Emergence of extended states at zero in the spectrum of sparse random graphs

Definition 2.1

A left (resp. right) Zinbiel algebra is a vector space $\mathcal{A}$ endowed with a bilinear product $\ast$ satisfying, for all $x,y,z\in\mathcal{A}$ ,

\displaystyle(x\ast y)\ast z=x\ast(y\ast z)+x\ast(z\ast y),

(2.1.1)

\displaystyle(\quad\mbox{resp.}\quad\;x\ast(y\ast z)=(x\ast y)\ast z+(y\ast x)% \ast z\;\quad),

(2.1.2)

or, equivalently,

\displaystyle(x,y,z)=x\ast(z\ast y)

(2.1.3)

\displaystyle(\quad\mbox{resp.}\quad\;(x,y,z)=-(y\ast x)\ast z)\quad),

(2.1.4)

where, $\forall x,y,z\in\mathcal{A},(x,y,x)=(x\ast y)\ast z-x\ast(y\ast z)$ is the associator associated to $\ast$ .

Definition 2.13

[ Ni_B ] Let $\displaystyle(\mathcal{G},\cdot)$ and $\displaystyle(\mathcal{H},\circ)$ be two commutative associative algebras, and ${\displaystyle\rho:\mathcal{H}\rightarrow\mathfrak{gl}(\mathcal{G})}$ and ${\displaystyle\mu:\mathcal{G}\rightarrow\mathfrak{gl}(\mathcal{H})}$ be two ${\mathcal{K}}$ -linear maps which are representations of ${\mathcal{H}}$ and ${\mathcal{G}}$ , respectively, satisfying the following relations: $\mbox{for all }x,y\in\mathcal{G}$ and all ${\displaystyle a,b\in~{}\mathcal{H}},$

\displaystyle{\mu(x)(a\circ b)=(\mu(x)a)\circ b+\mu(\rho(a)x)b},

(2.4.1)

\displaystyle{\rho(a)(x\cdot y)=(\rho(a)x)\cdot b+\rho(\mu(x)a)y}.

(2.4.2)

Then, ${\displaystyle(\mathcal{G},\mathcal{H},\rho,\mu)}$ is called a matched pair of the commutative associative algebras ${\displaystyle\mathcal{G}}$ and ${\displaystyle\mathcal{H}},$ denoted by ${\displaystyle\mathcal{G}\bowtie^{\mu}_{\rho}\mathcal{H}}.$

In this case, ${\displaystyle(\mathcal{G}\oplus\mathcal{H},\ast)}$ defines a commutative associative algebra with respect to the product ${\ast},$ given, for all ${x,y\in\mathcal{G}}$ and all ${a,b\in\mathcal{H}}$ , by

${\displaystyle(x+a)\ast(y+b)=x\cdot y+\mu(a)y+\mu(b)x+a\circ b+\rho(x)b+\rho(y% )a}$ .

Definition 3.6 .

Let $\sim$ denote the generalization of the equivalence relation ( 3.1 ) to the case of directed metric graphs,

(G,\lambda,o)\sim(G^{\prime},\lambda^{\prime},o^{\prime})\Longleftrightarrow% \begin{cases}\exists\varphi:G\longrightarrow G^{\prime}\text{ homothety and }% \forall e\in E(G):\\ o(e)=(x,y)\Longrightarrow o^{\prime}(\varphi(e))=(\varphi(x),\varphi(y)).\end{cases}

Then for $n,s\in\mathbb{N}$ the moduli space of directed graphs is defined as

\mathcal{MDG}_{n,s}:=\left\{(G,\{v_{1},\ldots,v_{s}\},\lambda,o)\;\middle|\;% \begin{tabular}[]{@{}l@{}}G adm.\ with $s$ legs, $|G|=n$,\\ directed by $o:E\rightarrow V^{2}$, $\lambda$ regular\end{tabular}\right\}_{% \Big{/}\sim}.

Definition 2

(Modulo operations) The notation $x\bmod a$ denotes reducing $x\in\mathbb{Z}$ modulo the integer interval $[-a,a)$ . That is,

x\bmod a=x-b\cdot\left[a-(-a)\right],

where $b\in\mathbb{Z}$ is the (unique) integer such that

x-b\cdot\left[a-(-a)\right]\in[-a,a).

Definition 3 .

Let $(V,B)$ be an anti-Hermitian space. Define a right action $\leftharpoonup$ of the multiplicative monoid $(R,\cdot)$ on $\mathfrak{H}$ by

(v,x)\leftharpoonup y=(vy,\bar{y}\bar{1}^{-1}xy).

This action has the property that $(\alpha\overset{.}{+}\beta)\leftharpoonup y=\alpha\leftharpoonup y\overset{.}{% +}\beta\leftharpoonup y$ for any $\alpha,\beta\in\mathfrak{H}$ and $y\in R$ .

Definition 2.4 .

A mapping $f\colon X\to Y$ is topologically equivalent to $g\colon X^{\prime}\to Y^{\prime}$ if there exists homeomorphisms $\phi$ and $\psi$ such that

f=\psi^{-1}\circ g\circ\phi.

In other words the following diagram commutes:

\tikzcd X\arrow{r}{f}\arrow[swap]{d}{\phi}&Y\arrow{d}{\psi}\\ X^{\prime}\arrow{r}{g}&Y^{\prime}.

Definition 1 .

Algebra $(Q,\cdot,/,\backslash)$ with three binary operations that satisfies the following identities:

x\cdot(x\backslash y)=y,

(1)

(y/x)\cdot x=y,

(2)

x\backslash(x\cdot y)=y,

(3)

(y\cdot x)/x=y,

(4)

is called an equational quasigroup (often it is called a quasigroup).

Definition 2 .

A quasigroup $(Q,\cdot)$ is said to be Neumann quasigroup if in this quasigroup the identity

x\cdot(yz\cdot yx)=z

(5)

holds true [ 8 , 12 , 7 ] , [ 15 , p. 248] .

Definition 11 .

A bijection $\theta$ of a set $Q$ is called a right pseudoautomorphism of a quasigroup $(Q,\cdot)$ if there exists at least one element $c\in Q$ such that

\theta x\cdot(\theta y\cdot c)=(\theta(x\cdot y))\cdot c

(9)

for all $x,y\in Q$ , i.e., $(\theta,R_{c}\theta,R_{c}\theta)$ is an autotopy a quasigroup $(Q,\cdot)$ . The element $c$ is called a companion of $\theta$ [ 11 , 9 , 13 ] .

A bijection $\theta$ of a set $Q$ is called a left pseudoautomorphism of a quasigroup $(Q,\cdot)$ if there exists at least one element $c\in Q$ such that

(c\cdot\theta x)\cdot\theta y=c\cdot(\theta(x\cdot y))

(10)

for all $x,y\in Q$ , i.e., $(L_{c}\theta,\theta,L_{c}\theta)$ is an autotopy a quasigroup $(Q,\cdot)$ . The element $c$ is called a companion of left pseudoautomorphism $\theta$ [ 11 , 9 , 13 ] .

Definition 20 .

Quasigroup $(Q,\cdot)$ with identity

yz\cdot yx=xz

(15)

is called Schweizer quasigroup [ 7 , p. 313] .

Definition 3 .

We say that a holomorphic function $f$ on $\mathbb{H}$ is a mock modular form of weight $k$ and multiplier $\chi$ on $\Gamma$ , if and only if it exists a weight $2-k$ cusp form $g$ on $\Gamma$ such that the non-holomorphic completion of $f$ , defined as

\hat{f}(\tau)=f(\tau)-g^{\ast}(\tau)

satisfies $\hat{f}=\hat{f}\lvert_{k,\chi}\gamma$ for every $\gamma\in\Gamma$ . In the above, we defined the non-holomorphic Eichler integral

g^{\ast}(\tau):={C\int_{-\bar{\tau}}^{i\infty}(\tau^{\prime}+\tau)^{-k}% \overline{g(-\bar{\tau}^{\prime})}\,d\tau^{\prime}}

(7.33)

for $\tau\in\mathbb{H}$ .

Definition 4 ( $\mathsf{First\text{-}order~{}critical~{}points}$ ) .

A first-order critical point (stationary point) ${\bm{x}}$ of $f(\cdot)$ is any point that satisfies

\nabla f({\bm{x}})=\bm{0}.

Definition 8.2 .

A global $C$ -definable type is generically stable if

p(x)\otimes p(y)=p(y)\otimes p(x).

Given a definable set $X$ in $k$ , we let $\widehat{X}(C)$ denote the set of generically stable types on $X$ , that is, those types containing a formula which defines $X$ .

Definition 3.1 .

Let $H$ be an algebra generated by the elements $a$ , $b$ , $c$ , $d$ satisfying the relations

(4)

\displaystyle\begin{split}\displaystyle a^{6}=1,\quad b^{2}=0,\quad c^{2}=0,% \quad d^{6}=1,\quad a^{2}=d^{2},\quad ad=da,\quad bc=0=cb,\\ \displaystyle ab=\xi ba,\quad ac=\xi ca,\quad db=-\xi bd,\quad dc=-\xi cd,% \quad bd=ca,\quad ba=cd.\end{split}

Definition 4.1 .

The Hopf algebra $K$ is the algebra generated by the elements $a$ , $b$ , $c$ satisfying the relations

(27)

\displaystyle a^{6}=1,\quad b^{2}=0,\quad c^{2}=1,\quad ba=\xi ab,\quad ac=ca,% \quad bc=cb,

with the coalgebra structure and antipode given by

(28)		$\displaystyle\Delta(a)=a\otimes a+(\xi^{4}+\xi^{5})b\otimes ba^{3},\quad\Delta% (b)=b\otimes a^{4}+a\otimes b,\quad\Delta(c)=c\otimes c,$
(29)		$\displaystyle\epsilon(a)=1,\quad\epsilon(b)=0,\quad\epsilon(c)=1,\quad S(a)=a^% {5},\quad S(b)=\xi^{-2}ba,\quad S(c)=c.$

Definition 2.3 .

A negation map on a $\mathcal{T}$ -module $\mathcal{A}$ is a monoid isomorphism $(-):\mathcal{A}\to\mathcal{A}$ of order $\leq 2,$ written $a\mapsto(-)a$ , which also respects the $\mathcal{T}$ -action in the sense that

(-)(ab)=a((-)b),

for $a\in\mathcal{T},$ $b\in\mathcal{A}.$

Definition 4.28 .

A (Systemic) Morita context is a six-tuple $(\mathcal{A},\mathcal{A}^{\prime},\mathcal{M},\mathcal{M}^{\prime},\tau,\tau^{% \prime})$ where $\mathcal{A},\mathcal{A}^{\prime}$ are systems, $\mathcal{M}$ is an $\mathcal{A}-\mathcal{A}^{\prime}$ bimodule, $\mathcal{M}^{\prime}$ is an $\mathcal{A}^{\prime}-\mathcal{A}$ bimodule, and

\tau:\mathcal{M}\otimes_{\mathcal{A}^{\prime}}\mathcal{M}^{\prime}\to\mathcal{% A},\qquad\tau^{\prime}:\mathcal{M}^{\prime}\otimes_{\mathcal{A}}\mathcal{M}\to% \mathcal{A}^{\prime}

are morphisms, linear on each side over ${\mathcal{A}}$ and ${\mathcal{A}^{\prime}}$ respectively, which satisfy the following equations, writing $(x,x^{\prime})$ for $\tau(x,x^{\prime})$ and $[x^{\prime},x]$ for $\tau(x^{\prime},x)$ :

(i)

$(x,x^{\prime})y=x[x^{\prime},y].$
(ii)

$x^{\prime}(x,y^{\prime})=[x^{\prime},x]y^{\prime}.$

Definition 1 .

A Hilbert algebra is an algebra $(H,\rightarrow,1)$ of type $(2,0)$ which satisfies the following conditions for every $a,b,c\in H$ :

a)

$a\rightarrow(b\rightarrow a)=1$ ,
b)

$(a\rightarrow(b\rightarrow c))\rightarrow((a\rightarrow b)\rightarrow(a% \rightarrow c))=1$ ,
c)

if $a\rightarrow b=b\rightarrow a=1$ then $a=b$ .

Definition 2.4 .

A Frobenius form on an axial algebra $A$ is a (symmetric) bilinear form $(\cdot,\cdot):A\times A\to\mathbb{F}$ such that the form associates with the algebra product. That is, for all $x,y,z\in A$ ,

(x,yz)=(xy,z).

Definition 2.15 .

Let $\mathbb{M}^{{}^{\prime}}_{2}$ be the free abelian group generated by $\{\alpha,\beta\}$ for $\alpha,\beta\in\mathbb{P}^{1}(\mathbb{Q})$ . Let $\mathbb{M}_{2}$ be the quotient of $\mathbb{M}^{{}^{\prime}}_{2}$ by the relations

\{\alpha,\beta\}+\{\beta,\gamma\}+\{\gamma,\alpha\}=0.

We define the action $\mathop{\rm GL}\nolimits_{2}(\mathbb{Q})$ on $\mathbb{M}_{2}$ by

g\{\alpha,\beta\}=\{g(\alpha),g(\beta)\}\,\,\,\text{for all}\,\,g\in\mathop{% \rm GL}\nolimits_{2}(\mathbb{Q}).

Definition 2.1 .

A propositional assignment is a function $e:X\to[0,1]$ . This function can be naturally extended to a propositional Gödel evaluation over $\mathcal{L}_{0}(X)$ by the following recursive definitions:

•

$e(\bot)=0$
•

$e(\phi\land\psi)=e(\phi)\odot e(\psi)$
•

$e(\phi\rightarrow\psi)=e(\phi)\Rightarrow e(\psi)$

Definition 19 .

A residuated lattice $(L,\vee,\wedge,\odot,\rightarrow,0,1)$ is called prelinear if it satisfies the identity

(x\rightarrow y)\vee(y\rightarrow x)\approx 1.

Definition 6 .

Let ${\cal A}=(Q,L_{I},L_{O}^{\delta},T,q_{0})$ be an SA, $q\in Q$ , $Q^{\prime}\subseteq Q$ , $\mu\in L$ , $\rho\in L^{*}$ , and $\epsilon$ the empty sequence. Then we define:

\displaystyle{{q}\mathbin{\textit{after}}{\epsilon}}=\{q\}

\displaystyle{{q}\mathbin{\textit{after}}{\mu\rho}}=\begin{cases}{{T(q,\mu)}% \mathbin{\textit{after}}{\rho}}&\text{if }T(q,\mu){\downarrow\!}\\ \emptyset&\text{otherwise}\end{cases}

\displaystyle{{{\cal A}}\mathbin{\textit{after}}{\rho}}={{q_{0}}\mathbin{% \textit{after}}{\rho}}

\displaystyle\mathop{\textit{out}}({Q^{\prime}})=\bigcup_{q^{\prime}\in Q^{% \prime}}\mathop{\textit{out}}({q^{\prime}})

\displaystyle\mathop{\textit{traces}}({{\cal A}})=\{\rho^{\prime}\in L^{*}\mid% {{{\cal A}}\mathbin{\textit{after}}{\rho^{\prime}}}\neq\emptyset\}

Definition 5.12 .

Let $\alpha$ be a finite even sequence. We say that $\alpha$ is evenly-composite if there exists an even sequence $\beta$ such that

\alpha=\beta\oplus\beta\oplus\ldots\oplus\beta.

Otherwise we say that $\alpha$ is evenly-prime .

Definition 1.1 .

Let $n\geq 1$ and $q$ be a primitive $2n$ -th root of unity. The Hopf algebra $H_{4n}$ is defined as follows. As an algebra it generated by $z,x$ with relations

z^{2n}=1,\quad zx=qxz,\quad x^{2}=0

for any $a\in k$ .

The coalgebra structure is

	$\displaystyle\Delta(z)=z\otimes z+a(1-q^{-2})z^{n+1}x\otimes zx,\quad\Delta(x)% =x\otimes 1+z^{n}\otimes x;$
	$\displaystyle\epsilon(z)=1,\quad\epsilon(x)=0,$
	$\displaystyle S(z)=z^{-1},\quad S(x)=-z^{n}x.$

5.1 Definition .

Given $\alpha\in{\mathbb{R}}$ and a path $\xi$ , we define

\rho(\alpha,\xi)=\rho(\alpha,e)\qquad\hbox{ if $\xi=e$ .}

If $\xi=(e_{i})_{i=1}^{M}$ , for $M>1$ , we set $\bar{\xi}=(e_{i})_{i=1}^{M-1}$ and define

\rho(\alpha,\xi)=\rho(\rho(\alpha,\bar{\xi}),e_{M}).

Definition 2.15 .

Let $G$ be an augmented Lie group. An augmented right $G-$ space is a right $G-$ space $A$ equipped with a continuous map $\epsilon:A\longrightarrow C$ such that

\epsilon(a\cdot g)=\epsilon(a)\cdot\epsilon(g)

for all $a\in A$ and all $g\in G$ .

Definition 1 (Lifted matching) .

A matching $\mu\subseteq\mathcal{U}_{a}\times\mathcal{U}_{b}$ gives rise to a lifted matching $\ell(\mu)\subseteq\binom{\mathcal{U}_{a}}{2}\times\binom{\mathcal{U}_{b}}{2}$ ,

\ell(\mu)=\left\{(\alpha(w),\beta(w)):w\in\binom{\mu}{2}\right\}.

Definition 5.1 .

Let $\mathcal{A}$ be a Banach algebra and $a,b\in\mathcal{A}$ . The quasi-product $a\circ b$ of $a$ and $b$ in $\mathcal{A}$ is then defined by

(5.1)

\displaystyle a\circ b=a+b-ab.

Definition 5.2 .

Let $\mathcal{A}$ be a Banach algebra and $a\in\mathcal{A}$ . An element $a^{0}\in\mathcal{A}$ is called a quasi-inverse for $a$ if

a\circ a^{0}=a^{0}\circ a=0.

If an element $a\in\mathcal{A}$ has a quasi-inverse, then it is called quasi-invertible or quasi-regular and otherwise it is called quasi-singular . The sets of all quasi-invertible and quasi-singular elements of $A$ are denoted by $\mathrm{q-Inv}(\mathcal{A})$ and $\mathrm{q-Sing}(\mathcal{A})$ respectively.

Definition 1.1 .

A Hom-Lie algebra is a triple $(\mathfrak{g},[~{},~{}],\alpha)$ , where $[~{},~{}]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ is a bilinear map and $\alpha:\mathfrak{g}\rightarrow\mathfrak{g}$ a linear map satisfying

[x,y]=-[y,x]~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{% (skew-symmetry)}

\displaystyle\circlearrowleft_{x,y,z}[\alpha(x),[y,z]]=0~{}~{}~{}~{}~{}~{}~{}~% {}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{(Hom-Jacobi~{} condition)}

for all x, y ,z from $\mathfrak{g}$ , where $\displaystyle\circlearrowleft_{x,y,z}$ denotes summation over the cyclic permutations of $x,y,z$ .

Definition 3.1 .

( [ 23 ] )

A vector space $T$ together with a trilinear map $(x,y,z)\rightarrow[x,y,z]$ is called a Lie triple system (LTS) if

(1)

$[x,x,z]=0,$
(2)

$[x,y,z]+[y,z,x]+[z,x,y]=0$ ,
(3)

$[u,v,[x,y,z]]=[[u,v,x],y,z]+[x,[u,v,y],z]+[x,y,[u,v,z]],$

for all $x,y,z,u,v\in T$ .

Definition 3.2 .

( [ 29 ] ) A Hom-Lie triple system (Hom-LTS for short) is denoted by $(T,[\cdot,\cdot,\cdot],\alpha)$ , which consists of an $\mathbb{K}$ -vector space $T$ , a trilinear product $[\cdot,\cdot,\cdot]:T\times T\times T\rightarrow T$ , and a linear map $\alpha:T\rightarrow T$ , called the twisted map, such that $\alpha$ preserves the product and for all $x,y,z,u,v\in T$ ,

(1)

$[x,x,z]=0,$
(2)

$[x,y,z]+[y,z,x]+[z,x,y]=0$ ,
(3)

$[\alpha(u),\alpha(v),[x,y,z]]=[[u,v,x],\alpha(y),\alpha(z)]+[\alpha(x),[u,v,y]% ,\alpha(z)]+[\alpha(x),\alpha(y),[u,v,z]]$ .

Definition 5.4 ( 2-regularity of mappings).

Let $g:\mathbb{R}^{n}\to\mathbb{R}^{s}$ be twice Fréchet differentiable at $\bar{x}\in\mathbb{R}^{n}$ . We say that $g$ is 2-regular at the point $\bar{x}$ in the direction $v\in\mathbb{R}^{n}$ if for any $p\in\mathbb{R}^{s}$ the system

\nabla g(\bar{x})z+\nabla^{2}g(\bar{x})(v,w)=p,\;\nabla g(\bar{x})w=0

admits a solution $(z,w)\in\mathbb{R}^{n}\times\mathbb{R}^{n}$ .