In here we are interested in a noncommutative scalar \Phi^4 theory. As an example, we will take the underlying noncommutative space to be fuzzy {\bf CP}^n_l which is obtained by Berezin quantization \cite{Berezin:1974du} of the projective space {\bf CP}^n=SU(n+1)/U(n).
Fuzzy {\bf CP}^n_l is given by the spectral triple \cite{Frohlich:1993es,Connes:1994yd} (see also \cite{Dolan:2001gn})
\begin{eqnarray} {\bf CP}^n_l=({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l}). \end{eqnarray}
{\bf H}_{n,l} is the Hilbert space associated with the irreducible representation of su(n+1) which is given by the totally symmetrized tensor product of l fundamental representations, viz (l,0,0,...,0). The dimension of this representation is given by
\begin{eqnarray} N_{n,l}=\frac{(n+l)!}{n!l!}. \end{eqnarray}
The Hilbert space {\bf H}_{n,l} is thus acted on by the complete matrix algebra {\rm Mat}_{N_{n,l}} of finite dimension N_{n,l} with inner product defined by
\begin{eqnarray} (f,g)=\frac{1}{N_{n,l}}Tr f^{\dagger}g. \end{eqnarray}
The \Delta_{n,l} is the Laplace operator on fuzzy {\bf CP}^n_l defined, in terms of the generators L_i of su(n+1) in the representation (l,0,0,...,0), by the quadratic Casimir
\begin{eqnarray} \Delta_{n,l}(f)=[L_i,[L_i,f]]. \end{eqnarray}
Derivation on fuzzy {\bf CP}^n_l are given precisely by the adjoint action of the generators L_i, i.e.
\begin{eqnarray} {\rm ad}L_i(f)=[L_i,f]=(L_i^L-L_i^R)(f). \end{eqnarray}
This shows explicitly that the space of functions on fuzzy {\bf CP}^n_l, which is the matrix algebra {\rm Mat}_{d_{n,l}}, decomposes under the action of SU(n+1) as the direct sum of the irreducible representations (m,...,m) of SU(n + 1),viz
\begin{eqnarray} (l,...,0)\otimes \overline{(l,...,0)}=\oplus_{m=0}^l(m,0,...,0,m). \end{eqnarray}
Hence, the matrix algebra {\rm Mat}_{N_{n,l}} is the endomorphism {\bf Hom}(V_{n,l})=V_{n,l}\otimes V_{n,l}^* where V_{n,l} is the vector space associated with the representation (l,0,...,0).
The polarization tensors T_{m \sigma} which transform in the irreducible representation (m,...,m) are the eigenmatrices of the Laplace \Delta_{n,l} with eigenvalue \lambda_m and degeneracy d_{n,m} given by
\begin{eqnarray} \lambda_m=2m(m+n). \end{eqnarray}
And
\begin{eqnarray} d_{n,m}=\frac{n(2m+n)((m+n-1)!)^2}{(n!)^2(m!)^2}. \end{eqnarray}
Indeed, the index \sigma in T_{m \sigma} denotes the other quantum numbers required to specify the representation (m,...,m).
The above spectrum is precisely the spectrum of the Laplace operator on commutative {\bf CP}^n only cutoff at m=l. The commutative limit is therefore l\longrightarrow\infty in which the eigenmatrices T_{m \sigma} go over to the correct eigenfunctions Y_{m \sigma} (spherical harmonics) on {\bf CP}^n.
The coordinate functions X_i on fuzzy {\bf CP}^n_l are obtained from suitably
rescaling the generators L_i of su(n + 1) in the representation (l,0,...,0), namely X_i=a_l L_i. We have then the commutation relations
\begin{eqnarray} [X_i,X_j]=ia_l f_{ijk}X_k. \end{eqnarray}
The data contained in the spectral triple ({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l}) is sufficient to write down an Euclidean action for a scalar field theory on fuzzy {\bf CP}^n_l. A scalar field \Phi is a hermitian N\times N matrix in {\rm Mat}_{N_{n,l}}, i.e. N\equiv N_{n,l}. An action on fuzzy {\bf CP}^n_l is then given by
\begin{eqnarray} S=Tr(\Phi C_2\Phi+r\Phi^2+g\Phi^4). \end{eqnarray}
The partition function of the model is given by
\begin{eqnarray} Z=\int d\mu_D(\Phi)\exp(-\beta S[\Phi]). \end{eqnarray}
d\mu_D(\Phi) is Dyson measure on {\rm Mat}_{N}.
We can diagonalize the matrix \Phi as \Phi=U\Lambda U^{\dagger}. We also write \Phi=\Phi_{\mu}T_{\mu} or \Phi_{\mu}=Tr\Phi T_{\mu} where T_{\mu}, \mu=1,...,N^2 are the generators of u(N). The partition function then becomes
\begin{eqnarray} Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\ &\times &\int d\mu_H(U)\exp\big(-\beta K_{\mu\nu}Tr U\Lambda U^{\dagger}T_{\mu}Tr U\Lambda U^{\dagger}T_{\nu}\big). \end{eqnarray}
The Vandermonde \Delta^2(\Lambda) is given by
\begin{eqnarray} \Delta^2(\Lambda)=\prod_{i=1}^N(\lambda_i-\lambda_j)^2. \end{eqnarray}
And d\mu_H(U) is the Haar measure on U(N). The kinetic matrix K_{\mu\nu} is given on the other hand by
\begin{eqnarray} K_{\mu\nu}=-Tr[L_i,T_{\mu}][L_i,T_{\nu}]. \end{eqnarray}
We can now perform a hopping parameter expansion to perform the integral over U (by using the properties (A\otimes B)(C\otimes D)=(AC\otimes BD) and Tr(A\otimes B)=TrA TrB and the orthogonality relation of the Haar measure) as follows \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray} Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\ &\times &\sum_k\frac{(-\beta)^k}{k!}\int d\mu_H(U)\prod_{i=1}^kK_{\mu_i\nu_i}Tr U\Lambda U^{\dagger}T_{\mu_i}Tr U\Lambda U^{\dagger}T_{\nu_i}\nonumber\\ &=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\ &\times & \int d\mu_H(U)Tr \bigg((U\otimes...\otimes U)(\Lambda\otimes...\otimes\Lambda)(U^{\dagger}\otimes...\otimes U^{\dagger})(T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k})\bigg)\nonumber\\ &=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\ &\times &\sum_{\rho}\frac{1}{{\rm dim}(\rho)} Tr_{\rho}(\Lambda\otimes...\otimes\Lambda)Tr (T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k}). \end{eqnarray}
After a very long and involved calculation we find upto the order O(\beta^2) the multitrace action \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray} Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\exp(-\beta (S_1+S_2)), \end{eqnarray}
where
\begin{eqnarray} S_1={\cal J}_1=a_{2,0}Tr\Lambda^2+a_{1,1}(Tr\Lambda)^2. \end{eqnarray}
And
\begin{eqnarray} S_2&=&\frac{\beta}{2}({\cal J}_1^2-{\cal J}_2)\nonumber\\ {\cal J}_2&=&a_{4,0}Tr\Lambda^4+a_{3,1}Tr\Lambda^3 Tr\Lambda+a_{2,2}(Tr\Lambda^2)^2+a_{2,1,1}Tr\Lambda^2(Tr\Lambda)^2+a_{1,1,1,1}(Tr\Lambda)^4. \end{eqnarray}
Fuzzy {\bf CP}^n_l is given by the spectral triple \cite{Frohlich:1993es,Connes:1994yd} (see also \cite{Dolan:2001gn})
\begin{eqnarray} {\bf CP}^n_l=({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l}). \end{eqnarray}
{\bf H}_{n,l} is the Hilbert space associated with the irreducible representation of su(n+1) which is given by the totally symmetrized tensor product of l fundamental representations, viz (l,0,0,...,0). The dimension of this representation is given by
\begin{eqnarray} N_{n,l}=\frac{(n+l)!}{n!l!}. \end{eqnarray}
The Hilbert space {\bf H}_{n,l} is thus acted on by the complete matrix algebra {\rm Mat}_{N_{n,l}} of finite dimension N_{n,l} with inner product defined by
\begin{eqnarray} (f,g)=\frac{1}{N_{n,l}}Tr f^{\dagger}g. \end{eqnarray}
The \Delta_{n,l} is the Laplace operator on fuzzy {\bf CP}^n_l defined, in terms of the generators L_i of su(n+1) in the representation (l,0,0,...,0), by the quadratic Casimir
\begin{eqnarray} \Delta_{n,l}(f)=[L_i,[L_i,f]]. \end{eqnarray}
Derivation on fuzzy {\bf CP}^n_l are given precisely by the adjoint action of the generators L_i, i.e.
\begin{eqnarray} {\rm ad}L_i(f)=[L_i,f]=(L_i^L-L_i^R)(f). \end{eqnarray}
This shows explicitly that the space of functions on fuzzy {\bf CP}^n_l, which is the matrix algebra {\rm Mat}_{d_{n,l}}, decomposes under the action of SU(n+1) as the direct sum of the irreducible representations (m,...,m) of SU(n + 1),viz
\begin{eqnarray} (l,...,0)\otimes \overline{(l,...,0)}=\oplus_{m=0}^l(m,0,...,0,m). \end{eqnarray}
Hence, the matrix algebra {\rm Mat}_{N_{n,l}} is the endomorphism {\bf Hom}(V_{n,l})=V_{n,l}\otimes V_{n,l}^* where V_{n,l} is the vector space associated with the representation (l,0,...,0).
The polarization tensors T_{m \sigma} which transform in the irreducible representation (m,...,m) are the eigenmatrices of the Laplace \Delta_{n,l} with eigenvalue \lambda_m and degeneracy d_{n,m} given by
\begin{eqnarray} \lambda_m=2m(m+n). \end{eqnarray}
And
\begin{eqnarray} d_{n,m}=\frac{n(2m+n)((m+n-1)!)^2}{(n!)^2(m!)^2}. \end{eqnarray}
Indeed, the index \sigma in T_{m \sigma} denotes the other quantum numbers required to specify the representation (m,...,m).
The above spectrum is precisely the spectrum of the Laplace operator on commutative {\bf CP}^n only cutoff at m=l. The commutative limit is therefore l\longrightarrow\infty in which the eigenmatrices T_{m \sigma} go over to the correct eigenfunctions Y_{m \sigma} (spherical harmonics) on {\bf CP}^n.
The coordinate functions X_i on fuzzy {\bf CP}^n_l are obtained from suitably
rescaling the generators L_i of su(n + 1) in the representation (l,0,...,0), namely X_i=a_l L_i. We have then the commutation relations
\begin{eqnarray} [X_i,X_j]=ia_l f_{ijk}X_k. \end{eqnarray}
The data contained in the spectral triple ({\bf H}_{n,l}, {\rm Mat}_{N_{n,l}},\Delta_{n,l}) is sufficient to write down an Euclidean action for a scalar field theory on fuzzy {\bf CP}^n_l. A scalar field \Phi is a hermitian N\times N matrix in {\rm Mat}_{N_{n,l}}, i.e. N\equiv N_{n,l}. An action on fuzzy {\bf CP}^n_l is then given by
\begin{eqnarray} S=Tr(\Phi C_2\Phi+r\Phi^2+g\Phi^4). \end{eqnarray}
The partition function of the model is given by
\begin{eqnarray} Z=\int d\mu_D(\Phi)\exp(-\beta S[\Phi]). \end{eqnarray}
d\mu_D(\Phi) is Dyson measure on {\rm Mat}_{N}.
We can diagonalize the matrix \Phi as \Phi=U\Lambda U^{\dagger}. We also write \Phi=\Phi_{\mu}T_{\mu} or \Phi_{\mu}=Tr\Phi T_{\mu} where T_{\mu}, \mu=1,...,N^2 are the generators of u(N). The partition function then becomes
\begin{eqnarray} Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\ &\times &\int d\mu_H(U)\exp\big(-\beta K_{\mu\nu}Tr U\Lambda U^{\dagger}T_{\mu}Tr U\Lambda U^{\dagger}T_{\nu}\big). \end{eqnarray}
The Vandermonde \Delta^2(\Lambda) is given by
\begin{eqnarray} \Delta^2(\Lambda)=\prod_{i=1}^N(\lambda_i-\lambda_j)^2. \end{eqnarray}
And d\mu_H(U) is the Haar measure on U(N). The kinetic matrix K_{\mu\nu} is given on the other hand by
\begin{eqnarray} K_{\mu\nu}=-Tr[L_i,T_{\mu}][L_i,T_{\nu}]. \end{eqnarray}
We can now perform a hopping parameter expansion to perform the integral over U (by using the properties (A\otimes B)(C\otimes D)=(AC\otimes BD) and Tr(A\otimes B)=TrA TrB and the orthogonality relation of the Haar measure) as follows \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray} Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\nonumber\\ &\times &\sum_k\frac{(-\beta)^k}{k!}\int d\mu_H(U)\prod_{i=1}^kK_{\mu_i\nu_i}Tr U\Lambda U^{\dagger}T_{\mu_i}Tr U\Lambda U^{\dagger}T_{\nu_i}\nonumber\\ &=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\ &\times & \int d\mu_H(U)Tr \bigg((U\otimes...\otimes U)(\Lambda\otimes...\otimes\Lambda)(U^{\dagger}\otimes...\otimes U^{\dagger})(T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k})\bigg)\nonumber\\ &=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\sum_k\frac{(-\beta)^k}{k!}K_{\mu_1\nu_1}...K_{\mu_k\nu_k}\nonumber\\ &\times &\sum_{\rho}\frac{1}{{\rm dim}(\rho)} Tr_{\rho}(\Lambda\otimes...\otimes\Lambda)Tr (T_{\mu_1}\otimes T_{\nu_1}\otimes...\otimes T_{\mu_k}\otimes T_{\nu_k}). \end{eqnarray}
After a very long and involved calculation we find upto the order O(\beta^2) the multitrace action \cite{Saemann:2010bw,OConnor:2007ibg}
\begin{eqnarray} Z&=&\int \prod_{i=1}^Nd\lambda_i\Delta^2(\Lambda)\exp\big(-\beta Tr(r\Lambda^2+g\Lambda^4)\big)\exp(-\beta (S_1+S_2)), \end{eqnarray}
where
\begin{eqnarray} S_1={\cal J}_1=a_{2,0}Tr\Lambda^2+a_{1,1}(Tr\Lambda)^2. \end{eqnarray}
And
\begin{eqnarray} S_2&=&\frac{\beta}{2}({\cal J}_1^2-{\cal J}_2)\nonumber\\ {\cal J}_2&=&a_{4,0}Tr\Lambda^4+a_{3,1}Tr\Lambda^3 Tr\Lambda+a_{2,2}(Tr\Lambda^2)^2+a_{2,1,1}Tr\Lambda^2(Tr\Lambda)^2+a_{1,1,1,1}(Tr\Lambda)^4. \end{eqnarray}
The coefficients a can be found in \cite{Saemann:2010bw,OConnor:2007ibg}.
For example, we compute on {\bf CP}^1_l, where the highest weight l is related to the ordinary spin s by s=l/2, the following quantities N=l+1, Tr K\sim N^4, Tr K^2\sim 4N^6/3, Tr K^3\sim 2N^8 and a_{3,1}\sim 8/3.
Whereas we compute in four dimensions on {\bf CP}^2_l the following quantities N=(l+1)(l+2)/2, i.e. l^2\sim 2N and as a consequence Tr K\sim 8 N^3/3, Tr K^2\sim 8N^4, Tr K^3\sim 128 N^5/5 and a_{3,1}\sim -256/9N^2.
It was argued in \cite{Ydri:2017riq} that the non-vanishing of the multitrace term proportional to a_{3,1} is a necessary and a sufficient condition for a stable Ising phase and a stable emergent background geometry. It was also argued there that quantum gravity in (or more precisely quantized spaces of) dimensions higher than two can thus be obtained from pure multitrace matrix models along these same lines.
By neglecting all multitrace terms we find that the quartic matrix model N Tr(\tilde{r}\Phi^2+\tilde{g}\Phi^4) (where \tilde{r}=r/N^{3/2} and \tilde{g}=g/N^2) in the limit of very weak coupling \tilde{g}\longrightarrow 0 is dominated by the Wigner semi-circle law given by
\begin{eqnarray} \rho(\lambda)=\frac{\tilde{r}}{\pi}\sqrt{\frac{2}{\tilde{r}}-\lambda^2}. \end{eqnarray}
However, by including all multitrace terms we get noncommutative \Phi^4 on fuzzy {\bf CP}^n_l which is also dominated (or more precisely its eigenvalue sector) in the limit of weak coupling \tilde{g}\longrightarrow 0 by a Wigner semi-circle corresponding to the following simple model \cite{Steinacker:2005wj}
\begin{eqnarray} S=\frac{2N}{\alpha_0^2(m)}Tr\Phi^2. \end{eqnarray}
In other words, we must make in the above Wigner semi-circle law the following replacement
\begin{eqnarray} \tilde{r}\longrightarrow\frac{2}{\alpha_0^2(m)}. \end{eqnarray}
\alpha_0 is the maximum eigenvalue and it is given in terms of the cutoff \Lambda\sim l on fuzzy {\bf CP}^n_l and the mass m^2\sim r by the relation \cite{Steinacker:2005wj}
\begin{eqnarray} \alpha_0^2(m)=4c(m,\Lambda)\Lambda^{d-2}. \end{eqnarray}
We have explicitly
\begin{eqnarray} c(m,\Lambda)=\frac{1}{16\pi^2}\bigg(1-\frac{m^2}{\Lambda^2}\ln(1+\frac{\Lambda^2}{m^2})\bigg)~,~d=4,n=2. \end{eqnarray}
\begin{eqnarray} c(m,\Lambda)=\frac{1}{4\pi}\ln(1+\frac{\Lambda^2}{m^2})~,~d=2,n=1. \end{eqnarray}
Thus, by studying the multitrace approximation at weak coupling we can determine when the Wigner behavior goes from the behavior of the pure matrix model d=0 to the behavior of noncommutative matrix models at d=2 and d=4 as we vary for example the parameter a_{3,1}. The emergence of the correct behavior is an indication of a stable Ising phase and as a consequence a stable emergent background geometry.
\bibitem{Berezin:1974du}
F.~A.~Berezin, ``General Concept of Quantization,'' Commun.\ Math.\ Phys.\ {\bf 40}, 153 (1975). doi:10.1007/BF01609397
%%CITATION = doi:10.1007/BF01609397;
%% %307 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Frohlich:1993es}
\bibitem{Frohlich:1993es}
J.~Frohlich and K.~Gawedzki,
``Conformal field theory and geometry of strings,'' In *Vancouver 1993, Proceedings, Mathematical quantum theory, vol. 1* 57-97, and Preprint - Gawedzki, K. (rec.Nov.93) 44 p
[hep-th/9310187].
%%CITATION = HEP-TH/9310187;
%% %74 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Connes:1994yd}
\bibitem{Connes:1994yd}
A.~Connes, ``Noncommutative geometry,'' %%CITATION = INSPIRE-391003;
%% %344 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Dolan:2001gn}
\bibitem{Dolan:2001gn}
B.~P.~Dolan, D.~O'Connor and P.~Presnajder, ``Matrix phi**4 models on the fuzzy sphere and their continuum limits,''
JHEP {\bf 0203}, 013 (2002) doi:10.1088/1126-6708/2002/03/013 [hep-th/0109084].
%%CITATION = doi:10.1088/1126-6708/2002/03/013;
%% %69 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Saemann:2010bw}
\bibitem{Saemann:2010bw}
C.~Saemann,
``The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP^n,''
SIGMA {\bf 6}, 050 (2010)
doi:10.3842/SIGMA.2010.050
[arXiv:1003.4683 [hep-th]].
%%CITATION = doi:10.3842/SIGMA.2010.050;%%
%20 citations counted in INSPIRE as of 18 Jan 2020
%\cite{OConnor:2007ibg}
\bibitem{OConnor:2007ibg}
D.~O'Connor and C.~Saemann,
``Fuzzy Scalar Field Theory as a Multitrace Matrix Model,''
JHEP {\bf 0708}, 066 (2007)
doi:10.1088/1126-6708/2007/08/066
[arXiv:0706.2493 [hep-th]].
%%CITATION = doi:10.1088/1126-6708/2007/08/066;%%
%40 citations counted in INSPIRE as of 18 Jan 2020
%\cite{Steinacker:2005wj}\bibitem{Steinacker:2005wj}
H.~Steinacker,
``A Non-perturbative approach to non-commutative scalar field theory,''
JHEP {\bf 0503}, 075 (2005)
doi:10.1088/1126-6708/2005/03/075
[hep-th/0501174].
%%CITATION = doi:10.1088/1126-6708/2005/03/075;%%
%48 citations counted in INSPIRE as of 19 Jan 2020
%\cite{Ydri:2017riq}
\bibitem{Ydri:2017riq}
B.~Ydri, C.~Soudani and A.~Rouag,
``Quantum Gravity as a Multitrace Matrix Model,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 32}, no. 31, 1750180 (2017)
doi:10.1142/S0217751X17501809
[arXiv:1706.07724 [hep-th]].
%%CITATION = doi:10.1142/S0217751X17501809;%%
%2 citations counted in INSPIRE as of 19 Jan 2020
For example, we compute on {\bf CP}^1_l, where the highest weight l is related to the ordinary spin s by s=l/2, the following quantities N=l+1, Tr K\sim N^4, Tr K^2\sim 4N^6/3, Tr K^3\sim 2N^8 and a_{3,1}\sim 8/3.
Whereas we compute in four dimensions on {\bf CP}^2_l the following quantities N=(l+1)(l+2)/2, i.e. l^2\sim 2N and as a consequence Tr K\sim 8 N^3/3, Tr K^2\sim 8N^4, Tr K^3\sim 128 N^5/5 and a_{3,1}\sim -256/9N^2.
It was argued in \cite{Ydri:2017riq} that the non-vanishing of the multitrace term proportional to a_{3,1} is a necessary and a sufficient condition for a stable Ising phase and a stable emergent background geometry. It was also argued there that quantum gravity in (or more precisely quantized spaces of) dimensions higher than two can thus be obtained from pure multitrace matrix models along these same lines.
By neglecting all multitrace terms we find that the quartic matrix model N Tr(\tilde{r}\Phi^2+\tilde{g}\Phi^4) (where \tilde{r}=r/N^{3/2} and \tilde{g}=g/N^2) in the limit of very weak coupling \tilde{g}\longrightarrow 0 is dominated by the Wigner semi-circle law given by
\begin{eqnarray} \rho(\lambda)=\frac{\tilde{r}}{\pi}\sqrt{\frac{2}{\tilde{r}}-\lambda^2}. \end{eqnarray}
However, by including all multitrace terms we get noncommutative \Phi^4 on fuzzy {\bf CP}^n_l which is also dominated (or more precisely its eigenvalue sector) in the limit of weak coupling \tilde{g}\longrightarrow 0 by a Wigner semi-circle corresponding to the following simple model \cite{Steinacker:2005wj}
\begin{eqnarray} S=\frac{2N}{\alpha_0^2(m)}Tr\Phi^2. \end{eqnarray}
In other words, we must make in the above Wigner semi-circle law the following replacement
\begin{eqnarray} \tilde{r}\longrightarrow\frac{2}{\alpha_0^2(m)}. \end{eqnarray}
\alpha_0 is the maximum eigenvalue and it is given in terms of the cutoff \Lambda\sim l on fuzzy {\bf CP}^n_l and the mass m^2\sim r by the relation \cite{Steinacker:2005wj}
\begin{eqnarray} \alpha_0^2(m)=4c(m,\Lambda)\Lambda^{d-2}. \end{eqnarray}
We have explicitly
\begin{eqnarray} c(m,\Lambda)=\frac{1}{16\pi^2}\bigg(1-\frac{m^2}{\Lambda^2}\ln(1+\frac{\Lambda^2}{m^2})\bigg)~,~d=4,n=2. \end{eqnarray}
\begin{eqnarray} c(m,\Lambda)=\frac{1}{4\pi}\ln(1+\frac{\Lambda^2}{m^2})~,~d=2,n=1. \end{eqnarray}
Thus, by studying the multitrace approximation at weak coupling we can determine when the Wigner behavior goes from the behavior of the pure matrix model d=0 to the behavior of noncommutative matrix models at d=2 and d=4 as we vary for example the parameter a_{3,1}. The emergence of the correct behavior is an indication of a stable Ising phase and as a consequence a stable emergent background geometry.
References
%\cite{Berezin:1974du}\bibitem{Berezin:1974du}
F.~A.~Berezin, ``General Concept of Quantization,'' Commun.\ Math.\ Phys.\ {\bf 40}, 153 (1975). doi:10.1007/BF01609397
%%CITATION = doi:10.1007/BF01609397;
%% %307 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Frohlich:1993es}
\bibitem{Frohlich:1993es}
J.~Frohlich and K.~Gawedzki,
``Conformal field theory and geometry of strings,'' In *Vancouver 1993, Proceedings, Mathematical quantum theory, vol. 1* 57-97, and Preprint - Gawedzki, K. (rec.Nov.93) 44 p
[hep-th/9310187].
%%CITATION = HEP-TH/9310187;
%% %74 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Connes:1994yd}
\bibitem{Connes:1994yd}
A.~Connes, ``Noncommutative geometry,'' %%CITATION = INSPIRE-391003;
%% %344 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Dolan:2001gn}
\bibitem{Dolan:2001gn}
B.~P.~Dolan, D.~O'Connor and P.~Presnajder, ``Matrix phi**4 models on the fuzzy sphere and their continuum limits,''
JHEP {\bf 0203}, 013 (2002) doi:10.1088/1126-6708/2002/03/013 [hep-th/0109084].
%%CITATION = doi:10.1088/1126-6708/2002/03/013;
%% %69 citations counted in INSPIRE as of 15 Jan 2020
%\cite{Saemann:2010bw}
\bibitem{Saemann:2010bw}
C.~Saemann,
``The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CP^n,''
SIGMA {\bf 6}, 050 (2010)
doi:10.3842/SIGMA.2010.050
[arXiv:1003.4683 [hep-th]].
%%CITATION = doi:10.3842/SIGMA.2010.050;%%
%20 citations counted in INSPIRE as of 18 Jan 2020
%\cite{OConnor:2007ibg}
\bibitem{OConnor:2007ibg}
D.~O'Connor and C.~Saemann,
``Fuzzy Scalar Field Theory as a Multitrace Matrix Model,''
JHEP {\bf 0708}, 066 (2007)
doi:10.1088/1126-6708/2007/08/066
[arXiv:0706.2493 [hep-th]].
%%CITATION = doi:10.1088/1126-6708/2007/08/066;%%
%40 citations counted in INSPIRE as of 18 Jan 2020
%\cite{Steinacker:2005wj}\bibitem{Steinacker:2005wj}
H.~Steinacker,
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