled organism called Last Universal Common Ancestor (LUCA). From the
Darwinism point of view, this theory proposes that all the living beings are
evaluated from each other since the creation of the first living cell or the
ancestor. Darwin also said that human beings are the result of the evolution
of four-legged mammals like chimpanzees regarded as Homo Sapiens, however,
this may be in contrast with the new version of evolution. The newest version
proposes a completely different viewpoint in which it contains three different
ancestors: (1) Bacteria, (2) bacteria-like microbes (Archaea), and (3)
Eukaryotes, the group that includes plants and other species, such as animals
and humans. All three cases are independent of each other, and it is also
believed that today’s human life may be evaluated from a completely different
cell than from other animals. Plants and trees cells contain certain organic
materials like chlorophyll which is responsible to transform the sunlight into
organic nutritional products whereas animals and humans cells contain more
complex structures.
The evolution works in the direction of modifying the mechanism of the entire
physical living system to a new more advanced system to protect the living
being from external intense conditions. When a creature undergoes unexpected
natural stresses and anomalies, the natural selection would evaluate it to a
more stable version to have the needed defense and the protection against the
unwilling behaviors. Evolution is a very long-term process and its time scales
with millions of years. That is why the observations of fossils of dinosaurs
and other ancient remnants can be surprisingly beneficial. The evolution is
strongly correlated with external environmental conditions such as air
pressure, temperature, food resources, cosmic rays, etc. and it also depends
on the occurrence of natural disasters like earthquakes, asteroids,
hurricanes, etc.
The basic question one can ask is that on what scale does the evolution
happen? Yet, the answer is ambiguous and it is mostly considered in cellular
scales. In this paper, we want to discuss the subject on a special scale where
the quantum dynamic is dominant and all the dynamic is represented by
probability laws. It has been observed that cosmic rays can remarkably affect
evolution and it would be strange to see the existence of life is in hands of
probabilities. This also means that there is a state in which our existence
may not have occurred at all! If the evolution happens in a quantum mechanical
way, then its dynamics can be described by the space of possible states or
Hilbert space which contains finite possible states to be observed by the
observer. The observed states are our observables in daily life but this does
not mean that the other states do not exist. This is due to the fundamental
uncertainty principle of quantum mechanics.
This whole scenario is meanly pointing out that natural selection can occur
quantum mechanically and choose between an infinite number of possible states
with random distribution each with a specific probability weight. This is
known as Quantum Darwinism and first proposed by [5].
Consider a setup like fig. (1) where we are trying to observe the actual spin
of the electron beam. The spin is an intrinsic quantity of a particle that
determines the magnetic field of the particle and its magnetic interaction
with other particles with the same type of spin.
If the observer detects the z component at first then changes the setup to
measure another component like x, the former spin result will be completely
changed. If we measure the z component again it may give us a completely
different value. This is unexpected because spin is an intrinsic property. So
we conclude that the effects of the observer $SG_{x}$ have changed the history
of the particle.

Figure 1: Stern-Gerlach setups to determine each spatial component of the
spin. This shows that the contribution of $SG_{x}$ has cleaned the history of
the particle and at the end, both $z$ component appears. This is strange since
we are blocked $S_{z^{-}}$ in the first measurement. This directly leads to
the principle that one can not exactly measure two quantities of the quantum
system at the same time. Once we try to do so, the system will be disrupted.

Having said that the states of the system can be chosen accidentally with a
specific probability weight, so the evolution from one level to an upper
advanced level can also be picked up randomly. This is possible if we assume
that the early evolution series has happened quantum mechanically. However,
random matrix theory is also can be used for classical phenomena as well.
Random matrix theory describes many-body chaotic systems with strong coupling
between its partitions which are basically thermal systems and it has been
intended to study statistical behaviors. The random matrix theory is used in
various fields, such as string theory, two-dimensional gravity [6], conformal
field theory, number theory, condensed matter [7], integrable systems, glass
theory, [8], and stochastic processes [9]. This theory is also the basis of
many scientific applications of economics [10]. Moreover, a big development of
this theory has been made and specified to study the RNA folding [11, 12, 13,
14, 15, 16, 17]. It is quite wondering that the natural selection can be
completely random and more wondering is that this randomness can be exactly
analyzed and by choosing an appropriate distribution for matrix elements, one
can see on what conditions the genetic evolution takes place which can also be
checked in the laboratory.
The state of the living cell before the decomposition can be considered as a
random matrix and when the decomposition starts it will generate two other
random matrices between $m$ possible set of states (fig. (2)). This is what we
call in this paper natural selection and it does not always lead to an
evolutionary change in the DNA structure. Just if the environmental conditions
change intensely the decoupling may get a genetic disorder after the
decomposition as we expect from a strongly coupled chaotic system (see
sec.(2),(3)).
Another point is that if we write each state growth as a random matrix model
with matrices of size $L$ and the coupling constant $g$, then at the double
scaling limit ($L\rightarrow\infty$ and $g\rightarrow g_{c}$), each state
construct a group of manifolds with $h$ handles and $n$ boundaries [6].
Therefore, all $m$ states can be considered as the moduli space of Riemann
surfaces and one can find an appropriate $SL(2,\mathbb{R})$ transformation
between every two states [18, 21, 22, 23]. One can also be able to find
Eynard-Orantin [24, 25, 26, 27] or Mirzakhani’s recursion relation [18, 19,
20] between the manifolds of each state.

Figure 2: If the mechanism of decomposition of the living cell is assumed
quantum mechanically then there are $m$ possible states for the cell to
choose. Each has its growth and probability weight. In the next sections, we
explain that the states $1,2,...,m$ can be represented by a random matrix
theory.

The context of evolution is related to several scientific approaches. Since it
is very sensitive to initial environmental and internal conditions, it can be
considered as a system with chaotic behavior and can be studied in the context
of chaos theory. In section (2), we will give a general view of the chaos
theory and see how such systems can be mathematically analyzed and related to
quantum mechanics.

In section (3), we show that the exponential evolution of a quantum chaotic
system is similar to the exponential growth of bacterial or cellular binary
fission.

In section (4), First, we give a general view of the DNA’s structure and
discuss the probability of $n$ possible evolutions that can happen between $m$
states of the cell proliferation. Moreover, we will point out an important
idea that can explain natural selection as a phase transition in the large $L$
random matrix theory.

Finally, Remarks and conclusions will be given in section (5) to sum up the
discussions argued in this paper.

##  2 Chaos Theory

Chaos in specific systems is known as a periodic long-time behavior that is
very sensitive to initial conditions. Many physical systems follow this kind
of behavior and chaos is a universal feature. A system with a strong
correlation between its partitions, with a small variation on its physical
quantities, can be significantly changed to a different state in a long time.
A famous example is that if a hurricane is occurring at a specific time in one
region of the earth, flying a butterfly close to that region can evaluate the
system drastically. This evolution may cause a bigger hurricane in the
northern regions of the earth. This is known as the butterfly effect. These
systems are chaotic and predicting their evolution over time is a hard piece
of work. However, sometimes chaotic behavior of small partitions of a system
on a large scale and long times can lead to an equilibrium that is well-
defined without any erratic behavior. For example, a very hot gas into a box
in microscopic scale and short times may seem quite complicated to study but
on the other hand, from the statistical point of view, In macroscopic scales
and long times, the system ends up with an equilibrium state and completely
predictable. Although studying chaotic systems seem to be intense, physics and
mathematical developments have accomplished great progress in this background.
one of these developments is studying their energy spectrum. The energy
spectrum of chaotic systems is completely different from integrable systems in
which one has to use nonperturbative statistic models to investigate their
features. One of the best of these models is the Random Matrix Theory (RMT)
[46, 47, 48, 49].

In this section, we first present an overview of chaos theory in classical and
quantum mechanical contexts and then connect the subject with bacterial
growth. Generally, most multi-particle quantum systems with strong coupling
constants are chaotic. Consider a particle in phase space, at time zero in
point $q(0)$ with momentum $p(0)$ which travels with a specific velocity on an
arbitrary path. Now if at time zero, we make small variations $(\delta p(0))$
and $(\delta q(0))$ on particle’s momentum and position, these variations will
grow exponentially with time and the particle will take a completely different
path because of the sensitivity of the system on initial conditions (figure b
of (3)).

$\displaystyle\frac{\delta q(t)}{\delta q(0)}=Ce^{\lambda_{L}t}\,$ (2.1)

Where $\lambda_{L}$ is called the Lyapunov exponent and $C$ is a constant
value. By replacing the derivative with the variations, the left-hand side of
(2.1) would be equal to a Poisson bracket

$\displaystyle\frac{\delta x(t)}{\delta
x(0)}=\\{x(t),p(0)\\}=Ce^{\lambda_{L}t}\,$ (2.2)

For some reason, a direct modification of chaos to quantum systems is not
available [28].

Figure 3:  (a) By producing very small variations on initial conditions, these
variations will grow over time. If the system is integrable (blue
trajectories), small variations in initial conditions will lead to a slightly
different path with a little difference from the first one and in some cases
is neglectable. These paths are stable and diverge with a linear combination
of time. In a chaotic system (red trajectories), small variations will lead to
a very big difference in the future of particles and the particle would travel
on a completely different path. These paths are unstable and diverge
exponentially with time. (b) The ratio of the particle path in second zero
$(\delta q(0))$ to the difference in the same path in second $t$, $(\delta
q(t))$, grows exponentially with time (equation(2.1)).

Larkin and Ovchinnikov provided an intuitive definition of chaos in the semi-
classical regime [31]. To modify the problem to quantum systems we just need
to use operator commutation relation instead of Poisson bracket. Generally,
the commutator is an operator, so for observing the exponential growth we need
to calculate the expectation value of the commutator. The expectation value of
the commutator in the thermal state becomes zero due to phase cancelation. A
simple way to fix this is to consider the square of the commutator 222The
expectation value of equation (2.4) and other equations are in a thermal
state. This is indicated by $\beta$ indices in the right-hand side of the
expectation values and shows a thermal ensemble averaging:
$\displaystyle\langle F(x)\rangle_{\beta}=\frac{\int dx~{}F(x)~{}e^{-\beta
H(x)}}{\int dx~{}e^{-\beta H(x)}}\,.$ (2.3) The Lyapunov exponent depends on
temperature: This is a well-known statement from classical chaos which says
the regions of the phase space that does not get mixed, have different
Lyapunov exponents. If we were working with a microcanonical ensemble, the
energy would be conserved. One should also note that the definition of the
Lyapunov exponent given here, is not original in the sense that here the
exponent is local and does not have a temporal average. [31],

$\displaystyle\langle[x(t),p(0)]^{2}\rangle_{\beta}=C^{2}\hbar^{2}e^{2\lambda_{L}t}\,.$
(2.4)

The expectation value of the commutator saturates to a maximum value (fig.(4))
at a specific time which is called the scrambling time.

Figure 4: The commutation grows exponentially and in a specific time $t_{*}$,
the growth will saturate.

In the context of black hole physics, this is how long it takes for the
information to be spread on the area which is logarithmic in terms of the
total number of degrees of freedom of the system. For example, if an electron
goes toward the black hole’s horizon from a certain radius, by considering a
conductive surface for the horizon, the amount of time it takes for the
electron’s charge density to be scattered on the horizon’s area is of order
the scrambling time [29]. If we throw a diary toward the black hole, the
amount of time we have to wait for the first information to reach us in the
form of Hawking radiation (after Page’s time) is also equal to the scrambling
time [30].
Equation (2.4) includes a set of four-point functions that are known as Out-
of-Time Order correlation functions (OTO). A key point is that in a chaotic
system, the behavior of $OTO(t)$ functions is exponential with time. Consider
the following four-point function:

$\displaystyle OTO(t)\equiv\langle\psi|W(t)V(0)W(t)V(0)|\psi\rangle\,$ (2.5)

Where $V(0)$ and $W(t)$ are two hermitian operators
$(WW^{\dagger}=W^{2}=1,VV^{\dagger}=V^{2}=1)$ in time zero and $t$. Now we
define the function $C(t)$:

$\displaystyle
C(t)\equiv\langle\psi|-[W(t),V(0)]^{2}|\psi\rangle=\langle\psi|W(t)V(0)V(0)W(T)+V(0)W(t)W(t)V(t)-$
$\displaystyle-W(t)V(0)W(t)V(0)-V(0)W(t)V(0)W(t)|\psi\rangle=2[1-Re(OTO)]\,$
(2.6)

Where $OTO(t)$ and $C(t)$ parts evaluate with time as shown in figure (5).
Sometimes it is more convenient to work with $C(t)$ instead of using $OTO(t)$
directly. The reason behind the decaying of $OTO(t)$ function is that if
firstly we affect the operator $V(0)$ and $W(t)$ on the state of the system,
because of the sensitivity on the initial conditions, the state of the system
becomes completely indistinguishable from the previous one in time $t$.
Consequently, if operators $V(0)$ and $W(t)$ act again on the state of the
system, The correlation between the primary state and the secondary state
decreases over time.

Figure 5: The behavior of OTO(t) and C(t) functions with time.

Now to estimate time $t_{*}$, We will consider a specific system with $N$
particles, with the following interactive Hamiltonian

$\displaystyle
H=\frac{1}{3!}\sum\limits_{i<j<k}^{N}\gamma_{ijk}S_{i}S_{j}S_{k}\,,$ (2.7)

Where $S$ are spinor operators which can evaluate with time and interact with
other particles. The parameter $\gamma_{ijk}$ is a matrix picked randomly, so
(2.7) is an ensemble of Hamiltonians. The model with four-interactive spins is
deeply analyzed and it is called the Sachdev-Ye-Kitaev model [32, 33, 34].
If we let one spin operator evaluates with time, by using the Baker-Hausdorff
lemma we can write

$\displaystyle
S_{j}(t)=e^{iHt}S(0)e^{-iHt}=\sum\limits_{k=0}^{\infty}\frac{(it)^{k}}{k!}[H,[H,...,[H,S_{j}]]]\,,$
(2.8)

Where $t$ is time and $k$ is the order of expansion and the number of
Hamiltonian commutations with $S_{j}$. According to figure (6) in every order
of time, by commutating the operator $S_{j}$ with the Hamiltonian (2.7), the
particle first interacts with the other two particles, then with the other
four, and this rate will grow with time.

Figure 6: The commutation of the spin with Hamiltonian in each time step,
based on equation (2.8), the spin interacts with two particles (or decay to
two particles) and this behavior will grow exponentially with time until it
interacts with the Nth particle of the system in $t_{*}$.

Note that in each step there are a bunch of spinor matrices that are random
and independent due to the choice of $\gamma_{ijk}$. Since $\gamma_{ijk}$
depends on the orders of indices and the Hilbert space dimension $2^{N/2}$,
this growth will stop at the scrambling time. Although $S_{i}$ and $S_{j}$
will form in the first place, $S_{j}$ can be reproduced with different random
distributions after some steps.
If we consider the system is compact on a circle or a sphere and all particles
have the same distance from each other, the question one may ask is that when
the particle $S_{j}$ will interact with (or decay to) the $N$th particle? In
other words, how long does take for the information to be spread on the entire
system? It is clear from figure (6) that the operator growth is exponential
and the time should be equivalent to the logarithm of the number of particles
in base two for each temporal step. Therefore for a thermal system with
chaotic evolution, this time would be

$\displaystyle t_{*}=c\beta log_{2}(N/N_{0})=c\beta log_{2}(S)\,,$ (2.9)

Where $N$ is the total number of particles that directly determine the entropy
($N\sim S$). $\beta$ is the inverse of temperature which is added to maintain
the dimension of time and $c$ is a proportionality constant factor. This
behavior in $k$ temporal steps is also referred to as k-locality or operator
growth.

Maldacena, Shenker, and Stanford have shown that for every quantum chaotic
system, the Lyapunov exponent can not exceed a maximum value. This value is
called the chaos bound or Maldacena-Shenker-Stanford (MSS) bound and it is
given by [38],

$\displaystyle\lambda_{L}\leq\frac{2\pi k_{B}T}{\hbar}\,,$ (2.10)

Where $T$ is the temperature of the system. The classical limit of this
equation occurs when we take $\hbar\rightarrow 0$ and this means that
classical chaotic systems have no limit on the Lyapunov exponent. Now by
taking $k_{B}$=$\hbar$=1, the chaos bound becomes

$\displaystyle\lambda_{L}\leq\frac{2\pi}{\beta}\,.$ (2.11)

For a Schwarzschild black hole, it has been shown that the energy spectrum
near the horizon is chaotic and $\lambda_{L}$ is equal to the surface gravity
parameter $\kappa$ which is given by

$\displaystyle\lambda_{L}=\kappa=\frac{2\pi}{\beta}\,.$ (2.12)

For Shwartzchild black hole $\kappa=\frac{1}{4MG}$ and by substituting it in
equation (2.12) we would obtain Hawking temperature [35]. For $OTO(t)$ and
$t_{*}$ we have

$\displaystyle OTO(t)\propto\frac{1}{S}e^{\kappa
t}\,,~{}~{}~{}~{}~{}~{}t_{*}=\frac{\beta}{2\pi}ln(S)\,.$ (2.13)

Therefore the constant $c$ in equation (2.9) is equal to $\frac{1}{2\pi}$.
Here $S$ is the black hole entropy [35, 36, 37]. In conclusion, black holes
saturate the chaos bound which means black holes are one of the most chaotic
quantum systems known. That is why these systems are sometimes called fast
scramblers. This has been conjectured by Sekino and Susskind [39] and then
proved in [40, 41, 42, 43, 44, 45].
In the next section, we will see that this behavior also appears in the case
of bacterial or cellular proliferation, which motivates us toward using this
natural decomposition as a single choice of many choices on an ensemble.

##  3 Bacterial and Cellular Proliferation

In this section, we will consider binary fission with initial $N_{0}$ numbers
of bacterias. After $k$ generations the total number of bacteria would be
equal to $N$. This can be given by

$\displaystyle\frac{N}{N_{0}}=2^{k}\,,$ (3.1)

We will conventionally take $N_{0}=1$. In each generation, the number of
bacteria doubles until it gets the maximum total number $N_{max}$ (fig. (7)).

Figure 7: Bacterial or cellular binary proliferation. The mechanism is similar
to what happens in fig. (6). Each color corresponds to a different random
matrix.

The exponential growth phase will turn into the stationary phase at the
scrambling time (fig.(8)):

$\displaystyle t_{*}=\alpha\beta log_{2}(N_{max})\,,$ (3.2)

Where $\alpha$ is a constant of proportionality and can be experimentally
determined by the data of bacterial growth. This is reasonable since the
stationary phase is proportional to the total number of bacteria $N_{max}$.
Note that in the stationary phase, the amount of produced bacterias is roughly
the same as those who are dying and the overall number remains fixed.
Moreover, we know that as the temperature increase, the growth rate of the
organism is also increased and that is clear from fig. (8) in which the
scrambling time gets shorter when the temperature gets higher. However one has
to be careful about the amount of the temperature since in very high
temperatures the growth can be drastically drop down and this shows the side
detrimental effects on the living system.

Figure 8: The bacterial growth has an exponential behavior in log phase and it
will stop at the scrambling time $t$ and turn to the stationary phase. A
similar process happens in fig.(4). Note that we are did not consider the
death phase here.

The Lyapunov exponent hence can be written as:

$\displaystyle\lambda_{L}=\frac{\text{total number of
generations}}{\text{scrambling
time}}=\frac{k_{total}}{t_{*}}=\frac{k_{total}}{\alpha\beta
log_{2}(N_{max})}\,,$ (3.3)

The genetics of the bacteria may change in each step and produce different
kinds of bacteria through changing the genetic codes. This is not a stable
phenomenon and can reproduce early bacterias again in the next generations
(fig. (7)) which is also applied for the cells. This exponential growth was
one feature of the systems that were described by random matrix theory similar
to what is discussed in section (2), however, in the next section, we will
give another perspective that shows a genetic deformation can occur randomly
at each generation step through DNA reconstruction in the decomposition
process.

##  4 Random Matrix Theory and Evolution

Random matrix theory is a strong mathematical tool to study the statistical
behavior of chaotic systems such as the heavy nucleus, black holes,
hurricanes, or here a system like DNA. It has been beautifully developed for
physics applications by Wigner and Dyson and then by many other people like
Gaudin and Mehta. For useful reviews on this context see [46, 47, 48, 49].

A single DNA is constructed of two sugar phosphate backbones and four
nucleotides namely Adenine (A), Thymine (T), Cytosine (C), and Guanine (G)
(fig. (9) and (10)). If each bond between A, T, C, and G codes on a DNA
represent a force with a random value on a specific distribution with zero
mean and determined variance $\mathcal{V}$, the total DNA would make a huge
random matrix of the size of the DNA in terms of the A, T, C and G bonds which
can be changed randomly in each generation step. The force can be determined
by the potential energy ($F=-\nabla U$) therefore each covalent bond carries a
specific amount of energy and hence one can write the random matrix on an
energy basis and find the energy eigenvalues of each cell’s DNA by
diagonalization of the matrix. The codes have correlations among each other
and that is in common with the feature of the correlation between eigenvalues
in the random matrix theory.

Figure 9: Chemical structure of the fundamental nucleotides.

In the decomposition of the cell, the DNA will separate into two branches to
construct two other cells from the same gene. In the whole process, the matrix
elements will change randomly, each bond in a specific variance, and that will
lead to different matrices for both produced cells. The random matrix would
choose one of the $m$ possible states (see fig. (2)) to start the reproduction
process randomly. Each state has its probability weight and the one which has
greater probability is more likely to be naturaly selected.
If each state grows with $k_{i}$ generation where $i=1,...,m$ and each produce
$N_{max}^{(i)}=2^{k_{i}}$ particles at the scrambling time $t_{*}^{(i)}$, than
the probabilty of having $n$ genetic evolution is equal to:

$\displaystyle
P_{n}=\frac{n}{N_{max}^{(1)}+N_{max}^{(2)}+...+N_{max}^{(m)}}=\frac{n}{\sum_{i=1}^{m}2^{k_{i}}}$
(4.1)

For example if we consider all states have the same number of generations
$k_{i}=k=\frac{t_{*}}{\alpha\beta}$ therefore for n=1:

$\displaystyle P_{1}=\frac{2^{-\gamma T}}{m}$ (4.2)

where we denoted $\gamma=\frac{t_{*}}{\alpha}$ and replaced
$\beta=\frac{1}{T}$. In $T\rightarrow\infty$ limit the probability is simply
zero since in high temperatures the cell will no longer can still alive to be
prepared for evolution. In the $T\rightarrow 0$ limit, the cell can not be
evaluated to multiple partitions and it will be stuck on the first stage hence
the probability only depends on the number of states $m$, i.e.
$P=\frac{1}{m}$.

Figure 10: The structure of DNA (right) and the nucleotide bindings (left).

In DNA, the energy can differ from one bond to another. For example, an AT
bond on one part of the DNA can have slightly different energy than the AT
bond on the other part. That is possible since this energy depends on the
location of the nucleotides and the side effects of the potentials of other
objects. By seeing this bond energy as a random variable, the DNA can have
statistical distribution in terms of energy eigenvalues, and hence each DNA
can be seen as a $2\times L$ random matrix M (also see [50]):

$\displaystyle M=\begin{pmatrix}E_{AT}^{(1)}&E_{GC}^{(1)}\\\\[5.69046pt]
E_{AT}^{(2)}&E_{GC}^{(2)}\\\\[5.69046pt] \vdots&\vdots\\\\[5.69046pt]
E_{AT}^{(L)}&E_{GC}^{(L)}\\\ \end{pmatrix}$ (4.3)

Where $L$ is the length of the DNA, the size of the random matrix, and the
total number of bindings333Later we will also denote the number of degrees of
freedom by $L$ instead of $N$ to preserve the notation.. $E_{AT}$ and $E_{GC}$
are the Energy of A-T and G-C bindings respectively. For example, If the
length of DNA is $L=4$ as the case of fig. (10), this matrix would be:

$\displaystyle M=\begin{pmatrix}E_{AT}^{(1)}&0\\\\[5.69046pt]
0&E_{GC}^{(2)}\\\\[5.69046pt] 0&E_{GC}^{(3)}\\\\[5.69046pt] E_{AT}^{(4)}&0\\\
\end{pmatrix}$ (4.4)

One chromosome is built out of two random matrix DNAs and can be defined as:

$\displaystyle C\equiv
M.\widetilde{M}=\begin{pmatrix}E_{AT}^{(1)}&E_{GC}^{(1)}\\\\[5.69046pt]
E_{AT}^{(2)}&E_{GC}^{(2)}\\\\[5.69046pt] \vdots&\vdots\\\\[5.69046pt]
E_{AT}^{(L)}&E_{GC}^{(L)}\\\
\end{pmatrix}\begin{pmatrix}\widetilde{E}_{AT}^{(1)}&\widetilde{E}_{AT}^{(2)}&\cdots&\widetilde{E}_{AT}^{(L)}\\\\[5.69046pt]
\widetilde{E}_{GC}^{(1)}&\widetilde{E}_{GC}^{(2)}&\cdots&\widetilde{E}_{GC}^{(L)}\\\
\end{pmatrix}=\sum_{i=1}^{L}\begin{pmatrix}E_{AT}^{(i)}.\widetilde{E}_{AT}^{(i)}&E_{GC}^{(i)}.\widetilde{E}_{AT}^{(i)}\\\\[5.69046pt]
E_{AT}^{(i)}.\widetilde{E}_{GC}^{(i)}&E_{GC}^{(i)}.\widetilde{E}_{GC}^{(i)}\\\
\end{pmatrix}$ (4.5)

This is a $2\times 2$ random matrix and its elements change slightly in each
generation step. The off-diagonal elements can be seen as the interaction
between A-T and G-C bonds in two DNAs. At the moment of decomposition process,
one specific bond should not necessarily have the exact value of previous
energy 444To see this, one can also write the density of energy eigenvalues
$\rho(E)$ and $\rho(\widetilde{E})$ in a Gaussian orthogonal ensemble for two
DNA random matrices and calculate the pair correlation function
$\rho_{c}(E,\widetilde{E})=\langle\rho(E)\rho(\widetilde{E})\rangle$ and take
the limit $E-\widetilde{E}\rightarrow 0$.. We assume that the amount of energy
of each bond $E_{AT}$ and $E_{GC}$ can not exceed a minimum and a maximum
limit. If so the binding would lead to the destruction of one part of the DNA.
The main question of this paper is when the genetic evolution occurs in the
decomposition process in this theory? Before to start answering this question,
let us see when evolution happens from a biological point of view. When the
cell starts to decouple two individual cells, most of the time it will
preserve the original gene. The story becomes interesting when one nucleotide
(like A) exchanges with another nucleotide (like C) and causes a crisis that
changes the features of the living cell. This is what is referred to as
genetic evolution. Now the change $A\rightarrow C$ or $T\rightarrow G$ can
generate a drastic gap in the energy spectrum of the chromosome. This means
one eigenvalue has changed the statistics of the chosen random matrix theory
which can slightly deviate the shape of the density of eigenvalues
distribution function.
This evolution can be related to the phase transition concept in random matrix
theory since it also describes a great variation on a random system through
the distribution near the maximum eigenvalue. This has been widely studied and
we are not going to go through the complete details in this paper. This phase
transition should depend on the external environmental effects near the cell’s
DNA.
Robert May has argued in [51] that a random complex interacting system will
remain stable555The stability here means the perseverance of the species
against external conditions. The system is unstable when the species goes
toward extinction or the system is prepared to preserve its heredity by
evolution. When the evolution occurs by natural selection, the system moves
from an unstable to a more stable state. if the largest eigenvalue of the
distribution is less than an inverse of the coupling constant of the system
$g$:

$\displaystyle\lambda_{max}<(g)^{-1}$ (4.6)

Hence there is a critical value for $g$ in which on that point the system
would experience a phase transition. The probability distribution and its
derivative are depicted in fig. (11) and (12) where the solid lines are for a
low finite $L$ and the dashed line for large $L$. We are interested in a large
finite $L$ case where the phase transition can occur on a gap of order
$L^{-2/3}$. The probability distribution of the hill in the gap region is
purely Gaussian however the tails connected to that hill from the left and
right sides are not quite Gaussian and they are given by certain functions
that satisfy the Painlevé $II$ equations. These functions of the tails and the
probability density of the gap near $g_{c}$ have been well-studied by Craig
Tracy and Harold Widom [52, 53, 54]. This development is called Tracy-Widom
distribution and has plenty of useful practical applications.

Figure 11: The probability of having $\lambda_{max}<(g)^{-1}$ between $L$
eigenvalues in terms of $g^{-1}$. The three solid lines are for small $L$ and
the dashed line is for large but finite $L$. There is a critical value for $g$
where the continuous phase transition can occur from the unstable (red line)
to the stable (green line) phase. For large $L$, there is a small central
region near $g_{c}^{-1}$ with a width of order $L^{-2/3}$ where the Tracy-
Widom distribution applies.

A similar phase transition occurs in two-dimensional large-$L$ lattice gauge
theories666For a useful review of large-L gauge theories see [57]. in critical
coupling constant which is known as Gross-Witten-Wadia (GWW) phase transition
[55, 56]. In this case, the stable phase is the strong coupling phase and the
unstable phase is the weak coupling phase of Yang-Mills gauge theory and the
Tracy-Widom is the crossover function between these two areas in the double
scaling limit (for finite but large $L$).

Figure 12: The derivative of $P(g^{-1},L)$ or the probability density function
gives us the form of a delta function with a narrow Gaussian gap around the
critical value. The left and right tails connected to the Gaussian Tracy-Widom
distribution are described by two different functions.

Just like many natural phenomena, if we assume the distribution of the energy
bindings in a DNA has a Gaussian distribution, then the density of energy
eigenvalues for large $L$ can be given by the famous Wigner semi-circle law
(fig. (13)):

$\displaystyle\rho(\lambda)=\frac{1}{\pi}\sqrt{2-\lambda^{2}}$ (4.7)

The Tracy-Widom distribution apply near the edge of this semi-circle .i.e.
$\sqrt{2}$. In random matrix theory, there is a Colomb force between
eigenvalues in each sample. This force can differ by changing the coupling
constant of the theory and that will cause a slight deviation to the effective
potential where the eigenvalues condensated and therefore $g^{-1}$ can be seen
as a potential well where we can move it on the real axis line of the energy.
When the potential well is located out of the semi-circle (fig. (a) of (14)),
no phase transition will appear. Just as the potential hit the edge of the
semi-circle, it can affect the edge and change the Wigner distribution to the
Tracy-Widom distribution close to that region (fig. (b) of (14)):

$\displaystyle\rho^{*}_{s}(\lambda)=\frac{\sqrt{\lambda+L(s)}}{2\pi\sqrt{s-\lambda}}(s+L(s)-2\lambda),~{}~{}~{}~{}L(s)=\frac{2\sqrt{s^{2}+6}-s}{3},~{}~{}\text{where}~{}s=g^{-1}$
(4.8)

that causes a 3rd order phase transition from stable to an unstable regime or
vice versa.

Figure 13: Wigner semi-circle law for density of eigenvalues (blue curve). The
red curve shows the Tracy-Widom distribution near the edge which is a
perturbation around the critical value. Figure 14: (a) Far away from the edge,
the potential well does not affect the distribution and there would be no
phase transition. (b) As the coupling changes, the potential will hit the
semi-circle and cause a phase transition to a different distribution.

Now we turn to our former case which was the DNA random matrix. When the cell
decouples in several steps, the structure of DNA may encounter some disorders
from outlandish unwilling effects that put the life of the cell in danger of
extinction. Therefore, the nucleotides will encounter such instabilities and
decide to leave a stronger structure on the next steps of decouplings. Hence
one of the nucleotides, say A, exchanges with other nucleotides which provide
more powerful binding, and case the DNA will turn into the stable regime. The
coupling constant does not always change in that system just if the system
undergoes certain circumstances and tries to preserve its existence. Since
there is an obvious interaction between nucleotides, this change transforms
the coupling constant to a new value which just at the transition phase it
reaches its critical value.
Statistically, if the number of energy bindings of A-T and G-C is equal (but
assuming $E_{GC}>E_{AT}$), changing $A\rightarrow C$ will change the
distribution from the equilibrium point. Therefore, the interaction between
A-T and G-C will turn into a new regime with a less effective coupling
constant between molecules. That means the larger the population of G-C can
get, the more easer and more accessible it can interact with neighboring
eigenvalues. Therefore, when the coupling constant gets smaller, $g^{-1}$ gets
bigger and the potential well will move out of the main Gaussian distribution.
One can expect that the energy cost of the potential well to get out of the
semi-circle density is equal to the energy of exchanging A with C. This is
similar to the Gross-Witten-Wadia phase transition from strong coupling to
weak coupling regime which can turn the DNA into a more stable regime.
Since this phase transition occurs randomly in the decoupling process, it is
somehow equivalent to the natural selection:

$\displaystyle\text{Natural Selection}\equiv\text{GWW Phase Transition}$ (4.9)

This phase transition can change the distribution for a short time and then
keep the primary form of the distribution invariant. GWW phase transition
leads to a change in specific energy eigenvalue which must be preserved in the
whole process. The change in one energy eigenvalue (which has to do with the
side effects such as temperature, pressure, high energy rays, etc) directly
changes the other eigenvalues due to the repulsion force between every two
eigenvalues.
In the next section, we will give a summary of what has been presented in this
paper and mention key remarks that can be discussed in this context.

##  5 Conclusion and Remarks

We have given a general look at the time evolution of the random matrix spin
operator and saw that this evolution is similar to the binary bacterial or
cellular growth where each cell represents a Gaussian random matrix with zero
mean and specific variance value. This is one state that has been selected
from many other possible states. Each cell has many states to choose and
evaluate, so there are an infinite number of such states, each with its
probability weight. Environmental parameters such as temperature can play role
in the probability value. We can conclude that the behavior of the cell is a
random chaotic phenomenon that can be described in classical and quantum
mechanical viewpoints by random matrix theory. Through the idea that has been
presented here, one can probe the exact statistical parameters of the cell for
a long period and that is possible thanks to Tracy-Widom distribution. The
change in DNA bindings can directly change the overall coupling constant which
leads to a GWW phase transition from unstable to the stable regime and vice
versa.
One of the key points argued in this paper is that in the double scaling
limit, the matrix theory diagrams can characterize a Riemann surface for DNA
and one can find out what is the interpretation of phase transition in the
algebraic topology. It is of significant importance to calculate the free
energy of DNA in large $L$ so that it can be compared with the free energy
obtained for strings scattering amplitude corresponding to planar and non-
planar diagrams [58].
The GWW phase transition has been observed in the lab by laser experiments and
that can open a new window to observe and keep track of the variation of DNA’s
structure.
Studying the ecological treasures and connecting the effects and mechanisms
inside the cell to the mathematical developments of the random matrix theory,
can take a great step in tracking the number of lost circles in nature and
drawing a historical map of different evolutions. By analyzing the spectrum
and calculating the energy change in GWW phase transition one expects to see
in what exact statistical circumstances the cells are more likely to survive.
Although the random matrix theory has been used in studying many biological
applications, it can also guide us to answer fundamental questions with the
help of observations.

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