# Large deviation principle for the norm of the Laplacian
matrix of inhomogeneous Erdős-Rényi random graphs

Rajat Subhra Hazra  ,  Frank den Hollander  and  Maarten Markering
Mathematisch instituut, Universiteit Leiden, The Netherlands
<EMAIL_ADDRESS><EMAIL_ADDRESS>DPMMS, University of
Cambridge, United Kingdom<EMAIL_ADDRESS>
###### Abstract.

We consider an inhomogeneous Erdős-Rényi random graph $G_{N}$ with vertex set
$[N]=\\{1,\dots,N\\}$ for which the pair of vertices $i,j\in[N]$, $i\neq j$,
is connected by an edge with probability $r_{N}(\tfrac{i}{N},\tfrac{j}{N})$,
independently of other pairs of vertices. Here,
$r_{N}\colon\,[0,1]^{2}\to(0,1)$ is a symmetric function that plays the role
of a reference graphon. Let $\lambda_{N}$ be the maximal eigenvalue of the
Laplacian matrix of $G_{N}$. We show that if
$\lim_{N\to\infty}\|r_{N}-r\|_{\infty}=0$ for some limiting graphon
$r\colon\,[0,1]^{2}\to(0,1)$, then $\lambda_{N}/N$ satisfies a downward LDP
with rate $\binom{N}{2}$ and an upward LDP with rate $N$. We identify the
associated rate functions $\psi_{r}$ and $\widehat{\psi}_{r}$, and derive
their basic properties.

###### Key words and phrases:

Inhomogeneous Erdős-Rényi random graph, Laplacian matrix, largest eigenvalue,
graphons, large deviation principle, rate function.

###### 2000 Mathematics Subject Classification:

05C80, 60B20, 60C05, 60F10

The research in this paper was supported through NWO Gravitation Grant
NETWORKS 024.002.003.

##  1\. Introduction and main results

Section 1.1 provides background. Section 1.2 states the LDP for the empirical
graphon associated with inhomogeneous Erdős-Rényi random graphs. Section 1.3
looks at graphon operators, in particular, the Laplacian operator that is the
central object in the present paper. Sections 1.4–1.5 state the downward,
respectively, upward LDP for the largest eigenvalue of the Laplacian matrix
and present basic properties of the associated rate functions. Section 1.6
places the various theorems in their proper context.

###  1.1. Background

Spectra of matrices associated with a graph play a crucial role in
understanding the geometry of the graph. Given a finite graph on $N$ vertices,
two important matrices are the _adjacency matrix_ $A_{N}$ and the _Laplacian
matrix_ $L_{N}=D_{N}-A_{N}$, where $D_{N}$ is the diagonal matrix whose
elements are the degrees of the vertices. In this paper we focus on the
largest eigenvalue of $L_{N}$ when the underlying graph is a _dense
inhomogeneous_ Erdős-Rényi random graph. The largest eigenvalue of $A_{N}$
satisfies a large deviation principle (LDP). This fact is an immediate
consequence of the LDP for the empirical graphon derived in [16] and [25] in
combination with the contraction principle, because the norm of the adjacency
graphon operator is bounded and continuous on the space of graphons. The rate
function is given in terms of a variational formula involving the rate
function of the LDP for the empirical graphon. In [11] we analysed this
variational formula, identified the basic properties of the rate function, and
identified its scaling behaviour near its unique minimiser and its two
boundary points.

The extension of the LDP to $L_{N}$ poses new challenges, because $L_{N}$ is a
more delicate object than $A_{N}$. For one, the upward and the downward large
deviations for the largest eigenvalue of $L_{N}$ live on _different scales_ ,
and the norm of the Laplacian graphon operator lacks certain continuities
properties that hold for the norm of the adjacency graphon operator. Like for
$A_{N}$, it is not possible to explicitly solve the variational formulas for
the two associated rate functions. Nonetheless, we derive their basic
properties and identify their scaling behaviour near their unique minimisers.
Our proofs require an analysis of the functional analytic properties of the
Laplacian operator, in combination with variational tools.

The literature on the eigenvalues of the Laplacian of a random matrix is
limited. For a general Wigner matrix with independent centered entries
satisfying certain moment conditions, the empirical spectral distribution was
derived in [8] and was identified as the free additive convolution of the
semicircle law and the Gaussian distribution. For entries with a common mean
$m$ it was shown that the empirical spectral distribution converges to the
Dirac measure at $m$. Moreover, under a fourth moment condition it was shown
that the spectral norm is of order $\sqrt{2N\log N}$ when the entries are
centered, and of order $N$ when the entries are not centered. Although the
moment conditions were rather general, [8] did not cover all relevant cases of
Erdős-Rényi random graphs. This was resolved in [20], [18] for the case where
the average degree diverges with $N$. The results were extended to
inhomogeneous Erdős-Rényi graphs in [10], [12]. The empirical distribution in
the sparse homogeneous Erdős-Rényi graph was identified in [21]. Later, an
alternative proof through local weak convergence was provided in [6]. The
results on extreme eigenvalues are limited and challenging. Recently, it was
shown in [9] that for the Gaussian orthogonal ensemble the largest eigenvalue
of the Laplacian, after appropriate scaling and centering, converges to the
Gumbel distribution.

Large deviations for Erdős-Rényi random graphs were explored extensively in
various works, including [13], [14], [24] by utilising graphon theory,
specifically subgraph densities and maximal eigenvalues. For a comprehensive
review of the literature, we refer to [13]. Large deviation theory for random
matrices originated in [3], where the focus was on the empirical spectral
distribution of $\beta$-ensembles with a quadratic potential. The rate was
found to be $N^{2}$, and the rate function was identified by using a non-
commutative notion of entropy. Subsequently, the maximal eigenvalue of such
ensembles was investigated in [4]. In [5], large deviations of the empirical
spectral distribution were derived for random matrices with non-Gaussian
tails, while the study of the maximal eigenvalue in that context was carried
out in [1], [2]. However, the adjacency matrix and the Laplacian matrix of an
inhomogeneous Erdős-Rényi random graph fall outside these regimes. The goal of
the present paper is to understand the _large deviations of the largest
eigenvalue_ of an _inhomogeneous Erdős-Rényi graph_ under certain density
assumptions.

###  1.2. LDP for inhomogeneous Erdős-Rényi random graphs

####  1.2.1. Graphons

Let

(1.1)
$\mathcal{W}=\bigl{\\{}h\colon\,[0,1]^{2}\to[0,1]\colon\,h(x,y)=h(y,x)\,\,\forall\,x,y\in[0,1]\bigr{\\}}$

denote the set of graphons. Let $\mathcal{M}$ denote the set of Lebesgue
measure-preserving bijective maps $\phi\colon\,[0,1]\mapsto[0,1]$. For two
graphons $h_{1},h_{2}\in\mathcal{W}$, the _cut-distance_ is defined by

(1.2) $d_{\square}(h_{1},h_{2})=\sup_{S,T\subset[0,1]}\bigg{|}\int_{S\times
T}\mathrm{d}x\,\mathrm{d}y\,\big{[}h_{1}(x,y)-h_{2}(x,y)\big{]}\bigg{|},$

and the _cut-metric_ by

(1.3)
$\delta_{\square}(h_{1},h_{2})=\inf_{\phi\in\mathcal{M}}d_{\square}(h_{1},h_{2}^{\phi}),$

where $h_{2}^{\phi}(x,y)=h_{2}(\phi(x),\phi(y))$. The cut-metric defines an
equivalence relation $\sim$ on $\mathcal{W}$ by declaring $h_{1}\sim h_{2}$ if
and only if $\delta_{\square}(h_{1},h_{2})=0$, and leads to the quotient space
$\widetilde{\mathcal{W}}=\mathcal{W}/_{\sim}$. For $h\in\mathcal{W}$, we write
$\widetilde{h}$ to denote the equivalence class of $h$ in
$\widetilde{\mathcal{W}}$. The equivalence classes correspond to relabelings
of the vertices of the graph. The pair
$(\widetilde{\mathcal{W}},\delta_{\square})$ is a compact metric space [23].

####  1.2.2. LDP for empirical graphon

Consider a sequence of symmetric functions $r_{N}\colon[0,1]^{2}\to(0,1)$,
$N\in\mathbb{N}$, that are constant on the blocks
$[\frac{i-1}{N},\frac{i}{N})\times[\frac{j-1}{N},\frac{j}{N})$, $1\leq i,j\leq
N$, such that

(1.4) $\lim_{N\to\infty}\|r_{N}-r\|_{\infty}=0$

for some symmetric function $r\colon[0,1]^{2}\to(0,1)$ that plays the role of
a _reference graphon_. Let $G_{N}$ be the inhomogeneous Erdős-Rényi random
graph with vertex set $[N]=\\{1,\dots,N\\}$ for which the pair of vertices
$i,j\in[N]$, $i\neq j$, is connected by an edge with probability
$(r_{N})_{ij}$, independently of other pairs of vertices, where $(r_{N})_{ij}$
is the value of $r_{N}$ on the block
$[\frac{i-1}{N},\frac{i}{N})\times[\frac{j-1}{N},\frac{j}{N})$. The function
$r_{N}$ plays the role of a _block reference graphon_ converging to some
_reference graphon_ $r$ as $N\to\infty$. We assume that the reference graphon
$r\in\mathcal{W}$ in (1.4) satisfies

(1.5) $\log r,\log(1-r)\in L^{1}([0,1]^{2}),$

and that the analogue of (1.5) holds for $\log r_{N}$ and $\log(1-r_{N})$ as
well.

Let $A_{N}$ be the _adjacency matrix_ of $G_{N}$ defined by

(1.6) $A_{N}(i,j)=\left\\{\begin{array}[]{ll}1,&\text{if there is an edge
between vertex $i$ and vertex $j$},\\\ 0,&\text{otherwise}.\end{array}\right.$

Let $D_{N}$ be the diagonal degree matrix defined by
$D_{N}(i,i)=\sum_{j\in[N]\setminus i}A_{N}(i,j)$ and $D_{N}(i,j)=0$ for $i\neq
j$, and put $L_{N}=D_{N}-A_{N}$, which is the _Laplacian matrix_. Write
$\mathbb{P}_{N}$ to denote the law of $G_{N}$. Use the same symbol for the law
on $\mathcal{W}$ induced by the map that associates with the graph $G_{N}$ its
empirical graphon $h^{G_{N}}$, defined by

(1.7) $h^{G_{N}}(x,y)=\left\\{\begin{array}[]{ll}1,&\text{if there is an edge
between vertex $\lceil Nx\rceil$ and vertex $\lceil Ny\rceil$},\\\
0,&\text{otherwise}.\end{array}\right.$

Write $\widetilde{\mathbb{P}}_{N}$ to denote the law of
$\widetilde{h}^{G_{N}}$.

The following LDP, which is an extension of the celebrated LDP for homogeneous
ERRG derived in [14], is proved in [16] and [25]. (For more background on
large deviation theory, see, for instance, [15].)

######  Theorem 1.1.

[LDP for inhomogeneous ERRG, [16], [25]] Subject to (1.5), the sequence
$(\widetilde{\mathbb{P}}_{N})_{N\in\mathbb{N}}$ satisfies the large deviation
principle on $(\widetilde{\mathcal{W}},\delta_{\square})$ with rate
$\binom{N}{2}$, i.e.,

(1.8)
$\displaystyle\limsup_{N\to\infty}\binom{N}{2}^{-1}\log\widetilde{\mathbb{P}}_{N}(\mathcal{C})$
$\displaystyle\leq-\inf_{\widetilde{h}\in\mathcal{C}}J_{r}(\widetilde{h})$
$\displaystyle\forall\,\mathcal{C}\subset\widetilde{\mathcal{W}}\text{
closed},$
$\displaystyle\liminf_{N\to\infty}\binom{N}{2}^{-1}\log\widetilde{\mathbb{P}}_{N}(\mathcal{O})$
$\displaystyle\geq-\inf_{\widetilde{h}\in\mathcal{O}}J_{r}(\widetilde{h})$
$\displaystyle\forall\,\mathcal{O}\subset\widetilde{\mathcal{W}}\text{ open},$

where the rate function $J_{r}\colon\,\widetilde{\mathcal{W}}\to\mathbb{R}$ is
given by

(1.9) $J_{r}(\widetilde{h})=\inf_{\phi\in\mathcal{M}}I_{r}(h^{\phi}),$

where $h$ is any representative of $\widetilde{h}$ and

(1.10)
$I_{r}(h)=\int_{[0,1]^{2}}\mathrm{d}x\,\mathrm{d}y\,\,\mathcal{R}\big{(}h(x,y)\mid
r(x,y)\big{)},\quad h\in\mathcal{W},$

with

(1.11) $\mathcal{R}\big{(}a\mid
b\big{)}=a\log\tfrac{a}{b}+(1-a)\log\tfrac{1-a}{1-b}$

the relative entropy of two Bernoulli distributions with success probabilities
$a\in[0,1]$, $b\in(0,1)$ (with the convention $0\log 0=0$).

######  Remark 1.2.

Theorem 1.1 was proved in [16] under the assumption that $r$ is bounded away
from $0$ and $1$. In [25] this assumption was relaxed to (1.5), and it was
also shown that $J_{r}$ is a good rate function, i.e., $J_{r}\not\equiv\infty$
and $J_{r}$ has compact level sets. Note that (1.9) differs from the
expression in [16], where the rate function is the lower semi-continuous
envelope of $I_{r}(h)$. However, as shown in [25], under (1.5) the two rate
functions are equivalent, since $J_{r}(\widetilde{h})$ is lower semi-
continuous on $\widetilde{\mathcal{W}}$. $\spadesuit$

As an application of Theorem 1.1, it was shown in [11] that the largest
eigenvalue of the adjacency matrix satisfies the LDP with rate ${N\choose 2}$.
The rate function was analysed in detail for reference graphons that are
rank-1. In the present paper we focus on the LDP for the largest eigenvalue of
the Laplacian matrix $L_{N}$.

###  1.3. Graphon operators

For $h\in\mathcal{W}$, the graphon operator $\mathcal{T}_{h}$ is the integral
operator on $L^{2}([0,1])$ defined by

(1.12) $(\mathcal{T}_{h}u)(x)=\int_{[0,1]}\mathrm{d}y\,h(x,y)u(y),\qquad
x\in[0,1].$

Note that $\mathcal{T}_{h}$ is a compact operator. Define the _degree
function_ as

(1.13) $d_{h}(x)=\int_{[0,1]}\mathrm{d}y\,h(x,y),\qquad x\in[0,1].$

The _degree operator_ $\mathcal{D}_{h}$ is the multiplication operator on
$L^{2}([0,1])$ defined by

(1.14) $(\mathcal{D}_{h}u)(x)=d_{h}(x)u(x),$

The _Laplacian operator_ $\mathcal{L}_{h}$ is the linear integral operator on
$L^{2}([0,1])$ defined by

(1.15)
$(\mathcal{L}_{h}u)(x)=\int_{[0,1]}\mathrm{d}y\,h(x,y)[u(x)-u(y)],\qquad
x\in[0,1].$

Note that

(1.16) $\mathcal{L}_{h}=\mathcal{D}_{h}-\mathcal{T}_{h}.$

Recall that, given an operator $S$ on a Hilbert space, the _spectrum_ of $S$
is defined as

(1.17) $\sigma(S)=\\{\lambda\in\mathbb{C}\colon\,S-\lambda I\text{ is not
invertible}\\}.$

Let $\sigma_{d}(S)$ denote the _discrete spectrum_ of $S$, which consists of
all the isolated eigenvalues with finite algebraic multiplicity. The
_essential spectrum_ of $S$ is denoted by

(1.18) $\sigma_{\mathrm{ess}}(S)=\sigma(S)\setminus\sigma_{d}(S).$

The essential spectrum is closed, and the discrete spectrum can only have
accumulation points on the boundary of the essential spectrum. Since it is
known that compact operators do not affect the essential spectrum, we have

(1.19)
$\sigma_{\mathrm{ess}}(\mathcal{L}_{h})=\sigma_{\mathrm{ess}}(\mathcal{D}_{h}).$

The operator $\mathcal{L}_{h}$ is not as well-behaved as $\mathcal{T}_{h}$
with the cut-norm. In fact, even when a sequence of graphons
$(h_{n})_{n\in\mathbb{N}}$ converges in cut-norm to a graphon $h$, the
eigenvalues and eigenvectors of $\mathcal{L}_{h_{n}}$ may not converge to
those of $\mathcal{L}_{h}$, as was already observed in [17], [27]. Assuming as
in [11] that the reference graphon is rank-1 does not help. In fact, if we
assume that $r$ is rank-1 and is continuous, then $0$ is the only eigenvalue
of $\mathcal{L}_{r}$ and
$\sigma_{\mathrm{ess}}(\mathcal{L}_{r})=d_{r}([0,1])$, as shown in [17,
Proposition 5.11].

If $h$ is the empirical graphon of a graph $G$ with $N$ vertices, then
$N\|\mathcal{T}_{h}\|$ equals the largest eigenvalue of the adjacency matrix
of $G$, and $N\|\mathcal{D}_{h}\|$ equals the maximum degree of $G$. In fact,
for any graphon $h$ the spectrum of $\mathcal{D}_{h}$ equals the range of
$d_{h}$ and the operator norm of $\mathcal{D}_{h}$ equals the supremum norm
$d_{h}$, i.e.,

(1.20) $\|\mathcal{D}_{h}\|=\|d_{h}\|_{\infty},$

where $\|d_{h}\|_{\infty}$ is the $L^{\infty}$-norm of the function $d_{h}$.
This follows from the fact that $\mathcal{D}_{h}$ is a multiplication
operator. Let

(1.21) $\|\mathcal{L}_{h}\|=\sup_{\begin{subarray}{c}u\in L^{2}([0,1])\\\
\|u\|_{2}=1\end{subarray}}\|\mathcal{L}_{h}u\|_{2}$

be the operator norm of $\mathcal{L}_{h}$, where $\|\cdot\|_{2}$ denotes the
$L^{2}$-norm. Since $\mathcal{L}_{h}$ is a normal operator with a non-negative
spectrum, $\|\mathcal{L}_{h}\|$ also equals the supremum of the spectrum of
$\mathcal{L}_{h}$.

######  Proposition 1.3.

[Properties of the Laplacian operator]
(i) Let $h$ be a graphon and $\mathcal{L}_{h}$ be the Laplacian operator on
$L^{2}([0,1])$. Then $\mathcal{L}_{h}$ is a bounded operator, and
$h\mapsto\|\mathcal{L}_{h}\|$ is lower semi-continuous in the cut-metric.
(ii) Let $G$ be a graph with $N$ vertices and $h^{G}$ be the empirical graphon
associated with $G$ and let $L_{N}$ be the Laplacian matrix with spectral norm
$\|L_{N}\|$. Then it follows that

(1.22) $\frac{\|L_{N}\|}{N}=\|\mathcal{L}_{h^{G}}\|\qquad\forall\,N.$

######  Remark 1.4.

Note that $h\mapsto\|\mathcal{L}_{h}\|$ is not continuous in the cut-metric.
For example, consider the sequence of graphons $(h_{N})_{N\in\mathbb{N}}$ such
that $h_{N}$ is the empirical graphon of the $N$-star graph (i.e., $1$ vertex
connected by an edge to each of the $N-1$ other vertices, and no further
edges). Then $h_{N}\downarrow 0$ as $N\to\infty$ in the cut-metric, but
$\|\mathcal{L}_{h_{N}}\|=1$ for all $N\in\mathbb{N}$. $\spadesuit$

Proposition 1.3 is proven in Section 2.2.

###  1.4. Main theorems: downward large deviations

Let

(1.23) $\lambda_{N}=\|L_{N}\|=\sup_{\begin{subarray}{c}u\in L^{2}([0,1])\\\
\|u\|_{2}=1\end{subarray}}\|L_{N}u\|_{2}.$

be the maximal eigenvalue of $L_{N}$, where $\|\cdot\|_{2}$ denotes the
$L^{2}$-norm. Abbreviate

(1.24) $C_{r}=\|\mathcal{L}_{r}\|.$

Our goal is to show that $\lambda_{N}/N$ satisfies a downward LDP as
$N\to\infty$, with rate $\binom{N}{2}$ and with a rate function that can be
analysed in detail. We write $\mathbb{P}^{*}_{N}$ to denote the law of
$\lambda_{N}$.

######  Theorem 1.5.

[Downward LDP] Subject to (1.5),

(1.25)
$\lim_{N\to\infty}\binom{N}{2}^{-1}\log\mathbb{P}^{*}_{N}(\lambda_{N}/N\leq\beta)=-\psi_{r}(\beta),\qquad\beta\in[0,C_{r}],$

with

(1.26)
$\psi_{r}(\beta)=\inf_{\begin{subarray}{c}\widetilde{h}\in\widetilde{\mathcal{W}}\\\
\|\mathcal{L}_{\widetilde{h}}\|\leq\beta\end{subarray}}J_{r}(\widetilde{h})=\inf_{\begin{subarray}{c}h\in\mathcal{W}\\\
\|\mathcal{L}_{h}\|\leq\beta\end{subarray}}I_{r}(h).$

The second equality in (1.26) uses that
$\|\mathcal{L}_{r}\|=\|\mathcal{L}_{r^{\phi}}\|$ for any $\phi\in\mathcal{M}$,
as is evident after replacing $u$ by $u^{\phi^{-1}}$ in (1.21) given by
$u^{\phi^{-1}}(x)=u(\phi^{-1}(x))$. Since the maximal eigenvalue is invariant
under relabelling of the vertices, we need not worry about the equivalence
classes.

$C_{r}$$1$$0$$C^{0}_{r}$$\beta$$\psi_{r}(\beta)$ Figure 1. Graph of
$\beta\mapsto\psi_{r}(\beta)$.

Let

(1.27) $C_{r}^{0}=\int_{[0,1]^{2}}\log\tfrac{1}{1-r}.$

When $\beta=C_{r}$, the optimal graphon is the reference graphon $r$ almost
everywhere, for which $I_{r}(r)=0$, and no large deviation occurs. When
$\beta=0$, the optimal graphon is the zero graphon $\underline{0}\equiv 0$,
for which $I_{r}(\underline{0})=C_{r}^{0}$ (see Fig. 1).

######  Theorem 1.6.

[Properties of the rate function] Subject to (1.5):
(i) $\psi_{r}$ is continuous and strictly decreasing on $[0,C_{r}]$, with
$\psi_{r}(0)=C^{0}_{r}>0$ and $\psi_{r}(C_{r})=0$.
(ii) For every $\beta\in[0,C_{r}]$, the set of minimisers of the variational
formula for $\psi_{r}(\beta)$ in (1.26) is non-empty and compact in
$\widetilde{\mathcal{W}}$.

Under the _additional assumptions_ that

(1.28) $\displaystyle\lambda_{\max}(\mathcal{L}_{r})<\|d_{r}\|_{\infty},$
(1.29) $\displaystyle\text{$r$ is bounded away from 0 and $1$},$ (1.30)
$\displaystyle\text{$r$ is continuous},$

we are able to compute the behaviour of $\psi_{r}$ around $C_{r}$. Here
$\lambda_{\max}(\mathcal{L}_{r})$ denotes the largest eigenvalue of
$\mathcal{L}_{r}$. Note that (1.29) is stronger than (1.5). Recall that
$d_{r}(x)=\int_{[0,1]}\mathrm{d}y\,r(x,y)$, $x\in[0,1]$. It is easy to check
that, subject to (1.28),

(1.31) $C_{r}=\|d_{r}\|_{\infty}.$

######  Theorem 1.7.

[Scaling of the rate function] Subject to (1.28)–(1.29),

(1.32)
$\psi_{r}(\beta)\asymp\int_{S_{r}(\beta)}\mathrm{d}x\,\frac{1}{v_{r}(x)}\,(d_{r}(x)-\beta)^{2},\qquad\beta\uparrow
C_{r},$

where

(1.33) $S_{r}(\beta)=\\{x\in[0,1]\colon\,d_{r}(x)\geq\beta\\}$

and

(1.34) $v_{r}(x)=\int_{[0,1]}\mathrm{d}y\,r(x,y)[1-r(x,y)].$

The proofs of Theorems 1.5–1.7 are given in Sections 3.1–3.3.

###  1.5. Main theorems: upward large deviations

Put

(1.35)
$J_{r}(x,\beta)=\int_{[0,1]}\mathrm{d}y\,\mathcal{R}\big{(}\widehat{r}_{\beta}(x,y)\mid
r(x,y)\big{)},\qquad x\in[0,1],$

where

(1.36)
$\widehat{r}_{\beta}(x,y)=\frac{\mathrm{e}^{\theta(x,\beta)}r(x,y)}{\mathrm{e}^{\theta(x,\beta)}r(x,y)+[1-r(x,y)]}$

is a _Cramér-type transform_ of the reference graphon $r$, with a Langrange
multiplier function $\theta(x,\beta)$, $x\in[0,1]$, chosen such that
$\int_{[0,1]}\mathrm{d}y\,\widehat{r}_{\beta}(x,y)=\beta$, $x\in[0,1]$. Note
that $\widehat{r}_{\beta}$ does not need to be symmetric and therefore is not
necessarily a graphon. Under the _additional assumption_ that

(1.37) $r$ is non-negative definite (i.e., $\mathcal{T}_{r}$ is a non-negative
definite operator)

we are able to derive an upward LDP and identify the associated rate function
in term of $J_{r}$.

######  Theorem 1.8.

[Upward LDP] Subject to (1.29) and (1.37),

(1.38)
$\lim_{N\to\infty}N^{-1}\log\mathbb{P}^{*}_{N}(\lambda_{N}/N\geq\beta)=-\widehat{\psi}_{r}(\beta),\qquad\beta\in[C_{r},1],$

with

(1.39) $\widehat{\psi}_{r}(\beta)=\inf_{x\in[0,1]}J_{r}(x,\beta).$

$C_{r}$$1$$0$$C^{1}_{r}$$\beta$$\widehat{\psi}_{r}(\beta)$ Figure 2. Graph of
$\beta\mapsto\widehat{\psi}_{r}(\beta)$.

Define

(1.40)
$C_{r}^{1}=\widehat{\psi}_{r}(1)=\inf_{x\in[0,1]}\int_{[0,1]}\mathrm{d}y\,\log\frac{1}{r(x,y)}.$

When $\beta=C_{r}$, the Lagrange multiplier in (1.36) is
$\theta(x,C_{r})\equiv 0$, for which $J_{r}(x,C_{r})\equiv 0$, and no large
deviation occurs. When $\beta=1$, the Lagrange multiplier is
$\theta(x,1)\equiv\infty$, for which $\widehat{\psi}_{r}(1)=C_{r}^{1}$ (see
Fig. 2).

######  Theorem 1.9.

[Properties of the rate function] Subject to (1.29), $\widehat{\psi}_{r}$ is
continuous and strictly increasing on $[C_{r},1]$, with
$\widehat{\psi}_{r}(C_{r})=0$ and $\widehat{\psi}_{r}(1)=C_{r}^{1}>0$.

######  Theorem 1.10.

[Scaling of the rate function] Subject to (1.29)–(1.30) and (1.37),

(1.41) $\widehat{\psi}_{r}(\beta)\sim\widehat{K}_{r}(\beta-
C_{r})^{2},\qquad\beta\downarrow C_{r},$

with

(1.42)
$\widehat{K}_{r}=\frac{1}{2}\inf_{x\in\mathcal{D}_{r}}\frac{1}{v_{r}(x)},$

where $\mathcal{D}_{r}=\\{x\in[0,1]\colon\,d_{r}(x)=\|d_{r}\|_{\infty}\\}$.

The proofs of Theorems 1.8–1.10 are given in Sections 4.1–4.3.

###  1.6. Discussion

1. Theorems 1.5–1.7 establish the downward LDP and identify basic properties of the rate function $\psi_{r}$. Note that $\lim_{\beta\uparrow C_{r}}S_{r}(\beta)=\mathcal{D}_{r}$. Hence, if $|\mathcal{D}_{r}|=0$, then the scaling of $\psi_{r}$ near $C_{r}$ is _faster than quadratic_ , which suggest that $\widehat{\lambda}_{N}/N$ does _not_ satisfy a standard central limit theorem. On the other hand, if $|\mathcal{D}_{r}|>0$, then the scaling is quadratic and a standard central limit is expected to hold. Both questions are open. Several scenarios for the precise scaling are possible depending on how $d_{r}$ scales near its maxima.

2. Assumption (1.5) is basic because it underlies Theorem 1.1, which is the jump board for our downward LDP in Theorem 1.5 and the general properties of the downward rate function in Theorem 1.6. Assumption (1.28) is needed for the upper bound in the scaling of the downward rate function in Theorem 1.7. It is the most severe assumption in the present paper, although it is still satisfied for a large class of graphons, for instance, those $r$ that are rank-1 and continuous (see [17, Proposition 5.11]). Assumption (1.28) guarantees that, for all graphons $h$ close enough to $r$ in $L^{2}$-norm, $\|\mathcal{L}_{h}\|=\|d_{h}\|_{\infty}$. This allows us to reduce the analysis of the rate function for the LDP of the Laplacian norm to the analysis of the rate function for the maximum degree, which is considerably easier. Assumption (1.29) implies that $v_{r}(x)\geq\delta d_{r}(x)$, $x\in[0,1]$ for some $\delta>0$ (via (1.34)), and ensures that the integral in (1.32) is well-defined.

3. Theorems 1.8–1.10 establish the upward LDP and identify basic properties of the rate function $\widehat{\psi}_{r}$. The decay of $\widehat{\psi}_{r}$ towards zero is _quadratic_ , which suggests that $\widehat{\lambda}_{N}/N$ satisfies a standard central limit theorem. The interpretation of (1.42) is that the curvature of $\widehat{\psi}_{r}$ at its unique zero $C_{r}$ is the _inverse_ of the variance of the associated central limit theorem, in line with standard folklore of large deviation theory (see [7]). Since $v_{r}(x)$ can be viewed as the variance of the empirical distribution of the degrees of the vertices with label $\approx xN$ and the large deviations are controlled by $x\in\mathcal{D}_{r}$, the relation in (1.42) is intuitively plausible.

4. Assumption (1.37) is very similar to Assumption (1.28), since it implies $\lambda_{\max}(\mathcal{L}_{r})\leq\|d_{r}\|_{\infty}$. Again, it is needed to make sure that for our upward LDP in Theorem 1.8 _the largest degree is dominant_. In fact, we will see that $J(x,\beta)$ in (1.35) is the upward rate function for the degrees of the vertices with label $\approx xN$. Assumption (1.29) is needed to ensure that the general properties of the upward rate function in Theorem 1.9 hold, Assumptions (1.29)–(1.30) are needed to get the sharp scaling of the upward rate function in Theorem 1.10 as stated in (1.41)–(1.42).

5. Whereas the scaling constant in Theorem 1.10 is sharp, it is not sharp in Theorem 1.7. The proof of Theorem 1.7 shows that it lies between $1$ and $2$.

6. The fine properties of $\psi_{r}$ and $\widehat{\psi}_{r}$ remain elusive. Neither convexity nor analyticity are obvious.

7. In the proofs we derive an LDP for the maximum degree of $G_{N}$ and analyse its rate function. The results we obtain for the maximum degree are analogous to Theorems 1.5–1.10 and are of independent interest. See Remark 3.4.

##  2\. Proof of proposition

Section 2.1 shows that the degree operator is lower semi-continuous. Section
2.2 provides the proof of Proposition 1.3.

###  2.1. Lower semi-continuity

######  Lemma 2.1.

The map $h\mapsto\|\mathcal{D}_{h}\|$ is lower semi-continuous in the cut-
metric.

###### Proof.

Let $(\widetilde{f}_{n})_{n\in\mathbb{N}}\subset\widetilde{\mathcal{W}}$ be a
sequence converging to some $\widetilde{f}\in\widetilde{\mathcal{W}}$ in the
cut-metric $\delta_{\square}$. Without loss of generality we may assume that
it converges in the cut-distance $d_{\square}$ as well, i.e.,
$\lim_{n\to\infty}d_{\square}(f_{n},f)=0$. Assume that

(2.1) $\liminf_{n\to\infty}\|\mathcal{D}_{f_{n}}\|<\|\mathcal{D}_{f}\|.$

Then there exists a $\delta>0$ (independent of $n$) such that
$\|\mathcal{D}_{f}\|>\|\mathcal{D}_{f_{n}}\|+\delta$ for infinitely many
$n\in\mathbb{N}$. By (1.20), there exists a set $A\subset[0,1]$ of positive
measure such that

(2.2)
$d_{f}(x)>\|\mathcal{D}_{f}\|-\tfrac{\delta}{2}>\|\mathcal{D}_{f_{n}}\|+\tfrac{\delta}{2}\geq
d_{f_{n}}(x)+\tfrac{\delta}{2}\qquad\forall\,x\in A.$

Hence, by (1.13),

(2.3)
$\int_{A\times[0,1]}\,\mathrm{d}x\,\mathrm{d}y\,f(x,y)>\int_{A\times[0,1]}\,\mathrm{d}x\,\mathrm{d}y\,f_{n}(x,y)+\tfrac{\delta}{2}\lambda(A)$

with $\lambda$ the Lebesgue measure. By the definition of the cut-distance in
(1.2),

(2.4)
$d_{\square}(f_{n},f)\geq\int_{A\times[0,1]}\,\mathrm{d}x\,\mathrm{d}y\,[f(x,y)-f_{n}(x,y)]>\tfrac{\delta}{2}\lambda(A).$

Since $\lambda(A)$ and $\delta$ are independent of $n$, this inequality is a
contradiction with the fact that $\lim_{n\to\infty}d_{\square}(f_{n},f)=0$.
Hence (2.1) is impossible. ∎

###  2.2. Proof of Proposition 1.3

###### Proof.

(i) The first part of the statement follows from the fact that
$\mathcal{L}_{h}$ is the sum of two bounded operators. For the second part of
the statement, we need that $h\mapsto\|\mathcal{T}_{h}\|$ is continuous and
that $h\mapsto\|\mathcal{D}_{h}\|$ is lower semi-continuous. The former was
shown shown in [24, Lemma 3.6], the latter was shown in Lemma 2.1.

Let $g\in L^{2}([0,1]^{2})$. Then, by (1.16),

(2.5)
$\begin{split}&\liminf_{n\to\infty}\big{[}\|\mathcal{L}_{f_{n}}(g)\|_{2}^{2}-\|\mathcal{L}_{f}(g)\|_{2}^{2}\big{]}\\\
&\geq\liminf_{n\to\infty}\big{[}\|\mathcal{D}_{f_{n}}(g)\|_{2}^{2}-\|\mathcal{D}_{f}(g)\|_{2}^{2}\big{]}+\liminf_{n\to\infty}\big{[}\|T_{f_{n}}(g)\|_{2}^{2}-\|T_{f}(g)\|_{2}^{2}\big{]}\\\
&\qquad-2\limsup_{n\to\infty}\big{[}\langle\mathcal{D}_{f_{n}}(g),T_{f_{n}}(g)\rangle-\langle\mathcal{D}_{f}(g),T_{f}(g)\rangle\big{]}\\\
&\geq-2\limsup_{n\to\infty}\big{[}\langle\mathcal{D}_{f_{n}}(g),T_{f_{n}}(g)\rangle-\langle\mathcal{D}_{f}(g),T_{f}(g)\rangle\big{]}.\end{split}$

It remains to show that the last expression equals 0. Since the simple
functions are dense in $L^{2}([0,1]^{2})$, we may assume without loss of
generality that $g$ is simple, i.e.,
$g=\sum_{i=1}^{k}\alpha_{i}\mathbf{1}_{A_{i}}$. Estimate

(2.6)
$\begin{split}&\big{|}\langle\mathcal{D}_{f_{n}}(g),T_{f_{n}}(g)\rangle-\langle\mathcal{D}_{f}(g),T_{f}(g)\rangle\big{|}\\\
&=\left|\int_{[0,1]^{3}}\,\mathrm{d}y\,\mathrm{d}y^{\prime}\,\mathrm{d}x\,f_{n}(x,y)f_{n}(x,y^{\prime})g(x)g(y)-\int_{[0,1]^{3}}\,\mathrm{d}y\,\mathrm{d}y^{\prime}\,\mathrm{d}x\,f(x,y)f(x,y^{\prime})g(x)g(y)\right|\\\
&=\left|\int_{[0,1]^{3}}\,\mathrm{d}y\,\mathrm{d}y^{\prime}\,\mathrm{d}x\,[f_{n}(x,y)f_{n}(x,y^{\prime})-f(x,y)f(x,y^{\prime})]\,g(x)g(y)\right|\\\
&=\left|\sum_{1\leq i,j\leq
k}\alpha_{i}\alpha_{j}\int_{[0,1]^{3}}\,\mathrm{d}y\,\mathrm{d}y^{\prime}\,\mathrm{d}x\,[f_{n}(x,y)f_{n}(x,y^{\prime})-f(x,y)f(x,y^{\prime})]\,\mathbf{1}_{A_{i}}(x)\mathbf{1}_{A_{j}}(y)\right|\\\
&\leq\sum_{1\leq i,j\leq
k}|\alpha_{i}\alpha_{j}|\left(\left|\int_{[0,1]^{3}}\,\mathrm{d}y\,\mathrm{d}x\,\mathrm{d}y^{\prime}\,f_{n}(x,y^{\prime})[f_{n}(x,y)-f(x,y)]\,\mathbf{1}_{A_{i}}(x)\mathbf{1}_{A_{j}}(y)\right|\right.\\\
&\qquad\qquad\qquad\qquad+\left.\left|\int_{[0,1]^{3}}\,\mathrm{d}y^{\prime}\,\mathrm{d}x\,\mathrm{d}y\,f(x,y)[f_{n}(x,y^{\prime})-f(x,y^{\prime})]\,\mathbf{1}_{A_{i}}(x)\mathbf{1}_{A_{j}}(y)\right|\right).\end{split}$

Note that, for every $y^{\prime}\in[0,1]$, $0\leq
f_{n}(x,y^{\prime})\mathbf{1}_{A_{i}}(x),\mathbf{1}_{A_{j}}(y)\leq 1$ and, for
every $y\in[0,1]$, $0\leq f(x,y)\mathbf{1}_{A_{i}}(x)\mathbf{1}_{A_{j}}(y)\leq
1$. Hence, by the definition of the cut-distance, the above expression is
bounded from above by

(2.7) $\begin{split}\sum_{1\leq i,j\leq
k}|\alpha_{i}\alpha_{j}|\,2d_{\square}(f_{n},f),\end{split}$

which tends to zero as $n\to\infty$.

We have thus shown that
$\liminf_{n\to\infty}\|\mathcal{L}_{f_{n}}(g)\|_{2}\geq\|\mathcal{L}_{f}(g)\|_{2}$
for all simple functions $g\in L^{2}([0,1])$. Now let $\varepsilon>0$. Then
there exists a simple $g^{\prime}\in L^{2}([0,1])$ such that
$\|\mathcal{L}_{f}\|<\|\mathcal{L}_{f}(g)\|+\varepsilon$. By the previous
result,

(2.8)
$\liminf_{n\to\infty}\|\mathcal{L}_{f_{n}}\|\geq\liminf_{n\to\infty}\|\mathcal{L}_{f_{n}}(g^{\prime})\|_{2}\geq\|\mathcal{L}_{f}(g^{\prime})\|>\|\mathcal{L}_{f}\|-\varepsilon.$

Since this holds for arbitrary $\varepsilon>0$, the proof is complete.

(ii) Let $G$ be a graph with $N$ vertices. We prove that $\frac{1}{N}L_{N}$
and $\mathcal{L}_{h^{G}}$ have the same eigenvalues. First, let
$u=(u_{1},\ldots,u_{N})$ be an eigenvector of $\frac{1}{N}L_{N}$ with
eigenvalue $\lambda$. Let $u\colon[0,1]\to\mathbb{R}$, and define
$\overline{u}\colon[0,1]\to\mathbb{R}$ to be the step function that on
$[\frac{i-1}{N},\frac{i}{N})$, $1\leq i\leq N$, equals the average of $u$ over
that interval. Then $u$ is an eigenfunction of $\mathcal{L}_{h^{G}}$ with
eigenvalue $\lambda$ if and only if, for all $1\leq j\leq N$ and
$x\in[\frac{j-1}{N},\frac{j}{N})$,

(2.9) $(\mathcal{L}_{h^{G}}u)(x)=\frac{1}{N}\sum_{i\in[N]\setminus
j}A_{N}(i,j)u(x)-\frac{1}{N}\sum_{i\in[N]\setminus
j}A_{N}(i,j)\overline{u}_{i}=\lambda u(x),$

i.e.,

(2.10) $u(x)=\left(\frac{1}{N}\sum_{i\in[N]\setminus
j}A_{N}(i,j)-\lambda\right)^{-1}\frac{1}{N}\sum_{i\in[N]\setminus
j}A_{N}(i,j)\overline{u}_{i}.$

Similarly, $v=(v_{1},\ldots,v_{N})$ is an eigenvector of $\frac{1}{N}L_{N}$
with eigenvalue $\lambda$ if and only if

(2.11) $v_{j}=\left(\frac{1}{N}\sum_{i\in[N]\setminus
j}A_{N}(i,j)-\lambda\right)^{-1}\frac{1}{N}\sum_{i\in[N]\setminus
j}A_{N}(i,j)v_{i}.$

Since every eigenfunction of $\mathcal{L}_{h^{G}}$ is constant on the blocks
$[\frac{j-1}{N},\frac{j}{N})$, $1\leq j\leq N$, we can view each eigenfunction
of $\mathcal{L}_{h^{G}}$ as an eigenvector of $\frac{1}{N}L_{N}$, and vice
versa.

Finally, note that $\mathcal{L}_{h^{G}}$ is a normal operator and has a non-
negative spectrum. So the norm of $\mathcal{L}_{h^{G}}$ equals the supremum of
its spectrum. Furthermore, the essential spectrum of $\mathcal{L}_{h^{G}}$
equals the range of the degree function $d_{h^{G}}(x)$ [17, Proposition 5.11].
Since the largest eigenvalue of the Laplacian matrix is bounded from below by
the maximum degree [19], it follows that the largest eigenvalue of
$\mathcal{L}_{h^{G_{N}}}$ equals its norm. ∎

##  3\. Proof of theorems for downward LDP

Sections 3.1–3.3 provide the proof of Theorems 1.5–1.7, respectively.

###  3.1. Proof of Theorem 1.5

####  3.1.1. Upper bound

By Proposition 1.3(i), $h\mapsto\|\mathcal{L}_{h}\|$ is lower semi-continuous.
Hence the set
$\\{\widetilde{h}\in\widetilde{\mathcal{W}}\colon\,\|\mathcal{L}_{\widetilde{h}}\|\leq\beta\\}$
is closed in $\widetilde{\mathcal{W}}$. By Proposition 1.3(ii),
$\frac{\|L_{N}\|}{N}=\|\mathcal{L}_{h^{G_{N}}}\|$, and so we can use Theorem
1.1 in combination with the contraction principle [15] to get

(3.1)
$\limsup_{N\to\infty}\binom{N}{2}^{-1}\log\mathbb{P}_{N}\left(\frac{\|L_{N}\|}{N}\leq\beta\right)\leq-\inf_{\begin{subarray}{c}\widetilde{h}\in\widetilde{\mathcal{W}}\\\
\|\mathcal{L}_{\widetilde{h}}\|\leq\beta\end{subarray}}J_{r}(\widetilde{h})=-\inf_{\begin{subarray}{c}h\in\mathcal{W}\\\
\|\mathcal{L}_{h}\|\leq\beta\end{subarray}}I_{r}(h).$

####  3.1.2. Lower bound

The proof proceeds via a change of measure. The key is the following lemma.

######  Lemma 3.1.

Let $h^{G_{N}}$ be the empirical graphon corresponding to the inhomogeneous
Erdős-Rényi random graph $G_{N}$. Then
$\lim_{N\to\infty}\|\mathcal{L}_{h^{G_{N}}}\|=\|\mathcal{L}_{r}\|$ in
probability.

###### Proof.

Since $\mathcal{L}_{h}$ is the sum of $\mathcal{D}_{h}$ and $-\mathcal{T}_{h}$
(recall (1.16)), it suffices to prove the claim for these two operators
separately.

To show that $\mathcal{D}_{h^{G_{N}}}$ converges to $\mathcal{D}_{r}$ in
operator norm in probability, let $d_{r_{N}}$ be the degree function of the
block graphon $r_{N}$. Then

(3.2)
$\begin{split}\|d_{h^{G_{N}}}-d_{r}\|_{\infty}\leq\|d_{h^{G_{N}}}-d_{r_{N}}\|_{\infty}+\|d_{r_{N}}-d_{r}\|_{\infty}.\end{split}$

Because $\|r_{N}-r\|_{\infty}\downarrow 0$ as $N\to\infty$ by (1.4), the
second term vanishes. As for the first term,

(3.3) $\displaystyle\mathbb{P}_{N}(\|d_{h^{G_{N}}}-d_{r_{N}}\|_{\infty}\geq
t)$
$\displaystyle\leq\sum_{i\in[N]}\mathbb{P}_{N}\left(\left|\frac{1}{N}\sum_{i\in[N]\setminus
j}^{N}A_{N}(i,j)-\frac{1}{N}\sum_{i\in[N]\setminus j}^{N}r_{N,ij}\right|\geq
t\right)\leq 2N\mathrm{e}^{-2Nt^{2}}\downarrow 0$

by a straightforward application of Hoeffding’s inequality. We use that
$A_{N}(i,j)$ has Bernoulli distribution with mean $(r_{N})_{ij}$. Hence
$\|d_{h^{G_{N}}}-d_{r_{N}}\|_{\infty}\to 0$ as $N\to\infty$ in probability,
and so

(3.4)
$\begin{split}\|\mathcal{D}_{h^{G_{N}}}-\mathcal{D}_{r}\|=\|d_{h^{G_{N}}}-d_{r}\|_{\infty}\to
0\text{ in probability}.\end{split}$

To show that $T_{h^{G_{N}}}$ converges to $T_{r}$ in operator norm in
probability, it suffices to note that

(3.5) $\|T_{h^{G_{N}}}-T_{r}\|\leq\sqrt{2}\,d_{\square}(h^{G_{N}},r)\downarrow
0\quad\text{ as }N\to\infty\text{ in probability}.$

The first inequality was shown in [24, Lemma 3.6]. Convergence in the cut-
metric was shown in [13, Lemma 5.11]. ∎

We are now ready to prove the lower bound.

######  Proposition 3.2.

Let $\beta\in[0,C_{r}]$, and let $h\in\mathcal{W}$ be such that
$\|\mathcal{L}_{h}\|<\beta$. Then

(3.6)
$\liminf_{N\to\infty}\frac{2}{N^{2}}\log\mathbb{P}_{N}(\|\mathcal{L}_{h^{G_{N}}}\|<\beta)\geq-
I_{r}(h).$

###### Proof.

The proof comes in three steps.

1. We begin by giving the proof for the case when $h$ is a _continuous_ graphon. Denote the law of an inhomogeneous ERRG with reference graphon $r_{N}$ by $\mathbb{P}_{N,r_{N}}$ instead of $\mathbb{P}_{N}$, and the law of an inhomogeneous ERRG with reference graphon $h_{N}$ by $\mathbb{P}_{N,h_{N}}$, where $h_{N}$ is the block graphon that is obtained by averaging $h$ over blocks of size $1/N$. Let $x=\|\mathcal{L}_{h}\|$, and define

(3.7)
$U_{\varepsilon}^{x}=\\{f\in\mathcal{W}\mid\|\mathcal{L}_{f}\|\in(x-\varepsilon,x+\varepsilon)\\}.$

Note that $h$ satisfies (1.4) by uniform continuity. Hence, by Lemma 3.1,

(3.8) $\mathbb{P}_{N,h_{N}}(U_{\varepsilon}^{x})\uparrow 1,\qquad N\to\infty,$

for all $\varepsilon>0$. Write

(3.9)
$\begin{split}\mathbb{P}_{N,r_{N}}(U_{\varepsilon}^{x})=\mathbb{P}_{N,h_{N}}(U_{\varepsilon}^{x})\,\frac{1}{\mathbb{P}_{N,h_{N}}(U_{\varepsilon}^{x})}\int_{U_{\varepsilon}^{x}}\exp\left(-\log\frac{\mathrm{d}\mathbb{P}_{N,h_{N}}}{\mathrm{d}\mathbb{P}_{N,r_{N}}}\right)\,\mathrm{d}\mathbb{P}_{N,h_{N}}.\end{split}$

By Jensen’s inequality,

(3.10)
$\log\mathbb{P}_{N,r_{N}}(U_{\varepsilon}^{x})\geq\log\mathbb{P}_{N,h_{N}}(U_{\varepsilon}^{x})-\frac{1}{\mathbb{P}_{N,h_{N}}(U_{\varepsilon}^{x})}\int_{U_{\varepsilon}^{x}}\left(\log\frac{\mathrm{d}\mathbb{P}_{N,h_{N}}}{\mathrm{d}\mathbb{P}_{N,r_{N}}}\right)\mathrm{d}\mathbb{P}_{N,h_{N}}.$

Using (3.8) and [13, Lemma 5.7], we obtain

(3.11)
$\liminf_{N\to\infty}\frac{2}{N^{2}}\log\mathbb{P}_{N,r_{N}}(U_{\varepsilon}^{x})\geq-\lim_{N\to\infty}\frac{2}{N^{2}}\int_{U_{\varepsilon}^{x}}\left(\log\frac{\mathrm{d}\mathbb{P}_{N,h_{N}}}{\mathrm{d}\mathbb{P}_{N,r_{N}}}\right)\mathrm{d}\mathbb{P}_{N,h_{N}}=-I_{r}(h).$

Note that [13, Lemma 5.7] was stated for the homogeneous ERRG, but the proof
may be extended to the inhomogeneous ERRG (see [26, Theorem 6.1]). Because
$U_{\varepsilon}^{x}\subseteq\\{f\in\mathcal{W}\mid\|\mathcal{L}_{f}\|<\beta\\}$
for $\varepsilon>0$ small enough, we conclude that

(3.12)
$\liminf_{N\to\infty}\frac{2}{N^{2}}\log\mathbb{P}_{N,r_{N}}(\|\mathcal{L}_{h^{G_{N}}}\|<\beta)\geq-
I_{r}(x).$

2. We next extend the proof to the case where $h$ is a _block graphon_ such that $\|\mathcal{L}_{h}\|<\beta$. Assume that $h$ is constant on the blocks $[\frac{i-1}{N},\frac{i}{N})\times[\frac{j-1}{N},\frac{j}{N})$, $1\leq i,j\leq N$. Then there exists a sequence of continuous graphons $(h_{k})_{k\in\mathbb{N}}$ that converges to $h$ in $L^{2}$ with $h_{k}\leq h$. Note that, for $f\in\mathcal{W}$,

(3.13) $\begin{split}\|\mathcal{L}_{f}\|&=\sup_{\|u\|_{2}\leq 1}\langle
g,\mathcal{L}_{f}(u)\rangle=\sup_{\|u\|_{2}\leq
1}\int_{[0,1]^{2}}\mathrm{d}y\,\mathrm{d}x\,u(x)f(x,y)[u(x)-u(y)]\\\
&=\tfrac{1}{2}\sup_{\|u\|_{2}\leq
1}\left[\int_{[0,1]^{2}}\mathrm{d}y\,\mathrm{d}x\,u(x)f(x,y)[u(x)-u(y)]+\int_{[0,1]^{2}}\mathrm{d}y\,\mathrm{d}x\,u(y)f(x,y)[u(y)-u(x)]\right]\\\
&=\tfrac{1}{2}\sup_{\|u\|^{2}\leq
1}\int_{[0,1]^{2}}\mathrm{d}x\,\mathrm{d}y\,f(x,y)[u(x)-u(y)]^{2},\end{split}$

where we use that $f$ is a symmetric function. From the above formula it is
immediate that, since $h_{k}\leq h$, we have
$\|\mathcal{L}_{h_{k}}\|\leq\|\mathcal{L}_{h}\|<\beta$. Then, by Lemma 3.2,

(3.14)
$\liminf_{N\to\infty}\frac{2}{N^{2}}\log\mathbb{P}_{N}(\|\mathcal{L}_{h^{G_{N}}}\|<\beta)\geq-
I_{r}(h_{k}),\qquad k\geq k_{0}.$

We conclude by noting that $I_{r}$ is continuous in $L^{2}([0,1]^{2})$,
implying that $I_{r}(h_{k})$ converges to $I_{r}(h)$ as $k\to\infty$.

3. We finally extend the proof to the case when $h$ is an _arbitrary_ graphon such that $\|\mathcal{L}_{h}\|<\beta$. Let $\overline{h}_{N}$ be the $N$-block approximant of $h$ for some $N$. Then $\overline{h}_{N}$ converges to $h$ in $L^{2}$ as $N\to\infty$ [13, Proposition 2.6]. Again, it suffices to prove

(3.15) $\|\mathcal{L}_{\overline{h}_{N}}\|\leq\|\mathcal{L}_{h}\|.$

First, suppose that
$\|\mathcal{L}_{\overline{h}_{N}}\|=\|d_{\overline{h}_{N}}\|_{\infty}$. For
$x\in[\frac{i-1}{N},\frac{i}{N})$,

(3.16)
$d_{\overline{h}_{N}}(x)=N\int_{[\frac{i-1}{N},\frac{i}{N})}\mathrm{d}x\,h(x)\leq\|d_{h}\|_{\infty}.$

It follows that
$\|\mathcal{L}_{\overline{h}_{N}}\|=\|d_{\overline{h}_{N}}\|_{\infty}\leq\|d_{h}\|_{\infty}\leq\|\mathcal{L}_{h}\|$.
Next, suppose that
$\|\mathcal{L}_{\overline{h}_{N}}\|>\|d_{\overline{h}_{N}}\|_{\infty}$. Then
$\|\mathcal{L}_{\overline{h}_{N}}\|=\lambda_{\max}(\mathcal{L}_{\overline{h}_{N}})$
and there exists an eigenfunction $u$ of $\mathcal{L}_{h}$ with $\|u\|_{2}=1$
such that
$\langle\mathcal{L}_{\overline{h}_{N}}u,u\rangle=\|\mathcal{L}_{\overline{h}_{N}}\|$.
In the proof of Proposition 1.3 it was shown that $u$ is constant on each of
the intervals $[\frac{i-1}{N},\frac{i}{N})$, $1\leq i\leq N$. Let $u_{i}$ be
the value of $u$ on $[\frac{i-1}{N},\frac{i}{N})$, $1\leq i\leq N$. Then

(3.17)
$\begin{split}\langle\mathcal{L}_{h}u,u\rangle=&\int_{[0,1]^{2}}\,\mathrm{d}x\,\,\mathrm{d}y\,h(x,y)[u(x)-u(y)]u(x)\\\
=&\sum_{i,j=1}^{N}\int_{[\frac{i-1}{N},\frac{i}{N})\times[\frac{j-1}{N},\frac{j}{N})}\,\mathrm{d}x\,\,\mathrm{d}y\,h(x,y)[u_{i}-u_{j}]u_{i}\\\
=&\sum_{i,j=1}^{N}\int_{[\frac{i-1}{N},\frac{i}{N})\times[\frac{j-1}{N},\frac{j}{N})}\,\mathrm{d}x\,\,\mathrm{d}y\,\overline{h}_{N}(x,y)[u_{i}-u_{j}]u_{i}=\langle\mathcal{L}_{\overline{h}_{N}}u,u\rangle.\end{split}$

The desired result now follows from the fact that
$\|\mathcal{L}_{h}\|=\sup_{\|v\|_{2}=1}\langle\mathcal{L}_{h}v,v\rangle\geq\langle\mathcal{L}_{h}u,u\rangle=\langle\mathcal{L}_{\overline{h}_{N}}u,u\rangle=\|\mathcal{L}_{\overline{h}_{N}}\|$.
∎

From Proposition 3.2, we obtain the lower bound

(3.18)
$\liminf_{N\to\infty}\frac{2}{N^{2}}\log\mathbb{P}_{N}(\|\mathcal{L}_{h^{G_{N}}}\|<\beta)\geq-\inf_{\begin{subarray}{c}h\in\mathcal{W}\\\
\|\mathcal{L}_{h}\|<\beta\end{subarray}}I_{r}(h).$

So it just remains to show

(3.19) $\inf_{\begin{subarray}{c}h\in\mathcal{W}\\\
\|\mathcal{L}_{h}\|<\beta\end{subarray}}I_{r}(h)=\inf_{\begin{subarray}{c}h\in\mathcal{W}\\\
\|\mathcal{L}_{h}\|\leq\beta\end{subarray}}I_{r}(h).$

Let $h\in\mathcal{W}$ with $\|\mathcal{L}_{h}\|=\beta$. Then
$\|\mathcal{L}_{(1-\varepsilon)h}\|=(1-\varepsilon)\beta<\beta$ for all
$\varepsilon>0$. Furthermore, $(1-\varepsilon)h$ converges to $h$ in $L^{2}$
as $\varepsilon\downarrow 0$ and $I_{r}$ is continuous in $L^{2}$, so
$I_{r}((1-\varepsilon)h)\rightarrow I_{r}(h)$ as $\varepsilon\downarrow 0$.
Hence, the left-hand side of (3.19) is bounded from above by the right-hand
side. The converse inequality is trivial, which settles the proof of the lower
bound.

###  3.2. Proof of Theorem 1.6

###### Proof.

(i) It was shown in the proof of Proposition 3.2 that if $h_{1}\leq h_{2}$,
then $\|\mathcal{L}_{h_{1}}\|\leq\|\mathcal{L}_{h_{2}}\|$. Via the same proof
as for [11, Theorem 1.5], it follows that $\psi_{r}$ is strictly decreasing on
$[0,C_{r}]$.

Let $t>0$, and let $(t_{n})_{n\in\mathbb{N}}$ be a strictly increasing
sequence converging to $t$. By the compactness of $\widetilde{\mathcal{W}}$,
the lower semi-continuity of $I_{r}$ and
$\widetilde{h}\mapsto\|\mathcal{L}_{\widetilde{h}}\|$, and the invariance of
$\widetilde{h}\mapsto\|\mathcal{L}_{h}\|$ under measure-preserving bijections,
there exists a sequence of graphons $(\widetilde{f}_{n})_{n\in\mathbb{N}}$
such that $\|\mathcal{L}_{f_{n}}\|\geq t_{n}$ and
$\psi_{r}(t_{n})=I_{r}(\widetilde{f}_{n})$ for all $n\in\mathbb{N}$. Again, by
the compactness of $\widetilde{\mathcal{W}}$, we may assume that the sequence
$(\widetilde{f}_{n})_{n\in\mathbb{N}}$ converges to some
$\widetilde{f}\in\widetilde{\mathcal{W}}$ in the cut-metric. Because
$\widetilde{h}\mapsto\|\mathcal{L}_{\widetilde{h}}\|$ is lower semi-
continuous, we have $\|\mathcal{L}_{\widetilde{f}}\|\geq t$. Once again
invoking the lower semi-continuity of $I_{r}$, we conclude that

(3.20)
$\liminf_{n\to\infty}\psi_{r}(t_{n})=\liminf_{n\to\infty}I_{r}(\widetilde{f}_{n})\geq
I_{r}(\widetilde{f})\geq\psi_{r}(t).$

Right-continuity of $\psi_{r}$ now follows by noting that $\psi_{r}$ is
strictly decreasing. The proof of the left-continuity of $\psi_{r}$ is the
same as the proof of (3.19).

(ii) This statement is immediate from the lower semi-continuity of
$h\mapsto\|\mathcal{L}_{h}\|$ and $I_{r}$, combined with the compactness of
$\widetilde{\mathcal{W}}$. ∎

###  3.3. Proof of Theorem 1.7

###### Proof.

We first prove a large deviation principle for the degree of a single vertex
$\approx xN$ with rate $N$ and with rate function $J_{r}(x,\beta)$ defined in
(1.35). We subsequently derive upper and lower bounds for $\psi_{r}(\beta)$ in
terms of this rate function, and use these to compute the scaling of
$\psi_{r}$ near its minimum.

####  3.3.1. LDP for single degrees

Let $d_{i}$ be the degree of vertex $i$. For $x\in[0,1]$, let
$i_{x}\in\\{1,\ldots,N\\}$ be the index such that
$x\in[\frac{i_{x}-1}{N},\frac{i_{x}}{N})$. A crucial ingredient is the LDP for
the family $N^{-1}d_{i_{x}}$, $N\in\mathbb{N}$, for fixed $x\in[0,1]$, which
we prove with the help of the Gärtner-Ellis theorem. To that end, note that
$d_{i_{x}}=\sum_{j\in[N]}A_{i_{x}j}$, with $A_{i_{x}j}=1$ if there is an edge
between $i$ and $j$ and $A_{i_{x}j}=0$ otherwise. Therefore the cumulant
generating function of $N^{-1}d_{i_{x}}$ is

(3.21)
$\begin{split}\Lambda_{N}(x,\theta)&=\log\mathbb{E}\left[\exp\left(\theta\frac{1}{N}\sum_{j\in[N]}A_{i_{x}j}\right)\right]=\sum_{j\in[N]\setminus
i}\log\mathbb{E}\left[\exp\left(\theta\frac{1}{N}A_{i_{x}j}\right)\right]\\\
&=\sum_{j\in[N]}\log\left((r_{N})_{i_{x}j}\,\mathrm{e}^{\frac{1}{N}\theta}+[1-(r_{N})_{i_{x}j}]\right)\\\
&=N\int_{[0,1]\setminus[\frac{i_{x}-1}{N},\frac{i_{x}}{N})}\mathrm{d}y\,\log\left(r_{N}(x,y)\mathrm{e}^{\frac{1}{N}\theta}+[1-r_{N}(x,y)]\right).\end{split}$

Since $\|r_{N}-r\|_{\infty}\rightarrow 0$, we have

(3.22)
$\Lambda_{r}(x,\theta)=\lim_{N\to\infty}\frac{1}{N}\Lambda_{N}(x,N\theta)=\int_{[0,1]}\mathrm{d}y\,\log\left(r(x,y)\mathrm{e}^{\theta}+[1-r(x,y)]\right).$

Since $\theta\mapsto\Lambda_{r}(x,\theta)$ is finite and differentiable on
$\mathbb{R}$, the Gärtner-Ellis theorem tells us that the family
$N^{-1}d_{i_{x}}$, $N\in\mathbb{N}$, satisfies the LDP on $[0,1]$ with rate
$N$ and with rate function $x\mapsto J_{r}(x,\beta)$ given by

(3.23)
$J_{r}(x,\beta)=\sup_{\theta\in\mathbb{R}}[\theta\beta-\Lambda_{r}(x,\theta)].$

The supremum is attained at the unique $\theta(x,\beta)$ such that

(3.24)
$\int_{[0,1]}\mathrm{d}y\,\widehat{r}_{\beta}(x,y)=\beta\qquad\forall\,x\in[0,1],$

with

(3.25)
$\widehat{r}_{\beta}(x,y)=\frac{\mathrm{e}^{\theta(x,\beta)}r(x,y)}{\mathrm{e}^{\theta(x,\beta)}r(x,y)+[1-r(x,y)]}.$

The Lagrange multiplier $\theta(x,\beta)$ exists because $r\in(0,1)$ almost
everywhere by (1.5). A simple computation shows that

(3.26)
$J_{r}(x,\beta)=\int_{[0,1]}\mathrm{d}y\,\mathcal{R}\big{(}\widehat{r}_{\beta}(x,y)\mid
r(x,y)\big{)}=\sup_{u}\int_{[0,1]}\mathrm{d}y\,\mathcal{R}(u(y)\mid r(x,y)),$

where the supremum is taken over all the measurable functions
$u\colon[0,1]\to[0,1]$ satisfying $\int_{[0,1]}\mathrm{d}y\,u(y)\leq\beta$.
The map

$\theta\mapsto\int_{[0,1]}\mathrm{d}y\,\frac{\mathrm{e}^{\theta}r(x,y)}{\mathrm{e}^{\theta}r(x,y)+1-r(x,y)}$

is a continuous bijection from $\mathbb{R}$ to $(0,1)$, again because
$r\in(0,1)$ almost everywhere, and so the inverse map
$\beta\mapsto\theta(x,\beta)$ is continuous.

####  3.3.2. Properties of $J_{r}$

We need the following lemma for the derivatives of $J_{r}$ with respect to
$\beta$. Henceforth we write $J^{\prime}_{r}=\frac{\partial
J_{r}}{\partial\beta}$ and
$\theta^{\prime}=\frac{\partial\theta}{\partial\beta}$, and use the upper
index $(k)$ for the $k$-th derivative.

######  Lemma 3.3.

$\beta\mapsto J_{r}(x,\beta)$ is analytic for every $x\in[0,1]$, with

(3.27) $J_{r}^{(k)}(x,\beta)=\theta^{(k-1)}(x,\beta).$

Moreover, subject to (1.29),

(3.28) $\sup_{\begin{subarray}{c}x\in[0,1]\\\
\beta\in[\varepsilon,1-\varepsilon]\end{subarray}}|J_{r}^{(k)}(x,\beta)|<\infty\qquad\forall\,\varepsilon>0.$

###### Proof.

Recall from Section 3.3.1 that

(3.29)
$J_{r}(x,\beta)=\int_{[0,1]}\mathrm{d}y\,\mathcal{R}\big{(}\widehat{r}_{\beta}(x,y)\mid
r(x,y)\big{)}=\theta(x,\beta)\beta-\int_{[0,1]}\mathrm{d}y\,\log\Big{(}1+\big{(}\mathrm{e}^{\theta(x,\beta)}-1\big{)}r(x,y)\Big{)}.$

Differentiating with respect to $\beta$ and using that
$d_{\widehat{r}_{\beta}}(x)=\beta$, we obtain

(3.30) $J_{r}^{\prime}(x,\beta)=\theta(x,\beta).$

Implicit differentiation of the equation

(3.31)
$\int_{[0,1]}\mathrm{d}y\,\frac{\mathrm{e}^{\theta}r(x,y)}{\mathrm{e}^{\theta}r(x,y)+(1-r(x,y))}=\beta$

gives

(3.32)
$\theta^{\prime}(x,\beta)=\left(\int_{[0,1]}\mathrm{d}y\,\frac{\mathrm{e}^{\theta(x,\beta)}r(x,y)(1-r(x,y))}{\left[\mathrm{e}^{\theta(x,\beta)}r(x,y)+(1-r(x,y))\right]^{2}}\right)^{-1}.$

Fix $\varepsilon>0$. Since $r$ is bounded away from 0 and 1, we see that
$\theta^{\prime}(x,\beta)$ is bounded for $x\in[0,1]$ and
$\beta\in[C_{r},1-\varepsilon]$ when $\theta(x,\beta)$ is. Iteratively
applying the chain, product and quotient rules of differentiation, we obtain
that $\theta^{(k)}$ is some polynomial of order $k$ in the variables

(3.33)
$\left(\int_{[0,1]}\mathrm{d}y\,\frac{\mathrm{e}^{\theta}r(x,y)(1-r(x,y))}{\left[\mathrm{e}^{\theta}r(x,y)+(1-r(x,y))\right]^{2}}\right)^{-1},\qquad\int_{[0,1]}\mathrm{d}y\,\frac{f\left(\mathrm{e}^{\theta},r,\theta^{\prime},\ldots,\theta^{(k-1)}\right)}{\left[\mathrm{e}^{\theta}r(x,y)+(1-r(x,y))\right]^{j}},$

with $f$ a polynomial and $j\in\mathbb{N}$. By induction, we obtain that $f$
is bounded when $\theta$ is bounded, and hence that $\theta^{(k)}$ is bounded.

Thus, it remains to show
$\sup_{x\in[0,1],\beta\in[\varepsilon,1-\varepsilon]}|\theta(x,\beta)|<\infty$.
Let

(3.34)
$r_{-}=\inf_{(x,y)\in[0,1]^{2}}r(x,y)>0,\qquad\widetilde{\theta}(\beta)=\log\frac{\beta(1-r_{-})}{(1-\beta)r_{-}}.$

Since the map
$r\mapsto\frac{\mathrm{e}^{\theta}r}{\mathrm{e}^{\theta}r+(1-r)}$ is
increasing, we have

(3.35)
$\int_{[0,1]}\mathrm{d}y\,\frac{\mathrm{e}^{\widetilde{\theta}}r(x,y)}{\mathrm{e}^{\widetilde{\theta}}r(x,y)+(1-r(x,y))}\geq\frac{\mathrm{e}^{\widetilde{\theta}}r_{-}}{\mathrm{e}^{\widetilde{\theta}}r_{-}+(1-r_{-})}=\beta.$

Hence, $\theta(x,\beta)\leq\widetilde{\theta}(\beta)<\infty$. Since
$\widetilde{\theta}(\beta)$ is independent of $x$ and
$\sup_{\beta\in[\varepsilon,1-\varepsilon]}|\widetilde{\theta}(\beta)|<\infty$,
all claims are settled. ∎

####  3.3.3. Lower bound

Let $\bar{h}$ be a minimizer of (1.26). Then
$\int_{[0,1]}\mathrm{d}y\,\bar{h}(x,y)\leq\|\mathcal{L}_{\bar{h}}\|\leq\beta$
for each $x\in S_{r}(\beta)$. Hence, by (3.26),

(3.36) $\int_{[0,1]}\mathrm{d}y\,\mathcal{R}\big{(}\bar{h}(x,y)\mid
r(x,y)\big{)}\geq J_{r}(x,\beta),$

which implies

(3.37)
$\begin{split}&\psi_{r}(\beta)=I_{r}(\bar{h})\geq\int_{S_{r}(\beta)}\mathrm{d}x\,J_{r}(x,\beta).\end{split}$

####  3.3.4. Upper bound

Let

(3.38) $h_{\beta}(x,y)=\begin{cases}r(x,y),&x,y\not\in S_{r}(\beta)\\\
\hat{r}_{\beta}(x,y),&x\in S_{r}(\beta)\text{ or }y\in S_{r}(\beta)\text{, but
}(x,y)\not\in S_{r}(\beta)^{2},\\\
\min\\{\hat{r}_{\beta}(x,y),\hat{r}_{\beta}(y,x)\\},&x,y\in
S_{r}(\beta).\end{cases}$

Then $h_{\beta}$ converges to $r$ in $L^{\infty}$ as $\beta\uparrow C_{r}.$ We
show that $\|\mathcal{L}_{h\beta}\|\leq\beta$ for $\beta$ close enough to
$C_{r}$. Note that $\|\mathrm{d}_{h_{\beta}}\|_{\infty}\leq\beta$ by
construction, so we only need to show
$\lambda_{\max}(\mathcal{L}_{h_{\beta}})\leq\beta$ for $\beta$ large enough.
Let $\lambda_{\beta}=\lambda_{\max}(\mathcal{L}_{h_{\beta}})$ and let
$\lambda^{\prime}$ be the limit of any convergent subsequence of
$(\lambda_{\beta})_{k\in\mathbb{N}}$. We show that $\lambda^{\prime}$ is an
eigenvalue of $\mathcal{L}_{h}$, so that $\limsup_{\beta\uparrow
C_{r}}\lambda_{\beta}\leq\lambda_{\max}(\mathcal{L}_{r})\leq\beta$ for $\beta$
close enough to $C_{r}$. Here we use Assumption (1.28).

Assume, by contradiction, that $\lambda^{\prime}$ is not an eigenvalue of
$\mathcal{L}_{r}$. Without loss of generality, we may assume that
$\lambda_{\beta}$ converges to $\lambda^{\prime}$. Consider
$F\colon\,L^{\infty}([0,1]^{2})\times\mathbb{R}\times L^{2}([0,1])\to
L^{2}([0,1])$ given by $F(g,\mu,u)=\mathcal{L}_{g}u-\mu u$. This map is
bounded and affine in each coordinate, and hence is continuously Fréchet
differentiable. The Fréchet derivative of $F$ at a point $(g,\mu,u)$ is given
by

(3.39) $((DF)(g,\mu,u))(f,\nu,w)=F(f,\nu,u)+F(g,\mu,w).$

Indeed, let $(f_{k},\nu_{k},w_{k})$ such that
$\|(f_{k},\nu_{k},w_{k})\|=\|f_{k}\|_{\infty}+|\nu_{k}|+\|w_{k}\|_{2}\rightarrow
0$ as $k\rightarrow\infty$. Then

(3.40)
$\begin{split}&\frac{\left\|F(g+f_{k},\mu+\nu_{k},u+w_{k})-F(g,\mu,u)-((DF)(g,\mu,u))(f_{k},\nu_{k},w_{k})\right\|}{\|(f_{k},\nu_{k},w_{k})\|}\\\
=&\frac{\|F(f_{k},\nu_{k},w_{k})\|}{\|(f_{k},\nu_{k},w_{k})\|}\leq\frac{\|\mathcal{L}_{f_{k}}w_{k}\|_{2}+\|\nu_{k}w_{k}\|_{2}}{\|f_{k}\|_{\infty}+|\nu_{k}|+\|w_{k}\|_{2}}\leq\frac{2\|f_{k}\|_{\infty}\|w_{k}\|_{2}+|\nu_{k}|\|w_{k}\|_{2}}{\|f_{k}\|_{\infty}+|\nu_{k}|+\|w_{k}\|_{2}}\rightarrow
0,\qquad k\rightarrow\infty.\end{split}$

Note that $F$ is not necessarily Fréchet differentiable as a function
$L^{2}([0,1]^{2})\times\mathbb{R}\times L^{2}([0,1])\to L^{2}([0,1])$. Since
$\lambda^{\prime}$ is not an eigenvalue of $\mathcal{L}_{r}$, the map

(3.41) $w\mapsto((DF)(r,\lambda^{\prime},0))(0,0,w)=F(r,\lambda^{\prime},w)$

has a trivial kernel and so is an isomorphism on its image space. So, by the
implicit function theorem for Banach spaces [22, Theorem I.5.9], there exists
a neighbourhood $U$ of $(r,\lambda^{\prime})\in
L^{\infty}([0,1]^{2})\times\mathbb{R}$ and a neighbourhood $V$ of
$\underline{0}\in L^{2}([0,1])$ such that $F(g,\mu,u)=0$ if and only if $u=0$
for all $(g,\mu,u)\in U\times V$. Since $h_{\beta}\to r$ in $L^{\infty}$ and
$\lambda_{\beta}\to\lambda^{\prime}$ as $\beta\uparrow C_{r}$, we have that
$(h_{\beta},\lambda_{\beta})\in U$ for $\beta\geq\beta_{0}$. However, since
$\lambda_{\beta}$ is an eigenvalue of $\mathcal{L}_{h_{\beta}}$, there exists
an eigenfunction $u_{\beta}\in L^{2}([0,1])$ with $u_{\beta}\neq 0$ such that
$F(h_{\beta},\lambda_{\beta},u_{\beta})=0$. Since $V$ is a neighbourhood of 0,
we can rescale $u_{\beta}$ such that it lies in $V$. This yields a
contradiction, and so $\lambda^{\prime}$ must be an eigenvalue of
$\mathcal{L}_{r}$.

Now since $\|\mathcal{L}_{h_{\beta}}\|\leq\beta$ for $\beta$ sufficiently
close to $C_{r}$, we have

(3.42) $\begin{split}\psi_{r}(\beta)&\leq I_{r}(h_{\beta})\\\
&=2\int_{S_{r}(\beta)\times([0,1]\setminus
S_{r}(\beta))}\,\mathrm{d}x\,\,\mathrm{d}y\,\mathcal{R}\big{(}\hat{r}_{\beta}(x,y)\mid
r(x,y)\big{)}\\\
&\qquad+\int_{S_{r}(\beta)^{2}}\,\mathrm{d}x\,\,\mathrm{d}y\,\mathcal{R}\big{(}\min\\{\hat{r}_{\beta}(x,y),\hat{r}_{\beta}(y,x)\\}\mid
r(x,y)\big{)}\\\ &\leq 2\int_{S_{r}(\beta)\times([0,1]\setminus
S_{\beta})}\,\mathrm{d}x\,\,\mathrm{d}y\,\mathcal{R}\big{(}\hat{r}_{\beta}(x,y)\mid
r(x,y)\big{)}+\int_{S_{r}(\beta)^{2}}\,\mathrm{d}x\,\,\mathrm{d}y\,\mathcal{R}\big{(}\hat{r}_{\beta}(x,y)\mid
r(x,y)\big{)}\\\
&\qquad+\int_{S_{r}(\beta)^{2}}\,\mathrm{d}x\,\,\mathrm{d}y\,\mathcal{R}\big{(}\hat{r}_{\beta}(y,x)\mid
r(x,y)\big{)}\\\
&=2\int_{S_{r}(\beta)}\,\mathrm{d}x\,J_{r}(x,\beta),\qquad\beta\uparrow
C_{r}.\end{split}$

For the last equality, we use that $r$ is symmetric and that
$S_{r}(\beta)^{2}$ is a symmetric domain.

######  Remark 3.4.

The above shows that, up to a constant, the last integral in (3.42) also
equals the rate function in the LDP for the maximum degree. The upper bound
can also be shown directly by the following computation.

The state space for the edges determining the adjacency matrix $(A_{ij})$ is
$\\{0,1\\}^{\binom{N}{2}}$, which we endow with the standard partial ordering.
The probability distribution of the edges is the product measure
$\prod_{\\{i,j\\}}\mathrm{BER}(r_{N}(\tfrac{i}{N},\tfrac{j}{N}))$, which is
log-convex. Note that the events $\\{N^{-1}\max_{1\leq i\leq N-1}d_{i}$
$\leq\beta\\}$ and $\\{N^{-1}d_{N}\leq\beta\\}$ are non-increasing in the
partial ordering. Hence we can use the FKG-inequality iteratively, to get

(3.43)
$\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\leq\beta\right)\geq\prod_{i\in[N]}\mathbb{P}(N^{-1}d_{i}\leq\beta).$

Consequently,

(3.44)
$\displaystyle\frac{1}{\binom{N}{2}}\log\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\leq\beta\right)\geq\frac{1}{\binom{N}{2}}\sum_{i=1}^{N}\log\mathbb{P}(N^{-1}d_{i}\leq\beta)$
$\displaystyle=\frac{2N}{N-1}\frac{1}{N}\sum_{i\in[N]}\frac{1}{N}\log\mathbb{P}(N^{-1}d_{i}\leq\beta)=\frac{2N}{N-1}\int_{[0,1]}\mathrm{d}x\,\frac{1}{N}\log\mathbb{P}(N^{-1}d_{i_{x}}\leq\beta).$

By the law of large numbers,
$\lim_{N\to\infty}N^{-1}d_{i_{x}}=\int_{[0,1]}\mathrm{d}y\,r(x,y)=d_{r}(x)$
$\mathbb{P}$-a.s., and so in the limit as $N\to\infty$ the last integral in
(3.44) may be restricted to the set
$S_{r}(\beta)=\\{x\in[0,1]\colon\,d_{r}(x)\geq\beta\\}$, i.e.,

$\liminf_{N\to\infty}\frac{1}{\binom{N}{2}}\log\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\leq\beta\right)\geq-2\int_{S_{r}(\beta)}\mathrm{d}x\,J_{r}(x,\beta),$

where we use that the family $N^{-1}d_{i_{x}}$, $N\in\mathbb{N}$, satisfies
the LDP on $[0,1]$ with rate $N$ and with rate function $x\mapsto
J_{r}(x,\beta)$, as shown in Section 3.3.1.  $\spadesuit$

####  3.3.5. Scaling of $J_{r}(x,\beta)$

Via Lemma 3.3 it is straightforward to show that
$J_{r}(x,d_{r}(x))=J^{\prime}_{r}(x,d_{r}(x))=0$ and
$J^{\prime\prime}_{r}(x,d_{r}(x))=\frac{1}{v_{r}(x)}$. So, by Taylor
expansion,

(3.45) $J_{r}(x,\beta)=\frac{1}{2v_{r}(x)}(\beta-
d_{r}(x))^{2}+\frac{1}{6}\theta^{\prime\prime}(x,\beta_{*})(\beta-
d_{r}(x))^{3}$

for some $\beta_{*}=\beta_{*}(x)\in[\beta,d_{r}(x)]$. Inserting this expansion
into the lower and upper bounds derived above and using that
$\theta^{\prime\prime}(x,\beta)$ is bounded uniformly in $x$ and $\beta$, we
obtain

(3.46)
$\psi_{r}(\beta)\asymp\int_{S_{r}(\beta)}\mathrm{d}x\,J_{r}(x,\beta)\sim\int_{S_{r}(\beta)}\mathrm{d}x\,\frac{1}{2v_{r}(x)}(\beta-
d_{r}(x))^{2},\qquad\beta\uparrow C_{r}.$

∎

##  4\. Proof of theorems for upward LDP

Sections 4.1–4.3 provide the proof of Theorems 1.8–1.10, respectively.

###  4.1. Proof of Theorem 1.8

###### Proof.

We first prove the LDP for the maximum degree. Afterwards we prove the LDP for
the maximal eigenvalue.

####  4.1.1. LDP for the maximum degree

For $x\in[0,1]$, let $i_{x}\in\\{1,\ldots,N\\}$ be the index such that
$x\in[\frac{i_{x}-1}{N},\frac{i_{x}}{N})$. Let $d_{i}$ be the degree of vertex
$i$. First note that we can sandwich

(4.1)
$\begin{split}\sup_{x\in[0,1]}\mathbb{P}(N^{-1}d_{i_{x}}\geq\beta)\leq\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\geq\beta\right)\leq\sum_{i\in[N]}\mathbb{P}(N^{-1}d_{i}\geq\beta)\leq
N\sup_{x\in[0,1]}\mathbb{P}(N^{-1}d_{i_{x}}\geq\beta).\end{split}$

Using the LDP for the single degrees derived in Section 3.3.1, we obtain

(4.2)
$\begin{split}&\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\geq\beta\right)=\lim_{N\to\infty}\sup_{x\in[0,1]}\frac{1}{N}\log\mathbb{P}\left(N^{-1}d_{i_{x}}\geq\beta\right)\\\
&=\sup_{x\in[0,1]}\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\left(N^{-1}d_{i_{x}}\geq\beta\right)=-\inf_{x\in[0,1]}J_{r}(x,\beta)=-\widehat{\psi}_{r}(\beta).\end{split}$

The convergence of $\Lambda(\theta)$ in (3.22) is uniform in $x\in[0,1]$
because $\|r_{N}-r\|_{\infty}\to 0$ as $N\to\infty$. Hence the convergence in
the LDP for $N^{-1}d_{i_{x}}$ is uniform in $x\in[0,1]$. This allows us to
swap the supremum and the limit in the second equality.

####  4.1.2. LDP for $\lambda_{\max}(L_{N})$

By Weyl’s interlacing inequalities, we have
$\lambda_{\max}(L_{N})\leq\lambda_{\max}(D_{N})-\lambda_{\min}(A_{N})$. We
also know that $\lambda_{\max}(D_{N})\leq\lambda_{\max}(L_{N})$. Hence

(4.3)
$\lambda_{\max}(D_{N})\leq\lambda_{\max}(L_{n})\leq\lambda_{\max}(D_{N})-\lambda_{\min}(A_{n}).$

Since $r$ is non-negative definite, the smallest eigenvalue of $T_{r}$ is 0.
Now let $(T_{n})_{n\in\mathbb{N}}$ be a sequence of self-adjoint operators
acting on $L^{2}([0,1])$ and converging in operator norm to some operator $T$.
Then

(4.4)
$\left|\lambda_{\min}(T_{n})-\lambda_{\min}(T)\right|=\left|\inf_{\|f\|_{2}=1}\langle
f,T_{n}f\rangle-\inf_{\|f\|_{2}=1}\langle
f,Tf\rangle\right|\leq\sup_{\|f\|_{2}=1}|\langle f,T_{n}f\rangle-\langle
f,Tf\rangle|\to 0$

as $n\to\infty$. Thus, the map $h\mapsto\lambda_{\min}(\mathcal{T}_{h})$ is
continuous in the cut norm. Hence, by the contraction principle,
$N^{-1}\lambda_{\min}(A_{N})=\lambda_{\min}(T_{h^{G_{N}}})$ converges to
$\lambda_{\min}(T_{r})=0$ at an exponential rate with coefficient $N^{2}$. By
(4.2) and (4.3) we can sandwich, for all $\varepsilon>0$,

(4.5) $\displaystyle-\widehat{\psi}_{r}(\beta)$
$\displaystyle=-\inf_{x\in[0,1]}J_{r}(x,\beta)=\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\geq\beta\right)$
$\displaystyle\leq\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\big{(}N^{-1}\lambda_{\max}(L_{N})\geq\beta\big{)}$
$\displaystyle\leq\lim_{N\to\infty}\frac{1}{N}\log\left[\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\geq\beta-\varepsilon\right)+\mathbb{P}\big{(}N^{-1}\lambda_{\min}(A_{N})\leq-\varepsilon\big{)}\right]$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N}\log\mathbb{P}\left(N^{-1}\max_{i\in[N]}d_{i}\geq\beta-\varepsilon\right)=-\inf_{x\in[0,1]}J_{r}(x,\beta-\varepsilon)=-\widehat{\psi}_{r}(\beta-\varepsilon).$

The result now follows from continuity of $\widehat{\psi}_{r}$, as stated in
Theorem 1.9. Note that the proof of Theorem 1.9 does not use Theorem 1.8. ∎

###  4.2. Proof of Theorem 1.9

####  4.2.1. Proof that $\widehat{\psi}_{r}$ is strictly increasing

Assume that $\inf_{x\in[0,1]}\theta(x,\beta)>0$ for all $\beta>C_{r}$. Then,
by Lemma 3.3, for $\beta_{2}>\beta_{1}>C_{r}$,

(4.6)
$\begin{split}\widehat{\psi}_{r}(\beta_{2})=&\inf_{x\in[0,1]}J_{r}(x,\beta_{2})\\\
\geq\,&\inf_{x\in[0,1]}J_{r}(x,\beta_{1})+\theta(x,\beta_{1})(\beta_{2}-\beta_{1})\\\
\geq\,&\widehat{\psi}_{r}(\beta_{1})+\inf_{x\in[0,1]}\theta(x,\beta_{1})(\beta_{2}-\beta_{1})>\widehat{\psi}_{r}(\beta_{1}).\end{split}$

For the first inequality we use that $J^{\prime}_{r}(x,\beta)=\theta(x,\beta)$
is increasing in $\beta$. Hence, it suffices to show
$\inf_{x\in[0,1]}\theta(x,\beta)>0$ for all $\beta>C_{r}$.

Note that the map
$r\mapsto\frac{\mathrm{e}^{\theta}r}{1+(\mathrm{e}^{\theta}-1)r}$ is concave.
Let
$\widetilde{\theta}(x,\beta)=\log\frac{\beta(1-d_{r}(x))}{(1-\beta)d_{r}(x)}$.
Then, by Jensen’s inequality, for all $x\in[0,1]$,

(4.7)
$\int_{[0,1]}\mathrm{d}y\,\frac{e^{\widetilde{\theta}(x,\beta)}r(x,y)}{1+(\mathrm{e}^{\widetilde{\theta}(x,\beta)}-1)r(x,y)}\leq\frac{\mathrm{e}^{\widetilde{\theta}(x,\beta)}d_{r}(x)}{1+(\mathrm{e}^{\widetilde{\theta}(x,\beta)}-1)d_{r}(x)}=\beta.$

Recall that $\theta(\beta,x)$ is chosen such that

(4.8)
$\int_{[0,1]}\mathrm{d}y\,\frac{\mathrm{e}^{\theta(x,\beta)}r(x,y)}{1+(\mathrm{e}^{\theta(x,\beta)}-1)r(x,y)}=\beta.$

This implies that
$\theta(x,\beta)\geq\widetilde{\theta}(x,\beta)\geq\log\frac{\beta(1-C_{r})}{(1-\beta)C_{r}}$.
We conclude the proof that $\widehat{\psi}_{r}$ is strictly increasing by
noting that this lower bound is strictly positive for $\beta>C_{r}$ and is
independent of $x$.

####  4.2.2. Continuity of $\widehat{\psi}_{r}$

Right-continuity of $\widehat{\psi}_{r}$ follows from the fact that the
infimum of continuous functions is right-continuous. By Lemma 3.3,
$\sup_{x\in[0,1]}\theta(x,\beta)<\infty$ for all $\beta\geq C_{r}$. Thus,

(4.9)
$\begin{split}\widehat{\psi}_{r}(\beta-\varepsilon)=&\inf_{x\in[0,1]}J_{r}(x,\beta-\varepsilon)\\\
\geq\,&\inf_{x\in[0,1]}[J_{r}(x,\beta)-\theta(x,\beta)\varepsilon]\\\
\geq\,&\widehat{\psi}_{r}(\beta)-\sup_{x\in[0,1]}\theta(x,\beta)\varepsilon\rightarrow\widehat{\psi}_{r}(\beta),\qquad\varepsilon\downarrow
0.\end{split}$

####  4.2.3. Value of $\widehat{\psi}_{r}$ at the boundary

We first show that $\widehat{\psi}_{r}(C_{r})=0$. By Lemma 3.3,

(4.10)
$\begin{split}\widehat{\psi}_{r}(C_{r})=&\inf_{x\in[0,1]}\left[J_{r}(x,d_{r}(x))+(C_{r}-d_{r}(x))\sup_{\beta\in[d_{r}(x),C_{r}]}\theta(x,\beta)\right]\\\
\leq&\,\inf_{x\in[0,1]}(C_{r}-d_{r}(x))\sup_{\beta\in[d_{r}(x),C_{r}]}\theta(x,\beta)=0,\end{split}$

where we use that $C_{r}=\sup_{x\in[0,1]}d_{r}(x)$ and that $\theta(x,\beta)$
is bounded uniformly in $x$ and $\beta$. Since $\hat{r}_{1}(x,y)\equiv 1$, we
have

(4.11)
$\widehat{\psi}_{r}(1)=\inf_{x\in[0,1]}J_{r}(x,1)=\inf_{x\in[0,1]}\int_{[0,1]}\mathrm{d}y\,\log\frac{1}{r(x,y)}.$

###  4.3. Proof of Theorem 1.10

Recall from Section 3.3.5 that

(4.12) $J_{r}(x,\beta)=\frac{1}{2v_{r}(x)}\,(\beta-
d_{r}(x))^{2}+\frac{1}{6}\,\theta^{\prime\prime}(x,\beta_{*})\,(\beta-
d_{r}(x))^{3},$

for some $\beta_{*}=\beta_{*}(x)\in[d_{r}(x),\beta]$. By Lemma 3.3,
$\theta^{\prime\prime}$ is bounded uniformly in $x$ and also uniformly in
$\beta$ bounded away from 1, so it immediately follows that

(4.13)
$\hat{\psi}_{r}(\beta)\leq\inf_{x\in\mathcal{D}_{r}}J_{r}(x,\beta)\leq\inf_{x\in\mathcal{D}_{r}}\frac{1}{2v_{r}(x)}(\beta-
C_{r})^{2}[1+o(1)],\qquad\beta\downarrow C_{r}.$

Let $x_{*}=x_{*}(\beta)=\arg\min_{x\in[0,1]}J_{r}(x,\beta)$. It is clear from
the above that $\beta-C_{r}\leq\beta-d_{r}(x_{*})=O(\beta-C_{r})$. Hence

(4.14)
$\hat{\psi}_{r}(\beta)=J_{r}(x_{*},\beta)\geq\frac{1}{2v_{r}(x_{*})}(\beta-
C_{r})^{2}+O((\beta-C_{r}))^{3}),\qquad\beta\downarrow C_{r}.$

By continuity of $r$, $\inf_{x\in\mathcal{D}_{r}}|x-x_{*}|\downarrow 0$ as
$\beta\downarrow C_{r}$. Since also $v_{r}$ is continuous, this implies that
$\frac{1}{2v_{r}(x_{*})}\geq\inf_{x\in\mathcal{D}_{r}}\frac{1}{2v_{r}(x)}[1+o(1)]$
as $\beta\downarrow C_{r}$. We conclude that

(4.15)
$\hat{\psi}_{r}(\beta)\geq\inf_{x\in\mathcal{D}_{r}}\frac{1}{2v_{r}(x)}(\beta-
C_{r})^{2}[1+o(1)],\qquad\beta\downarrow C_{r},$

which settles the claim.

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# Classical and quantum cosmology with two perfect fluids: stiff matter and
radiation

F. G. Alvarenga<EMAIL_ADDRESS>Departamento de Ciências Naturais,
CEUNES, Universidade Federal do Espírito Santo, CEP 29933-415, São Mateus, ES,
Brazil.  R. Fracalossi<EMAIL_ADDRESS>Departamento de Física, CCE,
Universidade Federal do Espírito Santo, CEP 29075-910, Vitória, ES, Brazil.
R. C. Freitas<EMAIL_ADDRESS>Departamento de Física, CCE,
Universidade Federal do Espírito Santo, CEP 29075-910, Vitória, ES, Brazil.
S. V. B. Gonçalves<EMAIL_ADDRESS>Departamento de Física, CCE,
Universidade Federal do Espírito Santo, CEP 29075-910, Vitória, ES, Brazil.

###### Abstract

In this work the homogeneous and isotropic Universe of Friedmann-Robertson-
Walker is studied in the presence of two fluids: stiff matter and radiation
described by the Schutz’s formalism. We obtain to the classic case the
behaviour of the scale factor of the universe. For the quantum case the wave
packets are constructed and the wave function of the universe is found.

PACS number(s): 98.80.-k, 98.80.Cq, 98.80.Qc

##  I Introduction

Quantum cosmology Israel ; Wiltshire  is one of the aspects explored in the
quantum gravity program. Its application to the Universe as a whole takes
place at very early times the universe was small indeed. For exemple, in the
Planck time, about $10^{-44}$ s after the Big Bang, its size was the order of
$10^{-33}$ cm suggesting that quantum effects dominated Atkatz . In general
the quantum cosmological models are built considering a finite number of
degrees of freedom in the presence of geometrical symmetry inherent to the
universe. The quantization of a system subjected to these conditions is called
quantization in a minisuperspace. Furthermore, with the quantum cosmology it
is possible to study a real specific system in which quantum effects of
gravity are fundamental. As the early Universe is the best of laboratories
where different quantum theories of gravitation can be tested, quantum
cosmology has as one of its motivations to serve as a powerful auxiliary tool
in the construction of a final and fundamental quantum theory of gravity.

Considering only the cosmological point of view, quantum cosmology plays an
important role in solving a serious problem of the standard cosmological
model: The existence of an initial singularity. Mechanisms such as quantum
tunneling used in more simple cosmological models or in sophisticated
approaches such as volume quantization provided by the loop formalism are
examples of consistent solutions of this problem Bojowald . In addition, the
theory allows us to establish the initial conditions for inflation, for
primordial perturbations and for spontaneous symmetry breaking Konstantin ;
Pad . That is, the quantum cosmology is a theory of initial conditions.

Minisuperspace quantization is a reasonable approximation to describes certain
quantum gravitational effects. It is expected however that the inclusion of
other degrees of freedom in the problem must refine the results obtained and
lead to a more accurate description of the phenomenon studied, even lead to
additional technical complications. This addition of new degrees of freedom
can be done in several different ways. In the hydrodynamic description of the
material content, consistent and obvious way to refine the description of the
Universe is to use not just one but two fluids Fracalossi ; Tovar .

In this paper we choose stiff matter ($\alpha=1$) and radiation ($\alpha=1/3$)
as matter content, which play an important role in the early Universe. The
energy density of this fluid in the cosmological gauge ($N(t)=1$) is
proportional to $1/a(t)^{4}$. As the energy density of the stiff matter
Zeldovich , in the same gauge, is proportional to $1/a(t)^{6}$, there must
have been a time before the radiation in which the stiff matter dominated. In
fact, the abundance of particles produced after the Big Bang due to expansion
and cooling of the Universe is an implication of the presence of this fluid
with $\alpha=1$ at this stage Kamionkowski ; Joyce ; Salati ; Pallis ; Gomez ;
Pallis2 ; Germano .

The material content is treated here according to the Schutz formalism Schutz1
; Schutz2 . With this description we can make the degrees of freedom of the
fluid plays the role of time in the theory, transforming the Wheeler-DeWitt
equation, which governs the dynamic behaviour of the system, in a Schrödinger-
like equation whose solution is a time-dependent wave function, among other
variables. So the complex problem of the absence of a variable linked to the
time evolution in quantum gravity is solved giving to time a purely
phenomenological character leading to a well-defined Hilbert space structure.

This paper is organized as follows: Section (II) presents the Schutz’s
formalism in a classical FRW cosmological model with stiff matter and
radiation. The analytical evolution of the scale factor of the universe,
$a(t)$, is obtained. We emphasize that, since this point, time is introduced
following the Schutz’s formalism. We present next, Section (III), the quantum
model and we obtain the equation that governs the dynamics of the model, the
Wheeler-DeWitt equation, which is solved. A detailed discussion and summary of
these results is provided in Section (IV).

##  II Classical FRW model with stiff matter and radiation described by the
Schutz’s formalism

We consider a homogeneous and isotropic Universe whose geometry is described
by the FRW metric, which is given by

$ds^{2}=-N(t)^{2}dt^{2}+a(t)\left(\frac{1}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}sin^{2}\theta
d\phi^{2}\right)\;\;\;,$ (1)

where $N(t)$ is the lapse function, $k$ is the space section curvature, which
can assume the values $-1,0,1$ and $a(t)$ is the scale factor of the Universe.

The action associated to the gravitational field is given by

$S_{g}=\int\sqrt{-g}Rd^{4}x+\int_{\partial M}d^{3}x\sqrt{h}h_{ij}K^{ij}\;,$
(2)

in units such that $(8\pi G/3)^{\frac{1}{2}}=1$, $R$ is the scalar of
curvature of spacetime, $h_{ij}$ is the 3-metric on the boundary $\partial M$
of the 4-manifold $M$ and $K_{ij}$ is the extrinsic curvature. Replacing the
metric (1) in (2) and using the ADM formalism of general relativity ADM , it’s
possible to write the gravitation Hamiltonian like

$H_{g}=-\frac{p_{a}^{2}}{24a}-6ka\;.$ (3)

We consider in this model a Universe whose material content is described by
two fluids: stiff matter and radiation. To describe the dynamics of these
fluids, we use the Schutz’s formalism Schutz1 ; Schutz2 . The matter content
will consist of a perfect fluid with pressure $p$, density of total mass-
energy $\rho$, and specific enthalpy $\mu$. In this formalism, the 4-velocity
is described from five velocity-potentials

$U_{\nu}=\frac{1}{\mu}(\phi_{,\nu}+\theta S_{,\nu}+\varphi\beta_{,\nu})\;,$
(4)

where only the specific entropy $S$ has any direct physical interpretation.
Moreover, $\varphi$ and $\beta$ are associated to rotational movements and are
nulls in homogeneous and isotropic models. This 4-velocity is normalized like

$g_{\alpha\beta}U^{\alpha}U^{\beta}=-1\;,$ (5)

and with this is possible to write the enthalpy in terms of the potentials

$\mu=\frac{1}{N}(\dot{\phi}+\theta\dot{S})\,\,.$ (6)

The Schutz’s representation is associated to a variational principle whose
action is especially simple

$S=\int_{M}d^{4}x\sqrt{-g}p\;.$ (7)

The pressure of the perfect fluid can be obtained from the barotropic equation
of state $p=\alpha\rho$. The basic thermodynamic relations of the fluid take
the form Rubakov

$\rho=\rho_{0}(1+\Pi)\,;\,\quad\mu=(1+\Pi)+\frac{p}{\rho_{0}}\,;\,\quad
T_{f}dS=d\Pi+pd\bigg{(}\frac{1}{\rho_{0}}\bigg{)}\,,\,$ (8)

where $T_{f}$ is the temperature and $\Pi$ is the specific internal energy.

If we write

$d\Pi+pd\bigg{(}\frac{1}{\rho_{0}}\bigg{)}=(1+\Pi)d[ln(1+\Pi)-\alpha\,ln\rho_{0}]\,\,,$
(9)

we can identify
$T_{f}=1+\Pi\,\,\mbox{and}\,\,S=ln(1+\Pi)/{\rho_{0}}^{\alpha}$.

After a few mathematical steps, the energy density and pressure reduces to

$\rho=\left(\frac{\mu}{\alpha+1}\right)^{1+\frac{1}{\alpha}}e^{-\frac{S}{\alpha}}\,\,\,\mbox{and}\,\,\,p=\alpha\left(\frac{\mu}{\alpha+1}\right)^{1+\frac{1}{\alpha}}e^{-\frac{S}{\alpha}}\,\,.$
(10)

If we use (6) and (10) in the matter action (7), it is possible to obtain the
Lagrangian of the fluid

$L_{f}=N^{-\frac{1}{\alpha}}a^{3}\,\frac{\alpha}{(\alpha+1)^{1+\frac{1}{\alpha}}}(\dot{\phi}+\theta\dot{S})^{1+\frac{1}{\alpha}}\,e^{-\frac{S}{\alpha}}\,.$
(11)

The fluid conjugated momenta are derived from the above Lagrangian, written in
terms of the canonical variables

$p_{\phi}=N^{-\frac{1}{\alpha}}a^{3}\,(\alpha+1)^{-\frac{1}{\alpha}}\,(\dot{\phi}+\theta\dot{S})^{\frac{1}{\alpha}}\,e^{-\frac{S}{\alpha}}\,,$
(12) $p_{S}=\frac{\partial L_{f}}{\partial\dot{S}}=\theta\,p_{\phi}\,,$ (13)

and

$p_{\theta}=0\,.$ (14)

By means of the Legendre transformation

$L_{f}=\dot{\phi}p_{\phi}+\dot{S}p_{S}-N{\cal H}_{f}\,\,,$ (15)

we obtain the matter Hamiltonian

$H_{f}=\frac{p_{\phi}^{\alpha+1}e^{S}}{a^{3\alpha}}\;.$ (16)

This can be put in a more suggestive form by means of the canonical
transformations

$T=-p_{S}e^{-S}p_{\phi}^{-(\alpha+1)}\;,$ (17)
${\overline{\phi}}=\phi-(\alpha+1)\frac{p_{S}}{p_{\phi}}\;,$ (18)
$p_{T}=p_{\phi}^{(\alpha+1)}e^{S}\;,$ (19)

and

${\overline{p}}_{\phi}=p_{\phi}\;.$ (20)

With this, the equation (16) is written in the form

$H_{f}=\frac{p_{T}}{a^{3\alpha}}\;.$ (21)

So, the total Hamiltonian will be

$\mathcal{H}=\mathcal{H}_{g}+\mathcal{H}_{f}\;,$
$\mathcal{H}=-\frac{p_{a}^{2}}{24a}-6ka+\frac{p_{T}}{a^{3\alpha}}\;.$ (22)

Note that the Hamiltonian is linear in one of the momenta and it’s possible to
introduce the variable $T$ as global phase time Hajicek .

The ADM formalism also shows that the lapse function $N$ acts like a Lagrange
multiplier of the system so that

$\mathcal{H}\approx 0\;,$ (23)

where the symbol $\approx$ which means zero weakly.

We now consider two non-interacting perfect fluids. Then, we can write

$H=N\mathcal{H}=N\left(-\frac{p_{a}^{2}}{24a}-6ka+\frac{p_{T}}{a^{3\alpha}}+\frac{p_{\sigma}}{a^{3\beta}}\right)\;.$
(24)

At this point, we particularize to the radiation fluid, ($\alpha=\frac{1}{3}$)
and stiff matter ($\beta=1$) and we consider the case of null curvature
($k=0)$ so that the equation (24) can be write in the form

$H=N\mathcal{H}=N\left(-\frac{p_{a}^{2}}{24a}+\frac{p_{T}}{a}+\frac{p_{\sigma}}{a^{3}}\right)\;.$
(25)

Using the Hamilton equations, we can write

$\dot{p_{a}}=-\frac{\partial(N\mathcal{H})}{\partial
a}=-N\left(\frac{p_{a}^{2}}{24a^{2}}-\frac{p_{T}}{a^{2}}-3\frac{p_{\sigma}}{a^{4}}\right)\;,$
(26) $\dot{p_{T}}=-\frac{\partial(N\mathcal{H})}{\partial
T}=0\;\;\rightarrow\;\;p_{T}=const.\;,$ (27)
$\dot{p_{\sigma}}=-\frac{\partial(N\mathcal{H})}{\partial\sigma}=0\;\;\rightarrow\;\;p_{\sigma}=const.\;,$
(28) $\dot{a}=\frac{\partial(N\mathcal{H})}{\partial
p_{a}}=-\frac{Np_{a}}{12a}\;,$ (29)
$\dot{T}=\frac{\partial(N\mathcal{H})}{\partial p_{T}}=\frac{N}{a}\;,$ (30)
$\dot{\sigma}=\frac{\partial(N\mathcal{H})}{\partial
p_{\sigma}}=\frac{N}{a^{3}}\quad.$ (31)

Let us choose $t=T$, so that $N=a$ in (30). Then, $T$ must be the conformal
time, so that the role of time in the theory is realized by the radiation
fluid. Using the gauge $N=a$, the Hamilton equations and the constraint
equation (23), we can obtain the equation of classical dynamic for the scale
factor $a$

${\dot{a}}^{2}-\frac{1}{6}\big{(}p_{T}+\frac{p_{\sigma}}{a^{2}}\big{)}=0\;,$
(32)

or

$\ddot{a}+\frac{1}{6}\frac{p_{\sigma}}{a^{3}}=0\;,$ (33)

Note that comparing equation (32) with the Friedmann’s equation obtained from
Einstein’s equation allow us to identify

$\displaystyle
p_{\sigma}=\frac{M_{\mathrm{Pl}}^{-2}}{2}\rho_{\mathrm{s}}^{*}\,,$ (34)
$\displaystyle p_{T}=\frac{M_{\mathrm{Pl}}^{-2}}{2}\rho_{\mathrm{r}}^{*}\,,$
(35)

where $M_{\mathrm{Pl}}$ is the reduce Planck mass and $\rho_{\mathrm{s}}^{*}$
and $\rho_{\mathrm{r}}^{*}$ are the stiff matter and radiation densities when
$a=a_{*}$.

We can now integrate equation (32) and find that

$a(\tau)=\sqrt{rA^{2}t^{2}+2At}\,$ (36)

with $r=\frac{\rho_{\mathrm{r}}^{*}}{\rho_{\mathrm{s}}^{*}}$ and
$A^{2}=\frac{M_{\mathrm{Pl}}^{-2}}{3}\rho_{\mathrm{s}}^{*}$.

The conformal time $t$ is related to the cosmic time $\tau$ via
$d\tau=N(t)dt=a(t)d\tau$ and with help of equation (36) we find

$\tau=\frac{\ln{\left(\sqrt{2rA^{3}}\right)}}{\sqrt{r^{3}A^{2}}}+\frac{a(\tau)}{2}\left(\frac{1}{rA}+t\right)-\frac{\ln{\left(rA^{2}t^{1/2}+\sqrt{2rA^{3}+r^{2}A^{4}t}\right)}}{\sqrt{r^{3}A^{2}}}\,.$
(37)

Using equations (36) and (37) we can plot the scale factor $a$ as a function
of the cosmic time $\tau$. In Figure 1 we show the classical behavior of the
scale factor for different values of stiff matter and radiation densities at a
fixed point $a_{*}=1$. The scale factor growth rate is more sensitive to the
stiff matter quantity, but since it value falls very rapidly the scale factor
will approach very fast the behavior of a universe filled with radiation. In
this scenario, the universe containing stiff matter and radiation is singular.

Figure 1: Classical behavior of a two fluids model: stiff matter and radiation

Of course, for small values of $t$ e $a$, the classical formalism do not take
into account the important quantum effects of this phase. It is therefore
necessary to quantize the theory obtained.

##  III Quantum model

The quantum model is constructed making the quantification of canonically
associated moments in equation (25) following the procedure proposed by Dirac
Dirac . The associated moments to $a$, $T$ e $\sigma$ coordinates are

$\hat{p}_{a}=-i\frac{\partial}{\partial
a}\;\;\;,\hat{p}_{T}=-i\frac{\partial}{\partial
T}\;\;\;,\hat{p}_{\sigma}=-i\frac{\partial}{\partial\sigma}\;.$

Thus, the Hamiltonian constraint $\mathcal{H}$ becomes the $\hat{H}$ operator,
which acts like the annihilator of the wave function of the universe $\Psi$.
Then,

$\left(-\frac{\partial^{2}}{\partial
a^{2}}-144ka^{2}+\frac{24i}{a^{3\alpha-1}}\frac{\partial}{\partial
t}+\frac{24i}{a^{3\beta-1}}\frac{\partial}{\partial\sigma}\right)\Psi(a,\sigma,t)=0\;.$
(38)

The above equation have the form of a Schrödinger equation. This is a
consequence of the Schutz’s formalism used to describe the material content of
the universe. Moreover, the scale factor of the universe is defined in the
domain $(0,\infty]$. This means that the operator $\hat{H}$ is not, in
general, self-adjoint. So in order to have a unitary evolution, the domain of
$\hat{H}$ operator defined above must be:
$D(\hat{H}):C_{c}^{\infty}(\mathbb{R}^{+}_{*}\times\mathbb{R}\times\mathbb{R}^{+};\frac{\partial}{\partial
a}\Psi_{(a=0)}=\alpha\Psi_{(a=0)})$.

The function $\Psi$ can be write in the form of a stationary wave, i.e.,

$\Psi(a,\sigma,t)=\psi(a,\sigma)e^{iEt}\;.$ (39)

So

$\left(-\frac{\partial^{2}}{\partial
a^{2}}-144ka^{2}-\frac{24E}{a^{3\alpha-1}}+\frac{24i}{a^{3\beta-1}}\frac{\partial}{\partial\sigma}\right)\psi(a,\sigma)=0\;.$
(40)

Separating variables, we can write

$\psi(a,\sigma)=R(a)\Sigma(\sigma)\;.$

Then,

$-\frac{1}{R(a)}\frac{\partial^{2}}{\partial
a^{2}}R(a)-144ka^{2}-24Ea^{1-3\alpha}=-24ia^{1-3\beta}\frac{1}{\Sigma(\sigma)}\frac{\partial}{\partial\sigma}\Sigma(\sigma)\;.$
(41)

We can adopt the solution of $\sigma$:

$\Sigma(\sigma)=Ae^{-n\sigma}\;,$ (42)

with $n\in\mathbb{C}$ and $A$ constant. Then

$\frac{d^{2}R(a)}{da^{2}}+(144ka^{2}+24Ea^{1-3\alpha}+24ina^{1-3\beta})R(a)=0\;.$
(43)

We restrict the analysis to $\alpha=1/3$, $\beta=1$ and $k=0$ case. With the
solution of (43), we can write the solution of (38) for the wave function
$\Psi$ as

$\Psi_{En}(a,\sigma,t)=\sqrt{a}\left[CJ_{\frac{\sqrt{1-96in}}{2}}(\sqrt{24E}a)+DY_{\frac{\sqrt{1-96in}}{2}}(\sqrt{24E}a)\right]e^{-n\sigma+iEt}\;,$
(44)

with $C$ and $D$ constants. Submitting the wave function to the boundary
condition $\Psi(0,\sigma,t)=0$, we obtain

$\Psi_{En}(a,\sigma,t)=C\sqrt{a}J_{\frac{\sqrt{1-96in}}{2}}(\sqrt{24E}a)e^{-n\sigma+iEt}\;.$
(45)

Although boundary conditions, the above wave function is not square
integrable. It is therefore necessary to construct the wave packets in $E$ and
$n$. With this, the wave packet can be written in the form

$\Psi_{En}(a,\sigma,t)=C\sqrt{a}\int dn\,g(n)e^{-n\sigma}\int
dEf(E)J_{\frac{\sqrt{1-96in}}{2}}(\sqrt{24E}a)e^{iEt}\;.$ (46)

If we do $\sqrt{24E}=q$ and make the choice

$f(q)=12q^{\frac{\sqrt{1-96in}}{2}}e^{-\gamma
q^{2}}\;,\,\,\,\gamma\in\mathbb{R}\quad,$

we can construct the wave packet in $E$, which is written in terms of
exponential functions

$\Psi_{qn}(a,\sigma,t)=\frac{C\,\sqrt{a}}{12}\int dn\,g(n)e^{-n\sigma}\int
dq\,q^{\frac{\sqrt{1-96in}}{2}+1}J_{\frac{\sqrt{1-96in}}{2}}(qa)e^{(-\gamma+\frac{it}{24})q^{2}}\;.$
(47)

The above integral can be solved analytically e gives abra

$\Psi_{n}(a,\sigma,t)=\frac{C\sqrt{a}}{2\gamma-\frac{it}{12}}e^{-\frac{a^{2}}{4\gamma-\frac{it}{6}}}\int
dng(n)e^{-n\sigma}\left(\frac{a}{2\gamma-\frac{it}{12}}\right)^{\frac{\sqrt{1-96in}}{2}}\;,$
(48)

subject to the following conditions $Re(\gamma-\frac{it}{24})>0$ e
$Re(\frac{\sqrt{1-96in}}{2})>-1$. The above integration can also be solved
analytically if we do $\frac{1-96in}{4}=u^{2}$ and $g(n)\rightarrow
h(u)=-\frac{e^{-\xi u^{2}}}{2i\sqrt{6\pi}Cu}$, $\xi\in\mathbb{R}$. So, we
obtain

$\displaystyle\Psi(a,\sigma,t)$ $\displaystyle=$
$\displaystyle\frac{6\sqrt{a}}{(24-it)\sqrt{24\xi-i\sigma}}\left(\frac{12ia}{t+24i\gamma}\right)^{\frac{12}{24\xi-i\sigma}}e^{\frac{i\sigma}{96}-\frac{6a^{2}}{24\gamma-
it}}\Bigg{[}1+$ (49) $\displaystyle+$
$\displaystyle\mathrm{erf}\left(\sqrt{\frac{6}{24\xi-i\sigma}}\mathrm{ln}\left(\frac{12ia}{t+24i\gamma}\right)\right)\Bigg{]}\;,$

where $\mathrm{erf}(x)$ is the error function (see Appendix). This type of
function has no closed form and their values for different arguments must be
obtained numerically.

According to the Figures 2 and 3, it is easy to verify that
$\Psi(a,\sigma,t)\rightarrow 0$ when the scale factor vanish, $a\rightarrow
0$, which is consistent with the imposed boundary conditions. In this case,
defining the probability density $P(a)=|\Psi(a)|^{2}$, it is noted that
$P(a)\rightarrow 0$ when $a\rightarrow 0$ Papa  so that the singularity,
observed in the classical version, is avoided at this quantum model.

Figure 2: The unormalized wave packet in the $a\times\sigma$ plane for fixed
$t$ with various combinations of the free parameters $\gamma$ and $\xi$.

Figure 3: The unormalized wave packet in the $a\times t$ plane for fixed
$\sigma$ with various combinations of the free parameters $\gamma$ and $\xi$.

Figure 4: The unormalized wave packet in the $\sigma\times t$ plane for fixed
$a$ and with various combinations of the free parameters $\gamma$ and $\xi$.

Figure 5: The unormalized wave packet as a function of only one of the three
variables, with the other two fixed and for various combinations of the free
parameters $\gamma$ and $\xi$.

Figure 6: The dependence of the unormalized wave packet in relation to the
free parameters $\gamma$ and $\xi$, for fixed values of the variables $a$,
$\sigma$ and $t$.

##  IV Conclusions

In this paper, we study the quantum FRW model with stiff matter and radiation
and we solve the Wheeler-DeWitt equation in minisuperspace to obtain the wave
function of the corresponding universe. The perfect fluid is described by the
Schutz’s canonical formalism, which allows to attribute dynamical degrees of
freedom to matter.

In the classical model, the universe of null curvature expands forever from an
initial singularity, as can be seen in equations (36), (37) and in Figure 1.
Indeed, this singular point can be avoided in quantum cosmology. In quantum
treatment the model of fluids was investigated on the basis of the associated
Wheeler-DeWitt equation. The introduction of a time variable
phenomenologically using the degrees of freedom of radiation fluid allows to
obtain a Hamiltonian constraint linear in one of the momenta and the Wheeler-
DeWitt equation can be reduced to a Schrödinger like equation, whose solution
is the wave function of the universe. Applying the boundary conditions, this
function was explicitly written and the wave packets was constructed
analytically.

The behavior of the wave packet (49) is showed in Figures 2, 3 and 4 for each
two pairs of variables $a$, $\sigma$ and $t$. The packet goes to $0$ as the
variables grow, but the decaying rate depends on the values of the third fixed
variable and the free parameters $\gamma$ and $\xi$. It is worth to note that
in equation (49) the variable $t$ appears many times accompanied by the
parameter $\gamma$ and $\sigma$ by $\xi$. This fact can be used as a manner to
regulate how fast the packet approaches zero for the directions $\sigma$ and
$t$. In Figure 5 we show the cross section of the wave packet and see how
different combinations of values for the variables and the parameters can
change the intensity and position of the peak, as the decaying rate. The
dependence in relation to the free parameter is showed in Figure 6.

The behavior of the wave packet, shown in Figures (2) and (3), makes it clear
that $\Psi(a,\sigma,t)$ tends to zero when $a$ tends to zero. Because of this,
the probability density $P(a)$ tends to zero when $a$ tends to zero, so that
the classical singularity is avoided, unlike what happens in the classical
case.

As perspective to future work, the investigation of new scenarios containing
other types of fluid is the natural way. It is hoped that this will highlight
the role played by new degrees of freedom added to cosmological models, as
well as contribute to the discussion about the quantization procedures applied
to cosmology.

## Acknowledgement

This work has received financial supporting from FAPES (Brazil).

##  Appendix A The error function

Error function $\mbox{erf}~{}x$ is given as abra

$\mbox{erf}~{}x=\frac{2}{\sqrt{\pi}}\int^{x}_{0}~{}e^{-t^{2}}~{}dt\quad.$ (50)

There is a direct connection between the error function and the Gaussian
function and the normalized Gaussian function, that are given by

$\displaystyle f(x)$ $\displaystyle=$ $\displaystyle
a~{}e^{\frac{-(x-b)^{2}}{2\sigma^{2}}}\quad,$ $\displaystyle g(x)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}~{}e^{\frac{-(x-b)^{2}}{2\sigma^{2}}}\quad,$
(51)

respectively, where $a,b$ are constants and $\sigma$ is the standard
deviation, sometimes called the Gaussian RMS width. The graph of a Gaussian is
a characteristic symmetric bell curve shape with the parameter $a$ being the
height of the curve’s peak, $b$ is the position of the center of the peak and
$\sigma$ controls the width of the bell.

With the substitution

$\displaystyle t^{2}$ $\displaystyle=$
$\displaystyle\frac{(x-b)^{2}}{2\sigma^{2}}\quad,$ $\displaystyle dt$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}\sigma}~{}dx\quad,$ (52)

we can integrated the normalized Gaussian function between $-x$ and $+x$ and
obtain

$\int^{+x}_{-x}~{}g(x)~{}dx=\frac{2}{\sqrt{\pi}}\int^{x}_{0}~{}e^{-t^{2}}~{}dt=\mbox{erf}~{}x\quad,$
(53)

since the normalized Gaussian is symmetric about the $y$-axis.

Some properties and approximations of the error function are

$\displaystyle\mbox{erf}~{}(-x)$ $\displaystyle=$
$\displaystyle-\mbox{erf}~{}x\quad,$ (54) $\displaystyle\mbox{erf}~{}0$
$\displaystyle=$ $\displaystyle 0\quad,$ (55)
$\displaystyle\mbox{erf}~{}(x^{*})$ $\displaystyle=$
$\displaystyle(\mbox{erf}~{}x)^{*}~{}~{}~{}~{}\mbox{where}~{}^{*}~{}\mbox{denotes
complex conjugation}\quad,$ (56) $\displaystyle\mbox{erf}~{}\infty$
$\displaystyle=$ $\displaystyle 1\quad,$ (57)
$\displaystyle\mbox{erf}~{}(-\infty)$ $\displaystyle=$ $\displaystyle-1\quad,$
(58) $\displaystyle\mbox{erf}~{}x$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi}}\int^{x}_{0}~{}\sum_{n=0}^{\infty}~{}\frac{(-1)^{n}t^{2n}}{n!}~{}dt~{}~{}~{}~{}\mbox{for}~{}x<<1\quad,$
(59) $\displaystyle\mbox{erf}~{}x$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi}}~{}\sum_{n=0}^{\infty}~{}\frac{(-1)^{n}x^{2n+1}}{(2n+1)n!}\quad,$
(60) $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi}}\biggl{(}x-\frac{x^{3}}{3}+\frac{x^{5}}{10}-\frac{x^{7}}{42}+...\biggr{)}~{}~{}~{}~{}\mbox{for}~{}x<<1\quad,$
$\displaystyle\mbox{erf}~{}x$ $\displaystyle=$ $\displaystyle
1-\frac{e^{-x^{2}}}{\sqrt{\pi}~{}x}\biggl{(}1-\frac{1}{2x^{2}}-...\biggr{)}~{}~{}~{}~{}\mbox{for}~{}x>>1\quad.$
(61)

The complementary error function, commonly denoted erfc $x$, is defined by

$\displaystyle\mbox{erfc}~{}x$ $\displaystyle=$ $\displaystyle
1-\mbox{erf}~{}x\quad,$ (62) $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}~{}e^{-t^{2}}~{}dt\quad.$

Some special values, approximations and results of erfc $x$ are

$\displaystyle\mbox{erfc}~{}(-x)$ $\displaystyle=$ $\displaystyle
2-\mbox{erfc}~{}x\quad,$ (63) $\displaystyle\mbox{erfc}~{}\infty$
$\displaystyle=$ $\displaystyle 0\quad,$ (64)
$\displaystyle\mbox{erfc}~{}(-\infty)$ $\displaystyle=$ $\displaystyle
2\quad,$ (65) $\displaystyle\mbox{erfc}~{}0$ $\displaystyle=$ $\displaystyle
1\quad,$ (66) $\displaystyle\mbox{erfc}~{}x$ $\displaystyle=$
$\displaystyle\frac{\Gamma(1/2,x^{2})}{\sqrt{\pi}}\quad,~{}~{}~{}~{}\mbox{where}~{}\Gamma(a,y)~{}\mbox{is
the gamma function}$ (67)
$\displaystyle\int_{0}^{\infty}~{}\mbox{erfc}~{}x~{}dx$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\pi}}\quad.$ (68)

We show in Figure (7) the behaviour of the special functions treated here in
this appendix.

Figure 7: The error and the complementary error functions.

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