# Composite operators of stochastic model A

D. Davletbaeva  Department of Theoretical Physics, St. Petersburg University,
St. Petersburg, Russia  M. Hnatič  Bogoliubov Laboratory of Theoretical
Physics, Joint Institute for Nuclear Research, Dubna, Russia  Faculty of
Science, Šafárik University, Košice, Slovakia  Institute of Experimental
Physics SAS, Košice, Slovakia  M. V. Komarova  Department of Theoretical
Physics, St. Petersburg University, St. Petersburg, Russia  T. Lučivjanský
Faculty of Science, Šafárik University, Košice, Slovakia  L. Mižišin
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear
Research, Dubna, Russia  M. Yu. Nalimov  Department of Theoretical Physics,
St. Petersburg University, St. Petersburg, Russia  Bogoliubov Laboratory of
Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia

###### Abstract

By means of the field-theoretic renormalization group, we study the damping of
the viscosity coefficient near the superfluid phase transition. We utilize the
fact that in the infrared region, the complex model used to describe the phase
transition belongs to the same universality class as the well-known stochastic
model A. This allows us a determination of the critical behavior of viscosity
using composite operators for model A. Our analysis is based on the
$\varepsilon$-expansion near the upper critical dimension $d_{c}=4$ of model
A. The critical exponent of viscosity is then calculated from the critical
dimensions of composite operators of massless two-component model A. In
particular, we present results for critical dimensions of a selected class of
composite operators with the canonical dimension $8$ to the leading order.

##  1 Introduction

The most relevant features of the system that affect its critical behavior are
gross properties such as symmetries of the order parameter and fields, the
tensor character of the order parameter, the number of components, etc. These
to a large extent determine to what kind of universality class a given model
belongs [1, 2, 3]. According to the classical monographs [4, 5] superfluid
critical dynamics should be properly captured by stochastic dynamical models E
or F. Fundamental dynamical variables of these models are slowly varying
fields that correspond to thermodynamic flows [4] and corresponding
theoretical models are conveniently expressed through them. However, for model
E, the renormalization group (RG) equation leads to the two candidates for an
infrared (IR) stable fixed point, and the question of which of them actually
determines the observed behavior of the bosonic gas remains open (unclear) [1,
6]. In addition, neither in model E nor in the more involved F model, it is
possible to calculate the critical exponent that determines the damping of
coefficient viscosity when approaching the critical point of the corresponding
phase transition from normal to the superfluid state.

In the recent papers [7, 8] it has been proposed to analyze the stability of
models E and F with respect to the influences of the additional hydrodynamic
velocity field, and due to density wave perturbations, respectively. It has
turned out that both models are sensitive to the inclusion of hydrodynamic
modes, which significantly affect the critical behavior and, in particular,
values of the critical exponents. Further, taking into account the
compressibility of the medium led to a paradoxical conclusion [9]: models E
and F of critical dynamics are unstable with respect to such perturbations,
and in the IR region effectively reduce to model A [1, 3, 4], which exhibits a
single IR stable fixed point. On the other hand, phenomenological
considerations are confirmed from a microscopic point of view [10, 11], where
a careful analysis of the Bose gas was carried out based on the formalism of
time-dependent Green functions at finite temperature. This formalism provides
additional support in favor of the aforementioned results.

The main conclusion of the corresponding analysis [9] is that model A also
correctly describes dynamical behavior in the vicinity of the lambda point.
Moreover, the original effective dynamical action determines critical
dimensions of IR irrelevant fields (e.g, velocity field ${\bm{v}}$, or field
$m$ that corresponds the density and temperature fluctuations) or possible
monomials constructed from them, in terms of dimensions of composite operators
of the two-component model A. In other words, once the most important IR
composite operators of the main fields are determined, it is possible to
analyze the damping of the viscosity coefficient using the composite operators
of model A.

By employing sophisticated RG techniques on model A [1, 4] it is possible to
calculate the critical exponent of viscosity in terms of the critical
dimensions of the composite operators with canonical dimension $8$. Composite
operators with lower canonical dimensions $2,4$, and $6$ are mixed with them,
and missing powers are necessarily compensated with the powers of a parameter
(chemical potential or momenta) responsible for the phase transition in the
original model. Our main aim here is to calculate critical dimensions of
operators of canonical dimension $8$ to the leading order in $\varepsilon=4-d$
expansion. Their determination allows us to obtain physically relevant
information about the damping of viscosity in the vicinity of the phase
transition from the normal state to the superfluid state.

##  2 Stochastic Model A

In the standard classification of specific dynamical models [4], the model A
can be regarded as the simplest model. For a long time, it was mainly used to
describe the critical behavior of an anisotropic ferromagnet or
antiferromagnet. The action functional for the generalized model A takes the
abbreviated form [1]

$\mathcal{S}(\varphi^{\prime},\varphi)=\alpha_{0}\varphi^{\prime
2}+\varphi^{\prime}\left(-\frac{\partial}{\partial
t}\varphi+\alpha_{0}\nabla^{2}\varphi-\alpha_{0}\tau_{0}\varphi-\frac{\alpha_{0}g_{0}}{6}\varphi^{3}\right),$
(1)

where fields $\varphi$ ($\varphi^{\prime}$) is $n$-component order parameter
(response) field, $\tau_{0}\propto(T-T_{c})$ is (unrenormalized) deviation
from critical temperature, $\alpha_{0}>0$ is kinematic (diffusion)
coefficient, and the $g_{0}$ is a coupling constant of the theory.

Correspondence between the original effective dynamical action [9, 10] and
generalized model A is straightforward and reads

$\psi^{+^{\prime}}=\varphi^{\prime}_{1},\psi^{{}^{\prime}}=\varphi^{\prime}_{2},\psi=\varphi_{1},\psi^{+}=\varphi_{2}.$
(2)

Therefore, the calculation of the critical behavior of viscosity is
effectively described by the two-component model A. We will, however, continue
to work with the generalized model A. The original effective action [9, 10]
does not contain the ”mass” term proportional to $\tau$ (deviation from
critical temperature $\tau=T-T_{c}$). Hence we also set $\tau=0$ in the
generalized model (1).

Figure 1: Graphical representation of Feynman rules for model A with general
notation for a composite operator $F$.

The calculation follows the standard procedure of quantum field theory [1, 2].
Feynman diagrammatic rules and all related diagrammatic elements can be
obtained in a straightforward manner from the action functional (1).
Propagators can be derived from the Gaussian part of the action and, in the
frequency-momentum representation, they read

$\displaystyle\langle\varphi\varphi^{\prime}\rangle=\frac{1}{-i\omega+\alpha
k^{2}},\quad\langle\varphi\varphi\rangle=\frac{2\alpha}{|-i\omega+\alpha
k^{2}|^{2}}.$ (3)

The interaction part consists of a single vertex
$\varphi^{\prime}\varphi^{3}$. All graphic elements of the perturbation theory
are depicted schematically in Fig. 1.

$Q$ | $\partial_{x}$ | $\partial_{t}$ | $\varphi$ | $\varphi^{\prime}$ | $\alpha$ | $g_{0}$
---|---|---|---|---|---|---
$d_{p}[Q]$ | $1$ | $0$ | $\frac{d}{2}-1$ | $\frac{d}{2}+1$ | $-2$ | $4-d$
$d_{\omega}[Q]$ | $0$ | $1$ | $0$ | $0$ | $1$ | $0$
$d[Q]$ | $1$ | $2$ | $\frac{d}{2}-1$ | $\frac{d}{2}+1$ | $0$ | $4-d$
Table 1: Canonical dimensions of fields and parameters for the model A.

The starting point of the field-theoretical RG approach is dimensional
analysis. As a rule, dynamical models of type (1) exhibit two independent
scales [3, 1]. This means that to each quantity $Q$, two independent canonical
dimensions can be assigned, i.e., the momentum dimension $d_{p}[Q]$ and the
frequency dimension $d_{\omega}[Q]$, respectively. They are determined from
the requirement that each term in the action functional (1) has to be
dimensionless [1, 2]. We employ the standard normalization conditions (for
canonical dimensions of momentum $p$ and frequency $\omega$) in the following
form

$\displaystyle d_{p}[p]$ $\displaystyle=-d_{p}[x]=1,$ $\displaystyle
d_{p}[\omega]$ $\displaystyle=d_{p}[t]=0,$ (4) $\displaystyle d_{\omega}[p]$
$\displaystyle=d_{\omega}[x]=0,$ $\displaystyle d_{\omega}[\omega]$
$\displaystyle=-d_{\omega}[t]=1.$ (5)

The total canonical dimension of a quantity $Q$ is given by a formula

$d[Q]=d_{p}[Q]+2d_{\omega}[Q],$ (6)

and all canonical dimensions for model A are summarized in Tab. 1. From which
we also infer that is that the upper critical dimension of the model is
$d_{c}=4$, for which the charge $g_{0}$ becomes dimensionless. This motivates
us to introduce the formally small parameter $\varepsilon$ of the theory [1,
2] as a deviation

$\varepsilon=d_{c}-d=4-d.$ (7)

We perform the entire RG analysis to the leading order of the perturbation
theory ($\sim g$) employing the minimal subtraction (MS) scheme [1, 2].

Since more information about the renormalization of generalized model A can be
found in the literature [1, 2, 4], we do not discuss it here. Rather, our main
aim is to analyze composite operators and their construction, and perform
related renormalization procedure.

##  3 Extended model

In the IR region, the original model with included hydrodynamic fluctuations
[9, 10] reduces to the generalized model A (1). In this reduction, the terms
containing the viscosity coefficient are not a part of model A. However, they
can be interpreted as an extension of the model by certain composite
operators. From the point of view of quantum field theory, the composite
operator $F$ is an arbitrary local scalar quantity (monomials) constructed
from fields and their derivatives. Let us call the model with included
composite operators extended. For such a model, the generating functional
takes the form

$G(A,A^{\prime},a)=c\int
D\varphi^{\prime}D\varphi\,\exp\left[\mathcal{S}(\varphi,\varphi^{\prime})+aF(\varphi,\varphi^{\prime})+A\varphi+A^{\prime}\varphi^{\prime}\right],$
(8)

where $a$ is a set of sources and the linear form
$aF(\varphi,\varphi^{\prime})$ is a convenient abbreviation for the expression
$\sum\int\,dxdt\ a_{i}(t,x)F_{i}(t,x;\varphi,\varphi^{\prime})$. Since all the
main theorems of the renormalization are formulated for an arbitrary local
interaction, they remain valid even for the extended model with interaction
part including the composite operators $(\alpha
g/6\varphi^{\prime}\varphi^{3}+aF)$ [1].

As has been mentioned previously, in the renormalization analysis the first
step is canonical dimension analysis. Here the original effective dynamical
model [9, 10] comes into consideration, for which the maximum value of the
total canonical dimension for the term with viscosity is $d_{max}[F]=8$.
However, during the renormalization of composite operators, other operators
with a lower canonical dimension also contribute to the corresponding
counterterms. And thus it is necessary to take into account operators with a
lower dimension. On the other hand, a monomial has to be a scalar quantity,
which is valid only for monomials with even dimensions. This reduces composite
operators to monomials with dimension $d=\\{2,4,6,8\\}$. Another
simplification comes from the fact that model A is studied at the critical
point $\tau=0$. This has the consequence that operators with different
canonical dimensions do not mix with each other in the renormalization
process. In other words, composite operators are divided into sets according
to their total canonical dimension, and operators in a given set do not give
contributions to operators in a different set during the renormalization
procedure. In this article, we focus on the RG analysis of the operators with
canonical dimension $8$.

Each composite operator is a scalar monomial, with two independent canonical
dimensions and its total canonical dimension is determined using Eq. (6),
which corresponds to the sum of the dimension of the fields and the
derivations defined in Tab. 1. The canonical dimension of the source $a_{i}$
can be determined from the condition that the action $\mathcal{S}$ is a
dimensionless quantity and thus we get

$d[a]+d[F]+d[x]+d[t]=d[\mathcal{S}]\equiv 0.$ (9)

Replacing $d\rightarrow\Delta$ yields a similar relation between critical
dimensions. In this way, we can determine the critical dimensions of sources
$\Delta_{a}$ once we know the critical dimension parameters and composite
operators $\Delta_{F}$.

The construction of all possible monomials of fields and their derivatives,
whose canonical dimension is $8$ leads to $74$ different scalar monomials. Let
us note that they contain a single derivative, which acts only on one of the
fields $\varphi^{\prime}$ or $\varphi$, respectively. Composite operators
containing multiple derivatives can be expressed through a suitable
combination of operators with a single derivative. It was necessary to take
into account that the fields are $n$-component and therefore we need to
distinguish the product between fields in the monomials. In actual
calculations, we work with a general $n$-component field and only at the end
of the calculation we set $n=2$.

The composite operators can be divided into different sets. One of them is
formed by composite operators, collectively denoted by $O$, containing exactly
two fields (e.g., $\varphi^{\prime}\partial_{t}\varphi^{\prime}$,
$\varphi^{\prime}\partial_{t}\partial^{2}\varphi$,
$\varphi\partial^{6}\varphi$, and so on). Operators $O$ display two important
properties in the leading order of perturbation theory. First, operators $O$
form a closed set with the canonical dimension $8$, which means that composite
operators with different numbers of fields do not mix with these $O$
operators. Second, the information about the renormalization of $O$ operators
can be obtained from the renormalization of operators with lower canonical
dimensions, i.e. those with canonical dimensions equal $6$, $4$, or $2$. These
two features can be proved by the direct inspection of Feynman diagrams and
lead to the fact that there is, in fact, no need for the renormalization of
$O$ operators. In other words, the composite operator $O$ can be excluded from
the renormalization procedure because its renormalization is extracted from an
operator with a lower canonical dimension.

As a result of these considerations, the final number of necessary composite
operators is reduced to a closed set of $41$ operators, which we summarized in
the following way

$\displaystyle F_{1}$
$\displaystyle=(\varphi^{\prime}\varphi^{\prime})(\varphi\varphi),$
$\displaystyle F_{2}$
$\displaystyle=(\varphi^{\prime}\varphi)(\varphi^{\prime}\varphi),$
$\displaystyle F_{3}$
$\displaystyle=(\varphi\partial^{2}\varphi^{\prime})(\varphi\varphi),$
$\displaystyle F_{4}$
$\displaystyle=(\varphi\partial_{i}\varphi^{\prime})(\varphi\partial_{i}\varphi),$
$\displaystyle F_{5}$
$\displaystyle=(\partial_{i}\varphi^{\prime}\partial_{i}\varphi)(\varphi\varphi),$
$\displaystyle F_{6}$
$\displaystyle=(\varphi^{\prime}\partial^{2}\varphi)(\varphi\varphi),$
$\displaystyle F_{7}$
$\displaystyle=(\varphi^{\prime}\varphi)(\varphi\partial^{2}\varphi),$
$\displaystyle F_{8}$
$\displaystyle=(\varphi^{\prime}\varphi)(\partial_{i}\varphi)^{2},$
$\displaystyle F_{9}$
$\displaystyle=(\varphi^{\prime}\partial_{i}\varphi)(\varphi\partial_{i}\varphi),$
$\displaystyle F_{10}$
$\displaystyle=(\varphi^{\prime}\partial_{t}\varphi)(\varphi\varphi),$
$\displaystyle F_{11}$
$\displaystyle=(\varphi^{\prime}\varphi)(\varphi\partial_{t}\varphi),$
$\displaystyle F_{12}$
$\displaystyle=(\varphi\partial_{t}\varphi^{\prime})(\varphi\varphi),$
$\displaystyle F_{13}$
$\displaystyle=(\varphi^{\prime}\varphi)(\varphi\varphi)^{2},$ $\displaystyle
F_{14}$ $\displaystyle=(\varphi\partial_{t}\varphi)(\varphi\varphi)^{2},$
$\displaystyle F_{15}$
$\displaystyle=(\varphi\partial_{t}^{2}\varphi)(\varphi\varphi),$
$\displaystyle F_{16}$
$\displaystyle=(\partial_{t}\varphi\partial_{t}\varphi)(\varphi\varphi),$
$\displaystyle F_{17}$
$\displaystyle=(\varphi\partial_{t}\varphi)(\varphi\partial_{t}\varphi),$
$\displaystyle F_{18}$
$\displaystyle=(\varphi\partial_{t}\partial^{2}\varphi)(\varphi\varphi),$
$\displaystyle F_{19}$
$\displaystyle=(\partial_{t}\varphi\partial^{2}\varphi)(\varphi\varphi),$
$\displaystyle F_{20}$
$\displaystyle=(\varphi\partial^{2}\varphi)(\varphi\partial_{t}\varphi),$
$\displaystyle F_{21}$
$\displaystyle=(\varphi\partial_{t}\varphi)(\partial_{i}\varphi\partial_{i}\varphi),$
$\displaystyle F_{22}$
$\displaystyle=(\partial_{i}\varphi\partial_{t}\varphi)(\varphi\partial_{i}\varphi),$
$\displaystyle F_{23}$
$\displaystyle=(\partial\varphi\partial_{t}\partial\varphi)(\varphi\varphi),$
$\displaystyle F_{24}$
$\displaystyle=(\varphi\partial_{t}\partial_{i}\varphi)(\varphi\partial_{i}\varphi),$
$\displaystyle F_{25}$ $\displaystyle=(\varphi\varphi)^{4},$ $\displaystyle
F_{26}$ $\displaystyle=(\varphi\partial^{2}\varphi)(\varphi\varphi)^{2},$
$\displaystyle F_{27}$
$\displaystyle=(\partial_{i}\varphi\partial_{i}\varphi)(\varphi\varphi)^{2},$
$\displaystyle F_{28}$
$\displaystyle=(\varphi\partial_{i}\varphi)^{2}(\varphi\varphi),$
$\displaystyle F_{29}$
$\displaystyle=(\varphi\partial^{4}\varphi)(\varphi\varphi),$ $\displaystyle
F_{30}$
$\displaystyle=(\partial^{2}\varphi\partial^{2}\varphi)(\varphi\varphi),$
$\displaystyle F_{31}$
$\displaystyle=(\varphi\partial^{2}\varphi)(\varphi\partial^{2}\varphi),$
$\displaystyle F_{32}$
$\displaystyle=(\partial_{i}\varphi\partial_{i}\partial^{2}\varphi)(\varphi\varphi),$
$\displaystyle F_{33}$
$\displaystyle=(\varphi\partial_{i}\partial^{2}\varphi)(\varphi\partial_{i}\varphi),$
$\displaystyle F_{34}$
$\displaystyle=(\partial_{i}\varphi\partial_{i}\varphi)(\varphi\partial^{2}\varphi),$
$\displaystyle F_{35}$
$\displaystyle=(\partial^{2}\varphi\partial_{i}\varphi)(\varphi\partial_{i}\varphi),$
$\displaystyle F_{36}$
$\displaystyle=(\partial_{i}\varphi\partial_{i}\varphi)(\partial_{j}\varphi\partial_{j}\varphi),$
$\displaystyle F_{37}$
$\displaystyle=(\partial_{i}\partial_{j}\varphi\partial_{i}\partial_{j}\varphi)(\varphi\varphi),$
$\displaystyle F_{38}$
$\displaystyle=(\varphi\partial_{i}\partial_{j}\varphi)(\varphi\partial_{i}\partial_{j}\varphi),$
$\displaystyle F_{39}$
$\displaystyle=(\partial_{i}\varphi\partial_{j}\varphi)(\varphi\partial_{i}\partial_{j}\varphi),$
$\displaystyle F_{40}$
$\displaystyle=(\partial_{i}\partial_{j}\varphi\partial_{i}\varphi)(\varphi\partial_{j}\varphi),$
$\displaystyle F_{41}$
$\displaystyle=(\partial_{i}\varphi\partial_{j}\varphi)(\partial_{i}\varphi\partial_{j}\varphi),$
(10)

where for simplicity we have denoted time derivative as ${\partial}/{\partial
t}\equiv\partial_{t}$ and spatial derivative as ${\partial}/{\partial{\bf
x}_{i}}\equiv\partial_{i}$. Summation over the same index is always implied,
e.g. $\partial_{i}\partial_{i}=\partial^{2}\equiv\nabla^{2}$. The canonical
dimensions for composite operators are listed in Tab. 2. The composite
operators are chosen in such a way that the derivative is always applied on
one of entering fields in a given composite operator. This to some extent
simplifies calculational procedure.

Further, any other operator of canonical dimension $8$ can be expressed by
proper linear combination (e.g.
$\partial_{t}[(\varphi\varphi^{\prime})(\varphi\varphi)]=F_{10}+2F_{11}+F_{12}$,
$\partial_{t}(\varphi\varphi)^{3}=6F_{14}$,
$(\varphi\varphi)\partial^{2}(\varphi\varphi^{\prime})=F_{3}+2F_{5}+F_{6}$,
etc.). Although the composite operators (10) form a closed set, for some of
them their renormalization can be related to other operators with lower
canonical dimension, for instance
$\partial^{2}[(\varphi\varphi)^{3}]_{R}=[\partial^{2}(\varphi\varphi)^{3}]_{R}=[\partial_{t}(\varphi\varphi)^{3}]_{R}$.
This property might be exploited to reduce the number of independent
operators, which we have used to provide an independent check in actual
calculations.

$F_{i}$ | $F_{1}-F_{9},F_{29}-F_{41}$ | $F_{10}-F_{12},F_{18}-F_{24}$ | $F_{13},F_{26}-F_{28}$
---|---|---|---
$d_{p}[F_{i}]$ | $2d$ | $2d-2$ | $3d-4$
$d_{\omega}[F_{i}]$ | $0$ | $1$ | $0$
$d[F_{i}]$ | $2d$ | $2d$ | $3d-4$
$F_{i}$ | $F_{14}$ | $F_{15}-F_{17}$ | $F_{25}$
$d_{p}[F_{i}]$ | $3d-6$ | $2d-4$ | $4d-8$
$d_{\omega}[F_{i}]$ | $1$ | $2$ | $0$
$d[F_{i}]$ | $3d-4$ | $2d$ | $4d-8$
Table 2: Canonical dimension of the composite operators $F_{i};i=1,\ldots,41$.
Figure 2: Feynman diagrams contributing to the composite operator 10 in the
leading order.

The model (1) without the term $aF$ is multiplicatively renormalizable and the
expression for multiplicative renormalization through terms of the first order
in $a$ can be generalized to the extended model. For a closed set of composite
operators, the relation between renormalized and unrenormalized takes the
general form

$[F_{i}(\varphi^{\prime},\varphi)]_{R}=\sum\limits_{k}\mathcal{Q}_{ik}F_{k}(\varphi^{\prime},\varphi).$
(11)

The mixing matrix $\mathcal{Q}$ is calculated directly from Feynman diagrams
involving composite operators (see Fig. 2). Moreover, the mixing matrix breaks
down into three parts in the leading order of the perturbation theory, in
which the composite operators mix with each other during RG procedure. Its
elements depend on the renormalization mass $\mu$ and the renormalized
parameters $\\{g,\alpha\\}$. For the closed set (10), the mixing matrix can be
schematically written as follows

$\mathcal{Q}(36\times
36)=\mathds{1}-\frac{g}{\varepsilon}\begin{pmatrix}\mathcal{Q}_{1}(13\times
13)&0&0\\\ ?&\mathcal{Q}_{2}(11\times 11)&0\\\ ?&?&\mathcal{Q}_{3}(17\times
17)\end{pmatrix},$ (12)

where $\mathds{1}$ is a unit matrix and
$\mathcal{Q}_{1},\mathcal{Q}_{2},\mathcal{Q}_{3}$ are block matrices with size
indicated in brackets (see Appendix). For the matrix, $\mathcal{Q}$, the rows
and columns correspond to the sequence of the composite operator $\\{F_{i}\\}$
from the set (10). The $"0"$ is a notation for the zero blocks in the matrix
and the symbol $"?"$ is part of the matrix, that does not need to be
considered.

Calculation of the diagrams is carried out in the MS scheme [1, 2]. The two-
loop diagrams (see Fig. 2) do not actually need to be calculated. In fact,
they contribute to the matrix (12) in the part marked with the symbol $"?"$.
The blocks of the mixing matrix $\mathcal{Q}$,

  * •

$\mathcal{Q}_{1}$ \- mixing of operators composed of
$\varphi^{\prime},\varphi,\partial_{t}$, and $\partial$,

  * •

$\mathcal{Q}_{2}$ \- mixing of operators composed of $\varphi,\partial_{t}$,
and $\partial$,

  * •

$\mathcal{Q}_{3}$ \- mixing of composite operators composed of
$\varphi,\partial$. This corresponds to the static $\varphi^{4}$ model.

As has been already stressed these blocks can be analyzed separately, which
substantially simplifies overall analysis.

Further, we continue with the calculation of the critical dimensions for the
set of composite operators. The first step is to determine renormalization
constants from mixing matrix $\mathcal{Q}$. The renormalization constants for
sources $a_{0}=aZ^{a}$ become a matrix and there is a relation with the mixing
matrix

$Z^{a}_{ij}=\mathcal{Q}_{ij}(Z_{\varphi})^{-n_{j}}(Z_{\varphi^{\prime}})^{-m_{j}},\quad
Z_{F}=(Z^{a})^{-1},$ (13)

where $n_{j}$ and $m_{j}$ is a number of fields $\varphi$, $\varphi^{\prime}$
appearing in monomials $F_{j}$, respectively. The renormalization constant
$Z_{F}$ is the inverse matrix of $Z^{a}$. The renormalization constants for
fields $Z_{\varphi},Z_{\varphi^{\prime}}$ do not have poles in the MS scheme
for the generalized model A in the leading order, which leads to the relations

$Z^{a}=\mathcal{Q},\quad Z_{F}=\mathcal{Q}^{-1}.$ (14)

where $\mathcal{Q}^{-1}$ is an inverse mixing matrix.

The information about critical scaling can be obtained from RG equations for
composite operators. They can be derived from the generating functional of
renormalized connected Green functions of the fields and composite operators
[1, 2]. Using RG method it can be shown that composite operators obey the
matrix equation

$\mathcal{D}_{RG}F_{R}=-\gamma_{F}F_{R},$ (15)

which in the component notation reads

$\mathcal{D}_{RG}F_{iR}=-\gamma_{F}^{ij}F_{jR}.$ (16)

Here, $\gamma_{F}$ is a matrix of anomalous dimension for a closed set of
composite operators.

The differential operator $\mathcal{D}_{RG}$ is RG operator in the MS scheme
and can be written in the the following form

$\mathcal{D}_{RG}\equiv\tilde{\mathcal{D}}_{\mu}=\mathcal{D}_{\mu}+\beta_{g}\partial_{g}-\gamma_{\alpha}\partial_{\alpha},$
(17)

where $\mathcal{D}_{\mu}=\mu\partial_{\mu}$ and $\beta_{g}$ is beta function
of coupling constant $g$. At the fixed point we have $g=g^{*}$, and the
contribution of the beta function vanishes. In time-coordinate representation,
the equation of critical scaling for the closed set of renormalized operators
$\\{F\\}_{R}$ takes the form

$\left[-\mathcal{D}_{x}+\Delta_{t}\mathcal{D}_{t}\right]F_{R}=\Delta_{F}F_{R},$
(18)

where $\Delta_{t}=-\Delta_{\omega}$ is critical dimension, and $\Delta_{F}$ is
a matrix of critical dimensions. The critical dimension of a composite
operator $\Delta_{F}$ can be obtained from the formula

$\Delta_{F}=d_{p}[F]+\Delta_{\omega}d_{\omega}[F]+\gamma_{F}^{*}=d[F]-\gamma_{\alpha}^{*}d_{\omega}[F]+\gamma_{F}^{*},$
(19)

where $\Delta_{\omega}=2-\gamma_{\alpha}^{*}$, and anomalous dimensions
$\gamma_{F}^{*}$ and $\gamma_{\alpha}^{*}$ are taken at the IR-stable fixed
point $(g=g^{*})$. The generalized model A exhibits a single with the
coordinate (see [1], chapter 4.2)

$g^{*}=\frac{3}{n+8}\varepsilon\bigg{|}_{n=2}=\frac{3}{10}\varepsilon.$ (20)

The matrix of anomalous dimension $\gamma_{F}$ is defined as

$\gamma_{F}\equiv
Z_{F}^{-1}\cdot(\widetilde{\mathcal{D}}_{\mu}Z_{F}),\quad\gamma_{F}=-\textrm{
UV-finite part}\left[\widetilde{\mathcal{D}}_{\mu}Z^{a}\right],$ (21)

where the second relation is valid in the MS scheme [1]. The relation can be
further simplified in the leading order of perturbation theory

$\gamma_{F}=\varepsilon g\partial_{g}Z^{a}=\varepsilon g\partial_{g}Q.$ (22)

The diagonal matrix of the canonical dimensions $d_{F}$ is known (see Tab. 2)
and matrix $\gamma_{F}^{*}$ is calculated from the relation (22) at the fixed
point $g=g^{*}$. If the $\gamma_{F}^{*}$ was diagonal as well, the equation of
critical scaling (18) would have the same structure and the diagonal elements
of the matrix $\Delta_{F}$ would be interpreted as the critical dimensions of
the corresponding operators $F_{iR}$. However, the blocks $\mathcal{Q}_{i}$
are non-zero square matrices, and the corresponding matrix of anomalous
dimension $\gamma_{F}^{*}$ at the fixed points is therefore non-diagonal (and
asymmetric). As a direct consequence, the operators $F_{iR}$ do not possess
definite critical dimensions. Instead, we can form certain linear combinations
of the latter and obtain quantities with definite critical dimensions. Let us
write

$F^{\prime}_{iR}=U_{ij}F_{jR},$ (23)

where $U$ is some matrix, which depends on parameters $\mu$ and $\alpha$. The
substitution $F_{R}=U^{-1}F_{R}^{\prime}$ in (18) yields an equation for new
set of composite operators $F_{R}^{\prime}$ with a new matrix [1]

$\Delta_{F}^{\prime}=U\Delta_{F}U^{-1},$ (24)

where the final expression is derived for the massless model A. The final
value of diagonal element $\Delta_{F}^{\prime}$ has been calculated
numerically for each block separately

$\displaystyle\Delta_{F_{1}}^{\prime}$
$\displaystyle=\\{\\{8-\frac{39}{5}\varepsilon\\},\\{8-4\varepsilon\\},\\{8-4\varepsilon\\},\\{8-3\varepsilon\\},\\{8-3\varepsilon\\},\\{8-\frac{14}{5}\varepsilon\\},$
$\displaystyle\\{8-\frac{12}{5}\varepsilon\\},\\{8-2.653\varepsilon\\},\\{8-3.346\varepsilon\\},\\{8-2.526\varepsilon\\},\\{8-3.674\varepsilon\\},$
$\displaystyle\\{8-2.589\varepsilon\\},\\{8-3.511\varepsilon\\}\\},$ (25)
$\displaystyle\Delta_{F_{2}}^{\prime}$
$\displaystyle=\\{\\{8-\frac{39}{5}\varepsilon\\},\\{8-4\varepsilon\\},\\{8-4\varepsilon\\},\\{8-3\varepsilon\\},\\{8-3\varepsilon\\},\\{8-\frac{14}{5}\varepsilon\\},$
$\displaystyle\\{8-\frac{12}{5}\varepsilon\\},\\{8-2.653\varepsilon\\},\\{8-3.346\varepsilon\\},\\{8-2.721\varepsilon\\},\\{8-3.179\varepsilon\\}\\}$
(26) $\displaystyle\Delta_{F_{3}}^{\prime}$
$\displaystyle=\\{\\{8-4\varepsilon\\},\\{8-\frac{12}{5}\varepsilon\\},\\{8-\frac{12}{5}\varepsilon\\},\\{8-\frac{12}{5}\varepsilon\\},\\{8-3\varepsilon\\},\\{8-3\varepsilon\\},$
$\displaystyle\\{8-3\varepsilon\\},\\{8-\frac{14}{5}\varepsilon\\},\\{8-\frac{64}{5}\varepsilon\\},\\{8-\frac{11}{5}\varepsilon\\},\\{8-\frac{29}{5}\varepsilon\\},\\{8-\frac{31}{5}\varepsilon\\},$
$\displaystyle\\{8-\frac{39}{5}\varepsilon\\},\\{8-3.527\varepsilon\\},\\{8-3.269\varepsilon\\},\\{8-2.583\varepsilon\\},\\{8-2.154\varepsilon\\}\\}$
(27)

where the values are rounded to the third decimal place. The critical
dimensions are calculated for the value $n=2$. Let us note that the first
seven values in the expressions (25) and (26), are, in fact, exact, whereas in
expression (27) the first thirteen values are exact. Finally, our main
interest is the behavior of viscosity. This can be related to the critical
dimensions of source terms of corresponding composite operators.

The critical dimension of the sources $\Delta_{a}^{\prime}$ can be determined
from the relation

$\Delta_{a}^{\prime}=d+\Delta_{\omega}-\Delta_{F}^{\prime}.$ (28)

It is worthwhile to note, the analysis becomes simple for the case when a
source is connected to the coefficient of viscosity. The presence of viscosity
as a source reduces the set of operators (rows and columns in the mixing
matrix) for the operators which are connected through the derivative with the
same number of fields, e.g.
$\partial_{t}(\varphi^{\prime}\varphi)(\varphi\varphi)=F_{10}+2F_{11}+F_{12}=0$.
The composite operators connected via derivative are linearly dependent and
one of the operators can be excluded from the analysis. This property will
provide a significant reduction in the calculation of the critical dimension
for the viscosity coefficient. For completeness, we keep all operators in the
work, and their form is reduced depending on the task.

We have calculated the critical dimensions of a set of composite operators
with the canonical dimension $8$ to the leading order. After verifying these
results, the next task consists of a determination of the critical dimensions
for operators with a lower canonical dimension and determining the behavior of
the viscosity coefficient in the leading order of perturbation theory.

## Conflict of Interest

The authors declare that they have no conflicts of interest.

### Acknowledgment

The work was supported by VEGA grant No. 1/0535/21 of the Ministry of
Education, Science, Research and Sport of the Slovak Republic and the
Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”
19-1-1-35-1.

## Appendix

Calculating the contributions of individual composite operators to the mixing
matrix $\mathcal{Q}$ is a laborious process, so we will simplify this
description to the present final form of the mixing matrix. More precisely,
these are the parts $\mathcal{Q}_{1}$, $\mathcal{Q}_{2}$ and $\mathcal{Q}_{3}$
that contribute to the final values of the critical dimensions
$\Delta_{F}^{\prime}$.

In the calculation, we have employed standard quantum-field methods [1, 2,
12]. To simplify the notation, we have redefined the coupling constant in the
calculations according to the prescription

$\frac{g}{(4\pi)^{d/2}}\longrightarrow g.$ (29)

The part of mixing matrix (12), which is denoted as $\mathcal{Q}_{1}$ has
following form

$\displaystyle\left(\setcounter{MaxMatrixCols}{13}\begin{smallmatrix}\frac{n+6}{3}&\frac{8}{3}&\frac{n+2}{8}&0&\frac{n+2}{4}&\frac{n+2}{8}&0&0&0&\frac{n+2}{4\alpha}&0&\frac{n+2}{4\alpha}&0\\\\[5.69054pt]
1&\frac{2(n+5)}{3}&\frac{3}{8}&\frac{1}{2}&\frac{1}{4}&\frac{1}{8}&\frac{1}{4}&0&0&\frac{1}{4\alpha}&\frac{1}{2\alpha}&\frac{3}{4\alpha}&0\\\\[5.69054pt]
0&0&\frac{5(n+8)}{12}&\frac{2}{3}&\frac{n+4}{6}&\frac{n+4}{12}&\frac{1}{3}&0&0&-\frac{n+4}{6\alpha}&-\frac{2}{3\alpha}&-\frac{n+8}{6\alpha}&0\\\\[5.69054pt]
0&0&\frac{1}{2}&\frac{7n+44}{12}&\frac{5}{4}&\frac{1}{6}&\frac{1}{3}&\frac{1}{4}&\frac{n+4}{4}&\frac{1}{6\alpha}&\frac{1}{3\alpha}&\frac{1}{2\alpha}&0\\\\[5.69054pt]
0&0&\frac{n+2}{6}&\frac{7}{3}&\frac{4n+15}{6}&\frac{n+2}{6}&0&\frac{1}{2}&1&\frac{n+2}{6\alpha}&0&\frac{n+2}{6\alpha}&0\\\\[5.69054pt]
-\frac{8}{9}&-\frac{16}{9}&-\frac{n+2}{12}&0&-\frac{n+2}{6}&\frac{3n+14}{12}&\frac{4}{3}&0&0&-\frac{n-2}{6\alpha}&\frac{4}{3\alpha}&-\frac{n+2}{6\alpha}&0\\\\[5.69054pt]
-\frac{4}{9}&-\frac{4(n+4)}{9}&-\frac{1}{4}&-\frac{1}{3}&-\frac{1}{6}&\frac{7}{12}&\frac{2n+11}{6}&0&0&\frac{1}{6\alpha}&\frac{n+3}{3\alpha}&-\frac{1}{2\alpha}&0\\\\[5.69054pt]
0&\frac{4(n+2)}{9}&0&\frac{1}{3}&\frac{1}{6}&0&\frac{n+2}{3}&\frac{4n+13}{6}&\frac{5}{3}&0&-\frac{n+2}{3\alpha}&0&0\\\\[5.69054pt]
\frac{4}{9}&\frac{8}{9}&0&\frac{n+4}{12}&\frac{1}{12}&\frac{1}{3}&\frac{2}{3}&\frac{13}{12}&\frac{5n+36}{12}&-\frac{1}{3\alpha}&-\frac{2}{3\alpha}&0&0\\\\[5.69054pt]
0&0&\frac{\alpha(n+2)}{12}&0&\frac{\alpha(n+2)}{6}&\frac{\alpha(n+2)}{12}&0&0&0&\frac{3n+14}{6}&\frac{8}{3}&\frac{n+2}{6}&0\\\\[5.69054pt]
0&0&\frac{\alpha}{4}&\frac{\alpha}{3}&\frac{\alpha}{6}&\frac{\alpha}{12}&\frac{\alpha}{6}&0&0&\frac{7}{6}&\frac{2n+11}{3}&\frac{1}{2}&0\\\\[5.69054pt]
0&0&-\frac{\alpha(n+8)}{12}&-\frac{2\alpha}{3}&-\frac{\alpha(n+4)}{6}&-\frac{\alpha(n+4)}{12}&-\frac{\alpha}{3}&0&0&\frac{n+4}{6}&\frac{2}{3}&\frac{n+8}{2}&0\\\\[5.69054pt]
0&0&0&0&0&0&0&0&0&0&0&0&n+14\\\\[5.69054pt] \end{smallmatrix}\right)$

The part of mixing matrix (12), which is denoted as $\mathcal{Q}_{2}$ has form

$\displaystyle\left(\setcounter{MaxMatrixCols}{11}\begin{smallmatrix}n+14&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&\frac{n+8}{2}&\frac{n+4}{6}&\frac{2}{3}&\frac{\alpha(n+8)}{12}&\frac{\alpha(n+4)}{12}&\frac{\alpha}{3}&0&0&\frac{\alpha(n+4)}{6}&\frac{2\alpha}{3}\\\\[5.69054pt]
0&\frac{n+2}{6}&\frac{3n+14}{6}&\frac{8}{3}&-\frac{\alpha(n+2)}{12}&-\frac{\alpha(n+2)}{12}&0&0&0&-\frac{\alpha(n+2)}{6}&0\\\\[5.69054pt]
0&\frac{1}{2}&\frac{7}{6}&\frac{2n+11}{3}&-\frac{\alpha}{4}&-\frac{\alpha}{12}&-\frac{\alpha}{6}&0&0&-\frac{\alpha}{6}&-\frac{\alpha}{3}\\\\[5.69054pt]
0&\frac{n+8}{6\alpha}&\frac{n+4}{6\alpha}&\frac{2}{3\alpha}&\frac{n+8}{4}&-\frac{n+4}{12}&-\frac{1}{3}&0&0&-\frac{n+4}{6}&-\frac{2}{3}\\\\[5.69054pt]
0&\frac{n+2}{6\alpha}&\frac{n+6}{6\alpha}&\frac{4}{3\alpha}&\frac{n+2}{12}&\frac{5n+18}{12}&\frac{4}{3}&0&0&\frac{n+2}{6}&0\\\\[5.69054pt]
0&\frac{1}{2\alpha}&\frac{1}{2\alpha}&\frac{n+5}{3\alpha}&\frac{1}{4}&\frac{3}{4}&\frac{2n+13}{6}&0&0&\frac{1}{6}&\frac{1}{3}\\\\[5.69054pt]
0&0&0&-\frac{n+2}{3\alpha}&0&0&\frac{n+2}{3}&\frac{4n+15}{6}&\frac{7}{3}&\frac{1}{2}&1\\\\[5.69054pt]
0&0&-\frac{1}{3\alpha}&-\frac{2}{3\alpha}&0&\frac{1}{3}&\frac{2}{3}&\frac{5}{4}&\frac{7n+44}{12}&\frac{1}{4}&\frac{n+4}{4}\\\\[5.69054pt]
0&-\frac{n+2}{6\alpha}&-\frac{n+2}{6\alpha}&0&\frac{n+2}{6}&\frac{n+2}{6}&0&\frac{1}{6}&\frac{1}{3}&\frac{4n+13}{6}&\frac{5}{3}\\\\[5.69054pt]
0&-\frac{1}{2\alpha}&-\frac{1}{6\alpha}&-\frac{1}{3\alpha}&\frac{1}{2}&\frac{1}{6}&\frac{1}{3}&\frac{1}{12}&\frac{n+4}{12}&\frac{13}{12}&\frac{5n+36}{12}\\\\[5.69054pt]
\end{smallmatrix}\right)$

The part of mixing matrix (12), which is denoted as $\mathcal{Q}_{3}$ has form

$\displaystyle\left(\setcounter{MaxMatrixCols}{17}\begin{smallmatrix}\frac{4n+80}{3}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&\frac{2n+28}{3}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&\frac{n+2}{3}&n+\frac{26}{3}&\frac{16}{3}&0&0&0&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&1&\frac{4}{3}&n+\frac{38}{3}&0&0&0&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&\frac{n+8}{3}&0&0&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&0&\frac{n+2}{3}&0&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&0&\frac{1}{3}&\frac{2}{3}&0&0&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&0&0&0&\frac{n+4}{3}&\frac{4}{3}&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&0&0&0&\frac{2}{3}&\frac{n+6}{3}&0&0&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&0&0&\frac{n+2}{3}&0&0&\frac{n+4}{3}&\frac{4}{3}&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&0&\frac{1}{3}&\frac{2}{3}&0&0&\frac{2}{3}&\frac{n+6}{3}&0&0&0&0&0&0\\\\[5.69054pt]
0&0&0&0&0&0&0&0&0&\frac{2n+8}{3}&\frac{8}{3}&\frac{2n+8}{3}&0&0&0&0&\frac{8}{3}\\\\[5.69054pt]
0&0&0&0&\frac{n+2}{6}&\frac{3n+2}{18}&-\frac{4}{9}&\frac{2n+4}{3}&0&-\frac{2}{9}&-\frac{4}{9}&0&\frac{6n+20}{9}&\frac{16}{9}&\frac{8}{9}&\frac{16}{9}&0\\\\[5.69054pt]
0&0&0&0&\frac{1}{2}&\frac{1}{18}&\frac{-n-1}{9}&\frac{2}{3}&\frac{4}{3}&\frac{-n-3}{9}&-\frac{2}{9}&0&\frac{10}{9}&\frac{4n+28}{9}&\frac{4n+12}{9}&\frac{8}{9}&0\\\\[5.69054pt]
0&0&0&0&0&0&\frac{n+2}{18}&\frac{1}{3}&\frac{2}{3}&\frac{1}{3}&\frac{2}{3}&\frac{-n-2}{18}&0&\frac{n+2}{9}&\frac{n+6}{3}&\frac{8}{3}&\frac{2n+4}{9}\\\\[5.69054pt]
0&0&0&0&0&\frac{1}{18}&\frac{1}{9}&\frac{1}{6}&\frac{n+4}{6}&\frac{1}{6}&\frac{n+4}{6}&\frac{1}{6}&\frac{1}{9}&\frac{2}{9}&1&\frac{2n+10}{3}&\frac{1}{3}\\\\[5.69054pt]
0&0&0&0&0&0&0&0&0&\frac{n+4}{9}&\frac{4}{9}&\frac{n+8}{9}&0&0&\frac{2n+8}{9}&\frac{8}{9}&\frac{2n+16}{9}\\\\[5.69054pt]
\end{smallmatrix}\right)$

## References

  * [1] A. N. Vasil’ev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics, [in Russian], PIYaF, St. Petersburg (1998); English trans., Chapman and Hall/CRC, Boca Raton, Fla (2004).
  * [2] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Oxford University Press, Oxford, 2002).
  * [3] U. C. Täuber, Critical Dynamics: A Field Theory Approach To Equilibrium And Non-Equilibrium Scaling Behavior, Cambridge University Press, Cambridge, 2014.
  * [4] P. C. Hohenberg, B. I. Halperin, Reviews of Modern Physics, 49(3), 435 (1977).
  * [5] R. Folk, G. Moser, Journal of Physics A: Mathematical and General, 39(24), R207 (2006).
  * [6] C. De Dominicis and L. Peliti, Phys. Rev. B 18, 353 (1978).
  * [7] M. Hnatich, M. V. Komarova, M. Yu. Nalimov, Theor. Math. Phys. 175, 779, (2013).
  * [8] M. Dančo, M. Hnatič, M. V. Komarova, T. Lučivjanský, M. Yu. Nalimov, Phys. Rev. E, 93(1), 012109 (2016).
  * [9] Yu A. Zhavoronkov, M. V. Komarova, Yu. G. Molotkov, M. Yu. Nalimov, J. Honkonen, Theor. Math. Phys. 200(2), 1237, (2019).
  * [10] J. Honkonen, M. V. Komarova, Yu. G. Molotkov, M. Yu. Nalimov, Nuclear Physics B 939, 105 (2019).
  * [11] J. Honkonen, M. V. Komarova, Yu. G. Molotkov, M. Yu. Nalimov, A. Trenogin, Phys. Rev. E, 106, 014126 (2022).
  * [12] D. J. Amit, V. Martín-Mayor, Filed Theory, the Renormalization Group, and Critical Phenomena, World Scientific Publishing Company, 2005.


fixed non-principal ultrafilter $\omega\in\partial\beta\mathbb{N}$, given a
conditional negative definite function $\psi:(\partial
G(X),\hat{\mu}_{\omega})\to\mathbb{R}$, there exists, by Proposition 3.22, a
sequence of maps $\psi_{i}$ from the graphs $X_{i}\times X_{i}$ to the real
numbers $\mathbb{R}$ that approximates $\psi$. We again view the lifts
$\psi_{i}$ as kernels $K_{i}$ along the graph sequence.

######  Lemma 4.24.

Let $\psi:({\partial G(X)},\hat{\mu}_{\omega})\to\mathbb{R}$ be a measurable
conditionally negative definite function such that the sequence of kernels
$(K_{i})_{i\in\mathbb{N}}$ approximates $\psi$ along a non-principal
ultrafilter $\omega\in\partial\beta\mathbb{N}$. Then, for all $\varepsilon>0$,
there exists a function $\rho_{2}^{\varepsilon}:\mathbb{R}_{+}\to\mathbb{R}$
such that
$\lim\limits_{i\to\omega}\hat{\mu}_{i}(B_{i})\leq\frac{\varepsilon}{2}$, where

$B_{i}=\\{(x_{i},y_{i})\in X_{i}\times
X_{i};K_{i}(x_{i},y_{i})\geq\rho_{2}^{\varepsilon}(d(x_{i},y_{i}))\\}.$

Moreover, if $\psi$ is measurably proper then, for all $\varepsilon>0$, exists
$\rho_{1}^{\varepsilon}:\mathbb{R}_{+}\to\mathbb{R}$ such that
$\lim\limits_{i\to\omega}\hat{\mu}_{i}(A_{i})\leq\frac{\varepsilon}{2}$, where

$A_{i}=\\{(x_{i},y_{i})\in X_{i}\times
X_{i};K_{i}(x_{i},y_{i})\leq\rho_{1}^{\varepsilon}(d(x_{i},y_{i}))\\}$

and the functions $\rho^{\varepsilon}_{1},\rho^{\varepsilon}_{2}$ go to
infinity at infinity. In particular,
$\lim\limits_{i\to\omega}\hat{\mu}_{i}(A_{i}\cup B_{i})\leq\varepsilon$ and
the maps $\rho_{1}^{\varepsilon}$, $\rho_{2}^{\varepsilon}$ are control
functions for $(K_{i})_{i\in\mathbb{N}}$ along the sequence of graphs
$X_{i}\times X_{i}\setminus(A_{i}\cup B_{i})$ with the induced length metric
of $X_{i}\times X_{i}$.

###### Proof.

Fix an $\varepsilon>0$ and non-principal ultrafilter
$\omega\in\partial\beta\mathbb{N}$, recall the notation of Proposition 3.22,
where $H_{\omega,k,L}$ is approximated by the sequence of sets
$H_{i,k,L}=\\{(x_{i},y_{i})\in X_{i}\times
X_{i};d(x_{i},y_{i})=k,K_{i}(x_{i},y_{i})\in[L,L+1)\\}$, i.e,
$H_{\omega,k,L}=[H_{i,k,L}]_{\omega}$ and
$S_{\omega,k}=\\{(x,y)\in\partial\beta X\times\partial\beta X;d(x,y)=k\\}$,
for $k\in\mathbb{N}$ and $L\in\mathbb{Z}$. Notice that
$S_{\omega,k}\subseteq\overline{E_{k}}$ has finite measure and
$S_{\omega,k}=\sqcup_{L=-\infty}^{\infty}H_{\omega,k,L}$. So, for any
$k\in\mathbb{N}$, there is a $L^{\varepsilon}_{k}\in\mathbb{N}$ such that

$\sum_{L=L^{\varepsilon}_{k}}^{\infty}\hat{\mu}_{\omega}(H_{\omega,k,L})\leq\dfrac{\varepsilon}{2^{k+1}}.$

For every $t\in\mathbb{R}_{+}$, define $\rho_{2}^{\varepsilon}(t)=L_{\lfloor
t\rfloor}^{\varepsilon}$, where $\lfloor t\rfloor$ is the integer part of $t$.
Notice that $B_{i}=\\{(x_{i},y_{i})\in X_{i}\times
X_{i};K_{i}(x_{i},y_{i})\geq\rho_{2}^{\varepsilon}(d(x_{i},y_{i}))\\}=\sqcup_{k\in\mathbb{N}}\sqcup_{L\geq
L_{k}^{\varepsilon}}H_{i,k,L}$, where $k=d(x_{i},y_{i})$. Thus,

$\displaystyle\lim_{i\to\omega}\hat{\mu}_{i}(B_{i})$
$\displaystyle=\lim_{i\to\omega}\hat{\mu}_{i}\bigg{(}\bigsqcup_{k\in\mathbb{N}}\bigsqcup_{L\geq
L_{k}^{\varepsilon}}^{\infty}H_{i,k,L}\bigg{)}$
$\displaystyle\leq\sum_{k\in\mathbb{N}}\sum_{L=L^{\varepsilon}_{k}}^{\infty}\hat{\mu}_{\omega}(H_{\omega,k,L})$
$\displaystyle\leq\sum_{k\in\mathbb{N}}\dfrac{\varepsilon}{2^{k+1}}$
$\displaystyle=\frac{\varepsilon}{2}.$

Assume now that $\psi$ is a measurably proper function, i.e, for every $C>0$,
the set $\\{\gamma\in\partial G(X);\psi(\gamma)\leq C\\}$ has finite measure.
For a given $C>0$, exist $k^{\prime}\in\mathbb{N}$ such that, for every $k\geq
k^{\prime}$, we have $\hat{\mu}_{\omega}(\\{\gamma\in
S_{\omega,{k}};\psi(\gamma)\leq C\\})\leq\varepsilon$, since we can decompose
$\partial G(X)$ as $\sqcup_{k\in\mathbb{N}}S_{\omega,k}$. We define the map
$\rho_{1}^{\varepsilon}$ for every positive real number $t$ as

$\rho_{1}^{\varepsilon}(\lfloor
t\rfloor)=\max\bigg{\\{}C>0;\hat{\mu}_{\omega}\bigg{(}\\{\gamma\in
S_{\omega,\lfloor t\rfloor};\psi(\gamma)\leq
C\\}\bigg{)}\leq\dfrac{\varepsilon}{2^{\lfloor t\rfloor+1}}\bigg{\\}}.$

Furthermore, the set $A_{i}=\\{(x_{i},y_{i})\in X_{i}\times
X_{i};K_{i}(x_{i},y_{i})\leq\rho_{1}^{\varepsilon}(d(x_{i},y_{i}))\\}$ can
described as
$\sqcup_{k\in\mathbb{N}}\sqcup_{L=-\infty}^{\rho_{1}^{\varepsilon}(k)-1}H_{i,k,L}$,
therefore

$\displaystyle\lim_{i\to\omega}\hat{\mu}_{i}(A_{i})$
$\displaystyle=\lim_{i\to\omega}\hat{\mu}_{i}\bigg{(}\bigsqcup_{k}\bigsqcup_{L=-\infty}^{\rho_{1}^{\varepsilon}(k)-1}H_{i,k,L}\bigg{)}$
$\displaystyle\leq\sum_{k\in\mathbb{N}}\sum_{L=-\infty}^{\rho_{1}^{\varepsilon}(k)-1}\hat{\mu}_{\omega}(H_{\omega,k,L})$
$\displaystyle\leq\sum_{k\in\mathbb{N}}\dfrac{\varepsilon}{2^{k+1}}$
$\displaystyle=\frac{\varepsilon}{2}.$

By the construction, it follow that

$\lim\limits_{i\to\omega}\hat{\mu}_{i}(A_{i}\cup B_{i})\leq\varepsilon$

Even more, the following inequality follows if we use the old length metrics
of the graphs $X_{i}$

$\rho_{1}^{\varepsilon}(d(x_{i},y_{i}))\leq
K_{i}(x_{i},y_{i})\leq\rho_{2}^{\varepsilon}(d(x_{i},y_{i})),$

for $x_{i},y_{i}\in X_{i}\times X_{i}\setminus(A_{i}\cup B_{i})$. The old
length metric $d_{i}$ of each graph $X_{i}$ is necessary here because after we
remove the points in $A_{i}\cup B_{i}$ from the graph $X_{i}$, we might had
disconnected the graph or increase the distance between points. Thus, the
first inequality of the control functions
$\rho_{1}^{\varepsilon}(d(x_{i},y_{i}))\leq
K_{i}(x_{i},y_{i})\leq\rho_{2}^{\varepsilon}(d(x_{i},y_{i}))$ might not hold
any longer. The second inequality still valid because $\rho_{2}^{\varepsilon}$
is monotonous and $d|_{X_{i}}(x_{i},y_{i})\leq d|_{X_{i}\times X_{i}\setminus
A_{i}\cup B_{i}}(x_{i},y_{i})$, for all $x_{i},y_{i}\in X_{i}\times
X_{i}\setminus(A_{i}\cup B_{i})$.

Moreover, since $\psi$ is measurably proper, when $k$ tends to infinity then
$\rho_{1}^{\varepsilon}(k)$ goes to infinity and so
$\rho_{2}^{\varepsilon}(k)$ goes to infinity.∎

Even in this situation, we can conclude that the sequence of graphs related to
a measurably a-T-menable coarse boundary groupoid $\partial G(X)$ is “almost
asymptotically coarsely embeddable” in the following sense.

######  Theorem 4.25.

Let $\mathcal{X}=\\{X_{i}\\}_{i\in\mathbb{N}}$ be a sequence of finite graphs
with bounded degree and $X$ the related space of graphs. If the coarse
boundary groupoid $(\partial G(X),\mu_{\omega})$ is a measurably a-T-menable
for every non-principal ultrafilter $\omega\in\partial\beta\mathbb{N}$, then,
for every $\varepsilon>0$, there exist
$\\{Z_{i}\\}_{i\in\mathbb{N}}\subset\\{X_{i}\\}_{i\in\mathbb{N}}$ such that
$\lim\limits_{i\to\infty}{\mu}_{i}(Z_{i})\leq\varepsilon$ and
$\\{X_{i}\backslash Z_{i}\\}_{i\in\mathbb{N}}$ is asymptotically coarsely
embeddable into a Hilbert space $\mathcal{H}$ when equipped with the
restriction of the metric from $X$.

###### Proof.

Given a fixed non-principal ultrafilter $\omega\in\partial\beta\mathbb{N}$ and
a measurably proper conditionally negative definite function $\psi$ from
$(\partial G(X),\mu_{\omega})$ to $\mathbb{R}$. By Lemma 3.22, we can
approximate $\psi:\partial G(X)\to\mathbb{R}$ by $K_{i}:X_{i}\times
X_{i}\to\mathbb{R}$ along the non-principal ultrafilter
$\omega\in\partial\beta\mathbb{N}$.

First, for some constant $a\geq 0$, take the set $Z_{i}^{a}=\\{x\in
X_{i}:b_{i}(x)>a\\}$, where
$b_{i}(x)=\max\\{0,\sigma(P_{x}^{R}T_{i}P_{x}^{R})\\}$ and $T_{i}$ is the
operator induced by $K_{i}$. Since the sequence of kernels
$(K_{i})_{i\in\mathbb{N}}$ approximates a conditionally negative definite
function, the sequence $(b_{i}(x))_{i\in\mathbb{N}}$ tends to zero for every
$x\in X_{i}$, then $\lim\limits_{i\to\omega}\mu_{i}(Z_{i}^{a})=0$. So, for
every $a>0$ and every $\varepsilon>0$, there is $n\in\mathbb{N}$ such that
$\mu_{i}(Z_{i}^{a})\leq\varepsilon$, for all $i\geq n$. Take
$a_{i}\coloneqq\max\\{a\geq 0;\mu_{i}(Z_{i}^{a})\leq\frac{1}{i}\\}$. In the
same way, the sequence $(a_{i})_{i\in\mathbb{N}}$ converges to zero when $i$
goes along the non-principal ultrafilter $\omega\in\partial\beta\mathbb{N}$.
Therefore, the measure of $\mu_{i}(Z_{i}^{a_{i}})$ also tends to zero along
$\omega\in\partial\beta\mathbb{N}$. For $Z_{i}^{\prime}\coloneqq
Z_{i}^{a_{i}}$ and by construction,

$\bigcup_{x\in X_{i}\backslash
Z_{i}^{\prime}}\sigma\big{(}P_{x}^{R}T_{i}P_{x}^{R}\big{)}\in(-\infty,a_{i}].$

In order to apply Lemma 4.22, we need to verify that the same holds but now
under the projection $Q_{x}^{R}:\mathbb{B}(\ell^{2}(X_{i}\backslash
Z_{i}^{\prime}))\to\mathbb{B}(\ell^{2}(B_{R}(x)\cap X_{i}\backslash
Z_{i}^{\prime}))_{0}$, where

$\mathbb{B}(\ell^{2}(B_{R}(x)\cap X_{i}\backslash
Z_{i}^{\prime}))_{0}=\bigg{\\{}f:B_{R}(x)\cap X_{i}\backslash
Z_{i}^{\prime}\to\mathbb{R};\sum\limits_{y\in B_{R}(x_{j})\cap X_{i}\backslash
Z_{i}^{\prime}}f(y)=0\bigg{\\}}.$

Clearly, $P_{x}^{R}Q_{x}^{R}=Q_{x}^{R}P_{x}^{R}=Q_{x}^{R}$, for all $x\in
X_{i}\backslash Z_{i}^{\prime}$. Given
$f\in\mathbb{B}(\ell^{2}(X_{i}\backslash Z_{i}^{\prime}))$, we compute that:

$\displaystyle\langle Q_{x}^{R}T_{i}Q_{x}^{R}f,f\rangle$
$\displaystyle=\langle Q_{x}^{R}P_{x}^{R}T_{i}P_{x}^{R}Q_{x}^{R}f,f\rangle$
$\displaystyle=\langle P_{x}^{R}T_{i}P_{x}^{R}Q_{x}^{R}f,Q_{x}^{R}f\rangle.$

In particular, $\langle
P_{x}^{R}T_{i}P_{x}^{R}f^{\prime},f^{\prime}\rangle\leq a_{i}$, for all
$f^{\prime}\in\mathbb{B}(\ell^{2}(B_{R}(x)\cap X_{i}\backslash
Z_{i}^{\prime}))_{0}$ then

$\bigcup_{x\in X_{i}\backslash
Z_{i}^{\prime}}\sigma\big{(}Q_{x}^{R}T_{i}Q_{x}^{R}\big{)}\in(-\infty,a_{i}].$

For a fixed $R>0$, we can apply Lemma 4.22. So, there exist a sequence of
kernels $(K_{i,R}^{\prime})_{i\in\mathbb{N}}$ that is $R$-locally
conditionally negative definite and $||K_{i,R}^{\prime}-K_{i}||_{max}\leq
C_{i,R}.$ With the same argument done in Theorem 5.10, for
$R_{i}\coloneqq\max\\{R:C_{i,R}\leq\frac{1}{R}\\}$, we obtain a sequence
$(K_{i,R_{i}}^{\prime}:X_{i}\setminus Z_{i}^{\prime}\times X_{i}\setminus
Z_{i}^{\prime}\to\mathbb{R})_{i\in\mathbb{N}}$ of conditionally negative
definite functions that approximates $\psi$ and an increasing sequence of real
numbers $(R_{i})_{i\in\mathbb{N}}$, we denote the functions only by
$(K_{i}^{\prime})_{i\in\mathbb{N}}$.

We finish the proof with the existence of the control functions along the
fixed ultrafilter. For a given $\varepsilon>0$, by Lemma 4.24, there exist
control functions $\rho_{1}^{\varepsilon}$ and $\rho_{2}^{\varepsilon}$ going
to infinity at infinity just along the sets $(X_{i}\setminus
Z_{i}^{\prime})\times(X_{i}\setminus Z_{i}^{\prime})\setminus(A_{i}\cup
B_{i})$. Take

$Z_{i}\coloneqq\\{x_{i}\in X_{i}\setminus Z_{i}^{\prime};\exists y_{i}\in
X_{i}\setminus Z_{i}^{\prime},(x_{i},y_{i})\in A_{i}\cup B_{i}\\}$

that clearly has measure smaller than $\varepsilon$ along the ultrafilter.

Thus, the sequence of graphs $\\{X_{i}\setminus Z_{i}\\}_{i\in\mathbb{N}}$ is
asymptotically coarsely embeddable into a Hilbert space, since the
restrictions of $K_{i}^{\prime}$’s to $X_{i}\setminus Z_{i}$ preserve the
conditionally negative definite condition and the control functions constrains
holds along $X_{i}\setminus Z_{i}$ with the old graph metric $d|_{X_{i}}$.

We’ve seen until this point that the sequence of graphs is asymptotically
coarsely embeddable into a Hilbert space along the non-principal ultrafilter
$\omega\in\partial\beta\mathbb{N}$. Now, we can use the similar arguments made
before to extend the result for all natural numbers. For that, suppose that
there is a $\varepsilon>0$ such that for all $\\{Z_{i}\\}_{i\in\mathbb{N}}$
with $\lim\limits_{i\to\infty}\mu_{i}(Z_{i})\leq\varepsilon$, the sequence
$\\{X_{i}\setminus Z_{i}\\}_{i\in\mathbb{N}}$ is not asymptotically coarsely
embeddable into a Hilbert space $\mathcal{H}$. So, for all $R>0$, there is a
$i_{R}\in\mathbb{N}$ such that $X_{i_{R}}\setminus Z_{i_{R}}$ is not
asymptotically coarsely embeddable. The set $I_{R}=\\{i_{R};R>0\\}$ is
infinite, then there is a non-principal ultrafilter
$\omega\in\partial\beta\mathbb{N}$ such that $I_{R}\subseteq\omega$. Thus,
$\\{X_{i}\setminus Z_{i}\\}_{i\in\mathbb{N}}$ is not asymptotically coarsely
embeddable into a Hilbert space, leading us in a contradiction.

This finish our proof, concluding that $\\{X_{i}\backslash
Z_{i}\\}_{i\in\mathbb{N}}$ is asymptotically coarsely embeddable into a
Hilbert space $\mathcal{H}$ in regard to the old length metric. ∎

One can observe two major differences to the corresponding result in the
amenable case. First, instead of discarding an asymptotically negligible set
from the graph sequence as in the definition of property almost-A, here we
discard a subset of arbitrary small but positive measure. This is a
consequence of the lifting result for measurably proper conditionally negative
definite functions used in the proof of this theorem (Lemma 4.24). This
lifting seems indeed optimal since – other than in the topological case – a
measurably proper function $\psi(x,y)$ can still remain small subsets of small
measure even the distance $d(x,y)$ increases. This suggests that there should
exist a sequence of graphs witnessing this phenomenon and therefore requiring
$\varepsilon>0$ in the conclusion of the above theorem; however, we were
unable to construct such a sequence directly and therefore leave it as an open
question.

######  Question 4.26.

Does there exist a graph sequence such that $(\partial
G(X),\hat{\mu}_{\omega})$ is measurably a-T-menable, but does not satisfy the
conclusion of the above theorem for $\varepsilon=0$? Notice that such an
example cannot come from a sofic approximation of a group, as explained in the
next section.

Another important difference to the amenable case is that here the technique
of the proof forces us to use the old metric even after discarding parts of
the graphs as opposed to the induced metric. This naturally leads to the
following question:

######  Question 4.27.

Is the sequence of graphs $\\{X_{i}\backslash Z_{i}\\}_{i\in\mathbb{N}}$
(considered _with their own length metric_) asymptotically coarsely embeddable
into a Hilbert space?

Being unable to answer the above question, we are hesitant to decide what the
right notion of being “almost asymptotically coarsely embeddable into a
Hilbert space” should be. As an _ad hoc_ notion for the rest of the paper, we
say that a sequence of graphs $\mathcal{X}=\\{X_{i}\\}_{i\in\mathbb{N}}$ is
_weakly asymptotically embedable into a Hilbert space_ if it satisfies the
conclusion of Theorem 4.25.

##  5\. Applications to sofic approximations

In this section we apply the results presented in the previous section to the
case where the sequence of graphs comes from a sofic approximation of a
finitely generated group, thus linking the analytic properties of the group
and the coarse geometry properties of the sequence of graphs. We take a
finitely generated sofic group $\Gamma$ with sofic approximation
$\mathcal{G}=\\{X_{i}\\}_{i\in\mathbb{N}}$. Let $X$ be the space of graphs
constructed from the sofic approximation.

First we apply Theorem 4.19 to rededuce the following result:

######  Corollary 5.1 ([Kai19, Theorem 5.6]).

Let $\Gamma$ be a sofic group with sofic approximation
$\mathcal{G}=\\{X_{i}\\}_{i\in\mathbb{N}}$. Then, the sofic approximation
$\\{X_{i}\\}_{i\in\mathbb{N}}$ has property A on average if and only if the
group $\Gamma$ is amenable.

###### Proof.

Suppose that $\Gamma$ is amenable, then, the topological groupoid and thus the
measured groupoid $\partial\beta X\rtimes\Gamma$ is amenable. According to
[AFS16, Theorem 3.11], there is an almost everywhere isomorphism between
$({\partial G(X)},\hat{\mu}_{\omega})$ and $(\partial\beta
X,\mu_{\omega})\rtimes\Gamma$. Thus, by the Theorem 4.19, the sofic
approximation $\\{X_{i}\\}_{i\in\mathbb{N}}$ has property A on average.

On the other hand, if the sofic approximation $\\{X_{i}\\}_{i\in\mathbb{N}}$
has property A on average, then, the sofic approximation has property
almost-A, and thus the coarse boundary groupoid $\partial G(X)$ is measurably
amenable. For the reduction to the core $Z$, the groupoid $Z\rtimes\Gamma$ is
also amenable with a $\Gamma$-invariant measure over $Z$. Therefore, the group
$\Gamma$ is amenable.∎

Mergin this corollary with all equivalences of Theorem 4.19 we obtain the
following equivalences:

$\begin{array}[]{ccc}\mathcal{G}\text{ is hyperfinite}&&\\\ \Updownarrow&&\\\
\mathcal{G}\text{ has property almost-A }&\iff&\Gamma\text{ is amenable}\\\
\Updownarrow&&\\\ \mathcal{G}\text{ has property A on average }&&\\\
\end{array}$

In the a-T-menable case, we could analogously apply Theorem 4.25 to deduce
that every sofic approximation of an a-T-menable group becomes coarsely
embeddable into a Hilbert space after removing an arbitrary small propotion of
the vertices (but keeping the original metric). However, we can do better
using the fact that a-T-menability of a sofic group gives us a conditionally
negative definite function on the boundary that only depends on the group.
Thus, the sofic approximation of a a-T-menable group is weakly asymptotically
coarsely embeddable into a Hilbert space with $\varepsilon=0$.

######  Proposition 5.2.

Let $\Gamma$ be a sofic finitely generated group with sofic approximation
$\mathcal{G}=\\{X_{i}\\}_{i\in\mathbb{N}}$. Then, the group $\Gamma$ is a-T-
menable if and only if there is a sequence of subgraphs
$\\{Z_{i}\\}_{i\in\mathbb{N}}\subset\\{X_{i}\\}_{i\in\mathbb{N}}$ of the sofic
approximation $\mathcal{G}$ such that
$\lim\limits_{i\to\infty}\frac{|Z_{i}|}{|X_{i}|}=0$ and $\\{X_{i}\setminus
Z_{i}\\}_{i\in\mathbb{N}}$ is asymptotically coarsely embeddable into a
Hilbert space $\mathcal{H}$ with the old metric of the sofic approximation.

###### Proof.

First, assume that the group $\Gamma$ is a-T-menable. From the proper
conditional negative definite function $\psi:\Gamma\to\mathbb{R}$, we can
induce a function $\psi^{\prime}$ on the groupoid $\partial\beta
X\rtimes\Gamma$ by $\psi^{\prime}(x,g)=\psi(g)$ that is also proper
conditional negative definite. In the measurable perspective, the function
$\psi^{\prime}$ is also measurably proper in $(\partial\beta
X,\mu_{\omega})\rtimes\Gamma$, for every non-principal ultrafilter
$\omega\in\partial\beta\mathbb{N}$. Even more, the groupoid $(\partial\beta
X,\mu_{\omega})\rtimes\Gamma$ is almost everywhere isomorphic to $(\partial
G(X),\hat{\mu}_{\omega})$. Thus, the coarse boundary groupoid $(\partial
G(X),\hat{\mu}_{\omega})$ is measurably a-T-menable with a function
$\psi^{\prime}$. Notice then, that the map $\psi^{\prime}$ is the same in
every fiber of $\partial G(X)$. So, the same phenomena that happens at Lemma
4.24 does not occur here, but we still need to discard a set of measure zero
from the boundary due to the almost everywhere isomorphism of the groupoids
and the measurable approximation.

By Proposition 3.22, for a fixed ultrafilter
$\omega\in\partial\beta\mathbb{N}$, there is an approximation of
$\psi^{\prime}:(\partial G(X),\hat{\mu}_{\omega})\to\mathbb{R}$ via a sequence
of functions $(K_{i}:X_{i}\times X_{i}\to\mathbb{R})_{i\in\mathbb{N}}$. Using
the same arguments from Theorem 4.25, there exist a sequence of conditionally
negative definite functions $(K_{i,R_{i}}^{\prime}:X_{i}\setminus Z_{i}\times
X_{i}\setminus Z_{i}\to\mathbb{R})_{i\in\mathbb{N}}$ that also approximates
the function $\psi^{\prime}$, when
$R_{i}^{\prime}=\max\\{R>0;C_{i,R}\leq\frac{1}{R}\\}$ is an increasing
sequence of real numbers and the measure of $\mu_{i}(Z_{i})$ tends to zero. We
denote this approximation only by $(K_{i}^{\prime})_{i\in\mathbb{N}}$.

By the conditions of the proper conditionally definite negative function
$\psi$, one can take control functions $\rho_{1}$, $\rho_{2}$ in a uniform
way, as in the construction of Lemma 4.20. Remember that the approximation of
$\psi$ by the sequence of kernels is a measurable approximation, so we still
need to discard an negligible subsets along the sequence of graphs to also fix
the conditional negative definiteness of the sequence. Thus, the sofic
approximation is weakly asymptotically coarsely embeddable into a Hilbert
space $\mathcal{H}$.

The converse follows from Theorem 2.15, by the fact that there is an almost
everywhere isomorphism between the coarse boundary groupoid associated to
$\\{X_{i}\\}_{i\in\mathbb{N}}$ and the one related to the sequence
$\\{X_{i}\setminus Z_{i}\\}_{i\in\mathbb{N}}$ and they are both a-T-menable.∎

Analogously to the general case of a graph sequence, we ask the following
question:

######  Question 5.3.

Is the sofic approximation $\mathcal{X}^{\prime}=\\{X_{i}\backslash
Z_{i}\\}_{i\in\mathbb{N}}$ (considered _with their own length metric_)
asymptotically coarsely embeddable into a Hilbert space?

## References

  * [AD02] Claire Anantharaman-Delaroche.  Amenability and exactness for dynamical systems and their c*-algebras.  Transactions of the American Mathematical Society, pages 4153–4178, 2002.
  * [AD13] Claire Anantharaman-Delaroche.  The Haagerup property for discrete measured groupoids.  In Operator algebra and dynamics, pages 1–30. Springer, 2013.
  * [AFS16] Vadim Alekseev and Martin Finn-Sell.  Sofic boundaries of groups and coarse geometry of sofic approximations.  arXiv preprint arXiv:1608.02242, 2016.
  * [AGŠ12] Goulnara Arzhantseva, Erik Guentner, and Ján Špakula.  Coarse non-amenability and coarse embeddings.  Geometric and Functional Analysis, 22(1):22–36, 2012.
  * [AR01] Claire Anantharaman and Jean Renault.  Amenable groupoids.  Contemporary Mathematics, 282:35–46, 2001.
  * [BO08] Nathanial Patrick Brown and Narutaka Ozawa.  C*-Algebras and Finite-Dimensional Approximations, volume 88.  American Mathematical Soc., 2008.
  * [Bun11] Jared Bunn.  Bounded Geometry and Property A for Non-metrizable Coarse Spaces.  2011\.
  * [Car15] Alessandro Carderi.  Groups, Actions and von Neumann algebras.  PhD thesis, Ecole Normale Supérieure de Lyon-ENS LYON, 2015.
  * [CCJ+12] Pierre-Alain Cherix, Michael Cowling, Paul Jolissaint, Pierre Julg, and Alain Valette.  Groups with the Haagerup property: Gromov’s a-T-menability, volume 197.  Birkhäuser, 2012.
  * [CGDLS18] Alessandro Carderi, Damien Gaboriau, and Mikael De La Salle.  Non-standard limits of graphs and some orbit equivalence invariants.  arXiv preprint arXiv:1812.00704, 2018.
  * [CKTD13] Clinton Conley, Alexander Kechris, and Robin Tucker-Drob.  Ultraproducts of measure preserving actions and graph combinatorics.  Ergodic Theory and Dynamical Systems, 33(2):334–374, 2013.
  * [CLP15] Valerio Capraro, Martino Lupini, and Vladimir Pestov.  Introduction to sofic and hyperlinear groups and Connes’ Embedding Conjecture, volume 1.  Springer, 2015.
  * [CWW13] Xiaoman Chen, Qin Wang, and Xianjin Wang.  Characterization of the Haagerup property by fibred coarse embedding into Hilbert space.  Bulletin of the London Mathematical Society, 45(5):1091–1099, 2013\.
  * [CWY13] Xiaoman Chen, Qin Wang, and Guoliang Yu.  The maximal coarse Baum–Connes Conjecture for spaces which admit a fibred coarse embedding into Hilbert space.  Advances in Mathematics, 249:88–130, 2013.
  * [Ele06] Gábor Elek.  The combinatorial cost.  arXiv preprint math/0608474, 2006.
  * [Ele10] Gábor Elek.  Parameter testing in bounded degree graphs of subexponential growth.  Random Structures & Algorithms, 37(2):248–270, 2010.
  * [Fre01] D.H Fremlin.  Measure Theory, Volume 1: The Irreducible Minimum.  The American Mathematical Monthly, 108(7):678, 2001.
  * [FS14] Martin Finn-Sell.  Fibred coarse embeddings, a-T-menability and the coarse analogue of the Novikov Conjecture.  Journal of Functional Analysis, 267(10):3758–3782, 2014.
  * [FSW14] Martin Finn-Sell and Nick Wright.  Spaces of graphs, boundary groupoids and the coarse Baum–Connes Conjecture.  Advances in Mathematics, 259:306–338, 2014.
  * [Goe09] Geoff Goehle.  Groupoid crossed products.  arXiv preprint arXiv:0905.4681, 2009.
  * [Gro99] Mikhael Gromov.  Endomorphisms of symbolic algebraic varieties.  Journal of the European Mathematical Society, 1(2):109–197, 1999\.
  * [Kai19] Tom Kaiser.  Combinatorial cost: a coarse setting.  Transactions of the American Mathematical Society, 2019.
  * [Khu14] Ana Khukhro.  Embeddable box spaces of free groups.  Mathematische Annalen, 360(1):53–66, 2014.
  * [KM04] Alexander Kechris and Benjamin Miller.  Topics in orbit equivalence, volume 1852.  Springer, Science and Business Media, 2004.
  * [KW12] Ana Khukhro and Nick J Wright.  Expanders and property A.  Algebraic & Geometric Topology, 12(1):37–47, 2012.
  * [Löh17] Clara Löh.  Geometric Group Theory, An introduction.  2017\.
  * [LPS88] Alexander Lubotzky, Ralph Phillips, and Peter Sarnak.  Ramanujan graphs.  Combinatorica, 8(3):261–277, 1988.
  * [Mar17] Andrew Marks.  A short proof of the Connes-Feldman-Weiss Theorem.  preprint, 2017.
  * [MOSS15] Masato Mimura, Narutaka Ozawa, Hiroki Sako, and Yuhei Suzuki.  Group approximation in Cayley topology and coarse geometry, Part III: Geometric property (T).  Algebraic & Geometric Topology, 15(2):1067–1091, 2015.
  * [MS19] Masato Mimura and Hiroki Sako.  Group approximation in Cayley topology and coarse geometry Part I: Coarse embeddings of amenable groups.  Journal of Topology and Analysis, pages 1–47, 2019.
  * [Now07] Piotr W Nowak.  Coarsely embeddable metric spaces without property A.  Journal of Functional Analysis, 252(1):126–136, 2007.
  * [Pes08] Vladimir G Pestov.  Hyperlinear and sofic groups: a brief guide.  Bulletin of Symbolic Logic, 14(4):449–480, 2008.
  * [Ren06] Jean Renault.  A groupoid approach to C*-algebras, volume 793.  Springer, 2006.
  * [Roe03] John Roe.  Lectures on coarse geometry.  Number 31. American Mathematical Soc., 2003.
  * [Sau02] Roman Sauer.  L2-invariants of groups and discrete measured groupoids.  2002\.
  * [STY02] Georges Skandalis, Jean-Louis Tu, and Guoliang Yu.  The coarse Baum–Connes Conjecture and groupoids.  Topology, 41(4):807–834, 2002.
  * [Wei00] Benjamin Weiss.  Sofic groups and dynamical systems.  Sankhya: The Indian Journal of Statistics, Series A, pages 350–359, 2000.
  * [Wil06] Rufus Willett.  Some notes on property A.  arXiv preprint math/0612492, 2006.
  * [Wil11] Rufus Willett.  Property A and graphs with large girth.  Journal of Topology and Analysis, 3(03):377–384, 2011.
  * [Wil15] Rufus Willett.  Random graphs, weak coarse embeddings, and higher index theory.  Journal of Topology and Analysis, 7(03):361–388, 2015.
  * [WY12] Rufus Willett and Guoliang Yu.  Higher index theory for certain expanders and Gromov monster groups, I.  Advances in mathematics, 229(3):1380–1416, 2012.
  * [Yu00] Guoliang Yu.  The coarse Baum–Connes Conjecture for spaces which admit a uniform embedding into Hilbert space.  Inventiones Mathematicae, 139(1):201–240, 2000.