The Tensor Harish-Chandra–Itzykson–Zuber Integral II: Detecting Entanglement in Large Quantum Systems

Abstract

We consider the recently introduced generalization of the Harish-Chandra–Itzykson–Zuber integral to tensors and discuss its asymptotic behavior when the characteristic size N of the tensors is taken to be large. This study requires us to make assumptions on the scaling with N of the external tensors. We analyze a two-parameter class of asymptotic scaling ansätze, uncovering several non-trivial asymptotic regimes. This study is relevant for analyzing the entanglement properties of multipartite quantum systems. We discuss potential applications of our results to this domain, in particular in the context of randomized local measurements.

Appendices

Asymptotic Regimes for D = 1, Proof of Theorem 3

The asymptotic expansion of the cumulants of the tensor HCIZ integral (3.3) for $D=1$ reads:

$$\begin{aligned}{} & {} C_n\bigl ( {N^{{\gamma _{{}_{\beta \epsilon } }}}}{\textrm{Tr}}(AUBU^* )\bigr )\\{} & {} \quad = N^{n( {\gamma _{{}_{\beta \epsilon } }}- 2 ) }\sum _{ \sigma ,\tau \in S_n} N^{s(\sigma , \tau ) +s_A(\sigma ) + s_B(\tau )} {\textrm{tr}}_{\sigma }({a}) \, {\textrm{tr}}_{\tau }({b}) f[\sigma , \tau ](1+O(1)) , \end{aligned}$$

where (2.4) and the asymptotic scaling ansatz are:

$$\begin{aligned} s(\sigma , \tau ) = \#(\sigma \tau ^{-1})- 2 \big [ |\Pi (\sigma , \tau )|-1 \big ] , \qquad \text {and} \qquad s_A(\sigma )=\beta _A \#(\sigma ) \;, \end{aligned}$$

and similarly for B. Using (4.7), we rewrite the exponent of N in a term as:

$$\begin{aligned} 2 + n({\gamma _{{}_{\beta \epsilon } }}- 1) - 2g(\sigma ,\tau ) - (1 - \beta _A ) \#(\sigma ) - (1 - \beta _B) \#(\tau ) , \end{aligned}$$

(A.1)

where $g(\sigma , \tau )$ is the genus of the embedded map $(\sigma , \tau )$ discussed in Sect. 4.4. The scaling always selects planar graphs $(\sigma ,\tau )$ and favors for $\beta < 1$ cyclic permutations, for $1<\beta$ the identity permutation, and for $\beta =1$ it is insensitive to the number of cycles of the permutations:

• 1. If $\beta _A=\beta _B=1$, (A.1) becomes $2 + n({\gamma _{{}_{\beta \epsilon } }}- 1) - 2g(\sigma , \tau )$, so that ${\gamma _{{}_{\beta \epsilon } }}=1$, ${\delta _{{}_{\beta \epsilon } }}= 2$, and the leading order graphs are $(\sigma ,\tau )$ planar.

• 2. If $\beta _A<\beta _B=1$, (A.1) is $\beta _A + 1 + n({\gamma _{{}_{\beta \epsilon } }}- 1) - 2\,g(\sigma , \tau ) - (1-\beta _A) (\#(\sigma )-1)$, so that ${\gamma _{{}_{\beta \epsilon } }}= 1$, ${\delta _{{}_{\beta \epsilon } }}= \beta _A+1$ and at leading order $g(\sigma , \tau )=0$ and $\#(\sigma )=1$. From Proposition 4.3, $\tau$ is non-crossing on $\sigma$ and from Theorem 1, $f[\sigma , \tau ] = \textsf{M}(\sigma \tau ^{-1})$.

• 3. If $\beta _A\le \beta _B<1$, we write (A.1) as:

  $$\begin{aligned} \beta _A + \beta _B + n({\gamma _{{}_{\beta \epsilon } }}-1 ) - 2g(\sigma , \tau ) - ( 1-\beta _A) (\#(\sigma )-1) - (1-\beta _B) (\#(\tau )-1) \;, \end{aligned}$$

  hence ${\delta _{{}_{\beta \epsilon } }}= \beta _A + \beta _B$, ${\gamma _{{}_{\beta \epsilon } }}= 1$ and at leading order $g(\sigma , \tau ) =0$ and $\#(\sigma )=\#(\tau )=1$. From Prop. 4.4$\sigma =\tau$ and from Theorem 1$f \bigl [\sigma , \tau \bigr ]=1$.

• 4. If $\beta _A< 1 <\beta _B$, we write (A.1) as:

  $$\begin{aligned} \beta _A + 1 + n({\gamma _{{}_{\beta \epsilon } }}+ \beta _B - 2 ) - 2g(\sigma , \tau ) - ( 1-\beta _A) (\#(\sigma )-1) - (\beta _B-1) (n- \#(\tau )) \;, \end{aligned}$$

  hence ${\delta _{{}_{\beta \epsilon } }}= \beta _A+1$, ${\gamma _{{}_{\beta \epsilon } }}= 2-\beta _B$, and at leading order $\#(\tau )=n$, i.e. $\tau = \textrm{id}$, and $\#(\sigma )=1$. From Theorem 1, $f[\sigma , {\textrm{id}}] = \textsf{M}(\sigma ) = \frac{(-1)^{n-1}}{n}\left( {\begin{array}{c}2n-2\\ n-1\end{array}}\right)$ since $\#(\sigma )=1$. As in the regime 2, the contribution is the same for all cycles.

• 5. If $\beta _A =1 <\beta _B$, we write (A.1) as $2 + n({\gamma _{{}_{\beta \epsilon } }}+ \beta _B - 2 ) - 2\,g(\sigma , \tau ) - (\beta _B-1) (n- \#(\tau ))$, hence ${\delta _{{}_{\beta \epsilon } }}= 2$, ${\gamma _{{}_{\beta \epsilon } }}= 2-\beta _B$, and at leading order $\#(\tau )=n$, i.e. $\tau = \textrm{id}$, and $\sigma$ is arbitrary.

• 6. If $1<\beta _A \le \beta _B$, we write (A.1) as:

  $$\begin{aligned} 2 + n({\gamma _{{}_{\beta \epsilon } }}+ \beta _A + \beta _B - 3 ) - 2g(\sigma , \tau ) - (\beta _A -1)(n - \#(\sigma )) - (\beta _B -1)(n - \#(\tau ) ) , \end{aligned}$$

  hence ${\delta _{{}_{\beta \epsilon } }}=2$, ${\gamma _{{}_{\beta \epsilon } }}=3-\beta _A-\beta _B$. At leading order, $\sigma =\tau ={\textrm{id}}$ and $f[{\textrm{id}}, {\textrm{id}}]$ follows from Theorem 1.

This concludes the proof of Theorem 3.

Proof of Theorem 6: Regimes with A Microscopic

We check the rest of the regimes in Theorem 6.

1.1 Regime II

Lemma B.1

For $\beta =1-\epsilon (D-1)>\frac{1}{D}$ and $\epsilon \ge 0$, the leading order graphs are the $({{\varvec{\sigma }}}, {{\varvec{\tau }}})$ with ${{\varvec{\sigma }}}$ connected $(D+1)$-melonic and ${{\varvec{\tau }}}\preceq {{\varvec{\sigma }}}$:

$$\begin{aligned}{} & {} \lim _{N\rightarrow +\infty } \frac{1}{N } C_n\Bigl ( N^{1 + \epsilon \frac{D(D-1)}{2}}{\textrm{Tr}}(AUBU^* )\Bigr )\\{} & {} \quad = \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}\, \in { {\varvec{S}} } _n \text { connected} }\\ { (D+1)\text {-melonic}} \end{array}} \, \sum _{ \begin{array}{c} {{{\varvec{\tau }}},\; {{\varvec{\tau }}}\preceq {{\varvec{\sigma }}}} \end{array}}\, {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, {\textrm{tr}}_{{{\varvec{\tau }}}^{-1}} ({b}) \, \prod _{c=1}^D \textsf{M}(\sigma _c \tau _c^{-1}). \end{aligned}$$

Proof

Using $\omega ({{\varvec{\tau }}})$ (4.5), $\Box _{{{\varvec{\tau }}}}$ (4.9), and ${\Delta }$ (4.8), the scaling in (5.1) reads:

$$\begin{aligned}&1 + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon \frac{D(D-1)}{2} + (1-\epsilon D)(D-1)\bigr ] - (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1) \\&\qquad \quad -\sum _c\bigl ( 2g(\sigma _c, \tau _c) +\#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr ) - \epsilon \omega ({{\varvec{\tau }}}) - \epsilon D \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}})\\&\quad -(1-\epsilon D){\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \;, \end{aligned}$$

thus $\Pi ({{\varvec{\sigma }}},{{\varvec{\tau }}})=1$ and from Proposition 4.4 we get ${{\varvec{\tau }}}\preceq {{\varvec{\sigma }}}$ and $\Pi ({{\varvec{\sigma }}}) = \Pi ({{\varvec{\sigma }}},{{\varvec{\tau }}}) =1$. As ${{\varvec{\tau }}}\preceq {{\varvec{\sigma }}}$, from Proposition 4.8 we get that ${\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}})=0$ if and only if ${{\varvec{\sigma }}}$ is $(D+1)$-melonic. On the other hand, ${\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}}) = 0$ ensures that $\Omega _{D+1}({{\varvec{\tau }}})=0$, hence $\omega ({{\varvec{\tau }}})=0$. $\square$

1.2 Regime III

Lemma B.2

For $0<\beta =\epsilon <1/D$, the leading order graphs are $({{\varvec{\sigma }}}, {{\varvec{\sigma }}})$ with ${{\varvec{\sigma }}}$ connected melonic:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^{\epsilon D} } C_n\Bigl ( N^{D- \epsilon \frac{D(D-1)}{2} }{\textrm{Tr}}(AUBU^* )\Bigr ) = \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}\, \in \, { {\varvec{S}} } _n \text {connected} }\\ {\text {melonic}} \end{array}} {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, {\textrm{tr}}_{{{\varvec{\sigma }}}^{-1}} ({b}). \end{aligned}$$

Proof

Using (4.5), (4.9) and (4.8), we rewrite (5.1) as:

$$\begin{aligned}&\epsilon D + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon \frac{D(D-1)}{2}\bigr ] -\sum _c\Bigl ( 2g(\sigma _c, \tau _c) +\#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\Bigr ) - \epsilon \omega ({{\varvec{\tau }}}) \\& - (2- \epsilon D) (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1) - \epsilon D\, \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}})\\&\qquad \quad -(1-\epsilon D)\sum _c\bigl [\#(\tau _c) - |\Pi (\sigma _c, \tau _c)|\bigr ] \; . \end{aligned}$$

From Proposition 4.4 we conclude that ${{\varvec{\tau }}}= {{\varvec{\sigma }}}$ and the lemma follows. $\square$

1.3 Regime IV

Lemma B.3

For $0\le \epsilon<\beta <1-\epsilon (D-1)$, the leading order graphs are the $({{\varvec{\sigma }}}, {{\varvec{\sigma }}})$ with ${{\varvec{\sigma }}}$ connected $(D+1)$-melonic, and:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^{\beta + \epsilon (D-1)} } C_n\Biggl ( \frac{N^D}{N^{(D-1)(\beta -\epsilon + \frac{\epsilon D}{2} )}}{\textrm{Tr}}(AUBU^* )\Biggr ) = \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}\,\in { {\varvec{S}} } _n \text { connected} }\\ { (D+1)\text {-melonic}} \end{array}} {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, {\textrm{tr}}_{{{\varvec{\sigma }}}^{-1}} ({b}) . \end{aligned}$$

Proof

Using the expressions for $\omega ({{\varvec{\tau }}})$ (4.5), $\Omega _{D+1}({{\varvec{\tau }}})$ (4.6) and $\Box _{{{\varvec{\tau }}}}$ (4.9), the scaling in N in (5.1) becomes:

$$\begin{aligned}&\beta + \epsilon (D-1) + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon \frac{D(D-1)}{2} + (\beta -\epsilon )(D-1)\bigr ] \\ {}&\qquad - (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1)- \epsilon \omega ({{\varvec{\tau }}}) -\sum _c\bigl ( 2g(\sigma _c, \tau _c) + \#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr ) \\ {}&\qquad \qquad - \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) -(\beta -\epsilon )\Omega _{D+1}({{\varvec{\tau }}}) -(1+\epsilon - \beta - \epsilon D)(|\Pi ({{\varvec{\tau }}})|-1) . \end{aligned}$$

The leading order graphs are connected. Moreover, since $1+\epsilon - \beta - \epsilon D>0$, they have $|\Pi ({{\varvec{\tau }}})|=1$, so that $\Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) = \sum _c\bigl [\#(\tau _c) - |\Pi (\sigma _c, \tau _c)|\bigr ] =0$. As $g(\sigma _c, \tau _c)=0$ and $\#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|=0$, from Proposition 4.4 we conclude that ${{\varvec{\tau }}}={{\varvec{\sigma }}}$. Taking into account that $\Omega _{D+1}({{\varvec{\tau }}})=0 \Rightarrow \omega ({{\varvec{\tau }}})=0$, we conclude that ${{\varvec{\sigma }}}={{\varvec{\tau }}}$ and ${{\varvec{\sigma }}}$ is a connected $(D+1)$-melonic graph. $\square$

1.4 Regime V

Lemma B.4

For $\epsilon > \beta$, $\epsilon > 0$ and $\beta < 1/D$, the leading order graphs are such that all the $\sigma _c,\tau _c$ are the same cycle:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^{\beta D} } C_n\Bigl ( N^{D- \epsilon \frac{D(D-1)}{2} }{\textrm{Tr}}(AUBU^* )\Bigr ) = (n-1)! \ {\textrm{tr}}({a}^n) \, {\textrm{tr}}({b}^n). \end{aligned}$$

Proof

For $\epsilon > \max (\beta , 0)$ and $\beta < 1/D$, we rewrite (5.1) as:

$$\begin{aligned}&\beta D + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon \frac{D(D-1)}{2}\bigr ] -\sum _c\bigl ( 2g(\sigma _c, \tau _c) + \#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr ) - (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1) \\&\qquad - \epsilon \omega ({{\varvec{\tau }}}) - \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) -(\epsilon - \beta )\bigl [\sum _c\#(\tau _c) - D|\Pi ({{\varvec{\tau }}})|\bigr ] - (1 - \beta D) (|\Pi ({{\varvec{\tau }}})|-1) \; . \end{aligned}$$

At leading order, $|\Pi ({{\varvec{\tau }}})|=1$, so that $\sum _c\#(\tau _c) - D|\Pi ({{\varvec{\tau }}})|=\sum _c\#(\tau _c) - D$, which vanishes if and only if for all c, $\tau _c$ is a cycle of length n. Since $\tau _c\preceq \sigma _c$, this imposes that $\sigma _c=\tau _c$, which implies $\Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}})=0$. Finally, from (4.5), $\omega ({{\varvec{\tau }}})=0$ and each $\tau _c$ is a cycle if and only if all the $\tau _c$ are equal.$\square$

1.5 Regime VII

Lemma B.5

For $\epsilon > \beta = 1/D$, the leading order graphs are such that ${{\varvec{\tau }}}= (\tau , \dots , \tau )$ for some $\tau \in S_n$, ${{\varvec{\sigma }}}$ is connected, ${{\varvec{\tau }}}\preceq {{\varvec{\sigma }}}$, and $\Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}},{{\varvec{\tau }}})=0$:

$$\begin{aligned}{} & {} \lim _{N\rightarrow +\infty } \frac{1}{N} C_n\Bigl ( N^{D- \epsilon \frac{D(D-1)}{2} }{\textrm{Tr}}(AUBU^* )\Bigr ) \\{} & {} \quad = \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}\, \in { {\varvec{S}} } _n}\\ {\text {connected}} \end{array}} \; \sum _{ \begin{array}{c} { {{\varvec{\tau }}}= (\tau ,\dots ,\tau ), \, \tau \in S_n, }\\ {{{\varvec{\tau }}}\preceq {{\varvec{\sigma }}}, \; \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}},{{\varvec{\tau }}})=0 } \end{array} } {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, {\textrm{tr}}_{{{\varvec{\tau }}}^{-1}} ({b}) \, \prod _{c=1}^D \textsf{M}(\sigma _c \tau _c^{-1}). \end{aligned}$$

The leading order graph include $({{\varvec{\sigma }}}, \textbf{id} )$ with ${{\varvec{\sigma }}}$ connected $(D+1)$-melonic and $\textbf{id}=({\textrm{id}}, \dots , {\textrm{id}}) \in { {\varvec{S}} } _n$.

Proof

For $\epsilon > \beta$ and $\beta =1/D$, we rewrite (5.1) as:

$$\begin{aligned}&1 + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon \frac{D(D-1)}{2}\bigr ] -\sum _c\bigl ( 2g(\sigma _c, \tau _c) + \#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr ) - (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1) \\&\qquad - \epsilon \omega ({{\varvec{\tau }}}) - \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) -(\epsilon - \frac{1}{D})\bigl [\sum _c\#(\tau _c) - D|\Pi ({{\varvec{\tau }}})|\bigr ] \; . \end{aligned}$$

The leading order graphs $({{\varvec{\sigma }}},{{\varvec{\tau }}})$ are connected, and from Proposition 4.4 they satisfy ${{\varvec{\tau }}}\preceq {{\varvec{\sigma }}}$ and $\Pi ({{\varvec{\sigma }}})=1$. From (4.5), we note that ${{\varvec{\tau }}}$ satisfies $\omega ({{\varvec{\tau }}})=0$ and $\sum _c\#(\tau _c) = D|\Pi ({{\varvec{\tau }}})|$ if and only if all $\tau _c =\tau$ from some $\tau \in S_n$.

The last assertion follows by noting that $\textbf{id} \preceq {{\varvec{\sigma }}}$; (4.6) and (4.8) imply that ${\Delta }({{\varvec{\sigma }}},\textbf{id} ) = \Omega _{D+1}({{\varvec{\sigma }}})$, and from Proposition 4.8, $\Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, \textbf{id} )=0$ if and only if ${\Delta }({{\varvec{\sigma }}},\textbf{id}) = 0$. $\square$

1.6 Regime VIII

Lemma B.6

For $\epsilon \ge \frac{1}{D}$, $\beta > \frac{1}{D}$, and $0\le \epsilon < \frac{1}{D}$, $\beta > 1- \epsilon (D-1)$, the leading order graphs are $({{\varvec{\sigma }}}, {{\varvec{\tau }}})$ with ${{\varvec{\sigma }}}$ connected $(D+1)$-melonic, and $\tau _c = {\textrm{id}}$ for every c:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N } C_n\Biggl ( \frac{N^{D+1}}{N^{\epsilon \frac{D(D-1)}{2} + \beta D}}{\textrm{Tr}}(AUBU^* )\Biggr ) = {\textrm{tr}}({b})^n \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}\, \in \, { {\varvec{S}} } _n\text { connected} }\\ { (D+1)\text {-melonic}} \end{array}} {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, \prod _{c=1}^D \textsf{M}(\sigma _c). \end{aligned}$$

Proof

We divide the proof for this regime in two regions: $\epsilon \ge 1/D$ and $\beta > 1/D$ on one hand, and $0\le \epsilon < 1/D$ and $\beta > 1- \epsilon (D-1)$ on the other hand.

For $\epsilon \ge 1/D$ and $\beta > 1/D$, we rewrite the scaling with N in (5.1) as:

$$\begin{aligned}&1 + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon \frac{D(D-1)}{2} + \beta D- 1\bigr ] -\sum _c\bigl ( 2g(\sigma _c, \tau _c) +\#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr ) - \epsilon \omega ({{\varvec{\tau }}})\\& - (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1) - \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \\&\qquad - (\beta - \frac{1}{D} )\sum _c(n- \#(\tau _c)) -(\epsilon - \frac{1}{D})(\sum _c \#(\tau _c) - D|\Pi ({{\varvec{\tau }}})|) \; . \end{aligned}$$

The leading order graphs are connected and such that for all c, $\#(\tau _c)=n$, that is, $\tau _c={\textrm{id}}$. From (4.9), we have $\Box _{{{\varvec{\sigma }}}}({{\varvec{\sigma }}},{{\varvec{\tau }}})=0$, that is, ${\Delta }({{\varvec{\sigma }}},{{\varvec{\tau }}}) = \Omega _{D+1}({{\varvec{\sigma }}})$, while from Proposition 4.8 we get $\Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}},{{\varvec{\tau }}}) = {\Delta }({{\varvec{\sigma }}},{{\varvec{\tau }}})$. It follows that at leading order, ${{\varvec{\sigma }}}$ is $(D+1)$-melonic and connected (as $\Pi ({{\varvec{\sigma }}},{{\varvec{\tau }}})=1$). For $0\le \epsilon < 1/D$ and $\beta > 1- \epsilon (D-1)$, we follow the same reasoning starting from rewriting (5.1) as:

$$\begin{aligned}&1+ n\Bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon \frac{D(D-1)}{2} + (1-\epsilon D)(D- 1) + (\beta - \epsilon + \epsilon D - 1)D\Bigr ] - (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1) \\&\qquad - \sum _c \bigl ( 2g(\sigma _c, \tau _c) + \#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr )- (\beta - \epsilon + \epsilon D - 1)\sum _c\bigl (n - \#(\tau _c)\bigr )\\ {}& - \epsilon \omega ({{\varvec{\tau }}}) - \epsilon D\, \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) - (1 - \epsilon D ){\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \; . \end{aligned}$$

$\square$

Proof of Theorem 7: Symmetric Regimes

We now check the rest of the regimes in Theorem 7.

1.1 Regime S-II

Lemma C.1

For $\beta =1-\epsilon (D-1)> 1/D$ and $\epsilon \ge 0$, we have:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^2 } C_n\bigl ( N^{2-D + \epsilon D (D-1)}{\textrm{Tr}}(AUBU^* )\bigr ) = \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}, {{\varvec{\tau }}}\, \in \, { {\varvec{S}} } _n }\\ { {\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}}) =0 }\\ {\forall c,\, g(\sigma _c, \tau _c)=0 } \end{array}} \, {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, {\textrm{tr}}_{{{\varvec{\tau }}}^{-1}} ({b}) \, f \bigl [\mathrm {{{\varvec{\sigma }}}, {{\varvec{\tau }}}}\bigr ] . \end{aligned}$$

Proof

Using $\omega ({{\varvec{\tau }}})$ from (4.5), $\Box$ from Proposition 4.8, $\Box _{{{\varvec{\tau }}}}$ from (4.9), and ${\Delta }$ from (4.8), we rewrite (5.1) as:

$$\begin{aligned}&2 + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon D(D-1) + 2(1-\epsilon D)(D-1)\bigr ] \\&\qquad \quad - 2 \sum _c g(\sigma _c, \tau _c) - \epsilon (\omega ({{\varvec{\sigma }}}) + \omega ({{\varvec{\tau }}})) - 2\epsilon D\, \Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}})- 2 (1-\epsilon D){\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \; . \end{aligned}$$

From Proposition 4.8, ${\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}})=0$ implies that both $\Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}})=0$ and $\Omega _{D+1}({{\varvec{\sigma }}})=\Omega _{D+1}({{\varvec{\tau }}}) =0$, which from Proposition 4.2 implies that $\omega ({{\varvec{\sigma }}})=\omega ({{\varvec{\tau }}})=0$. $\square$

1.2 Regime S-III

Lemma C.2

For $\beta _A=\beta _B=\epsilon _A=\epsilon _B=\epsilon$, with $0<\epsilon < 1/D$, the leading order graphs are the $({{\varvec{\sigma }}}, {{\varvec{\sigma }}})$, with ${{\varvec{\sigma }}}$ a connected melonic graph:

$$\begin{aligned} \begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^{2\epsilon D} } C_n\Bigl ( N^{D(1- \epsilon (D-1) }{\textrm{Tr}}(AUBU^* )\Bigr ) = \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}\, \in \, { {\varvec{S}} } _n \text { connected}} \\ {\omega ({{\varvec{\sigma }}})=0} \end{array}} {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, {\textrm{tr}}_{{{\varvec{\sigma }}}^{-1}} ({b}) \;. \end{aligned} \end{aligned}$$

Proof

In this regime, the scaling in (5.1) reads:

$$\begin{aligned}&2\epsilon D + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon D(D-1)\bigr ] - 2 \sum _c g(\sigma _c, \tau _c) - \epsilon \bigl (\omega ({{\varvec{\sigma }}}) + \omega ({{\varvec{\tau }}})\bigr )\\&\qquad - 2(1- \epsilon D) \bigl (|\Pi ({{\varvec{\sigma }}}, {{\varvec{\tau }}})|- 1\bigr ) \\&\qquad - 2\epsilon D\, \Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}}) -(1-\epsilon D)\sum _c\bigl [\#(\sigma _c) + \#(\tau _c) - 2 |\Pi (\sigma _c, \tau _c)|\bigr ] \; . \end{aligned}$$

The leading order graphs have $\omega ({{\varvec{\sigma }}})=0$. They also have $g(\sigma _c, \tau _c)=0$, $\#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|=0$, and $\#(\tau _c) - |\Pi (\sigma _c, \tau _c)|=0$, and hence have ${{\varvec{\sigma }}}= {{\varvec{\tau }}}$ (Proposition 4.4) and are connected. But then $|\Pi ({{\varvec{\tau }}})|= |\Pi ({{\varvec{\sigma }}})|=1$, so that $\Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) = \sum _c\bigl [\#(\tau _c) - |\Pi (\sigma _c, \tau _c)|\bigr ]$ and $\Box _{{{\varvec{\sigma }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) = \sum _c\bigl [\#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr ]$.$\square$

1.3 Regime S-IV

Lemma C.3

For $\beta _A=\beta _B=\beta$ and $\epsilon _A=\epsilon _B=\epsilon$, with $0\le \epsilon<\beta <1-\epsilon (D-1)$, the leading order graphs are the $({{\varvec{\sigma }}}, {{\varvec{\sigma }}})$ with ${{\varvec{\sigma }}}$ a connected $(D+1)$-melonic graph:

$$\begin{aligned}{} & {} \lim _{N\rightarrow +\infty } \frac{1}{N ^{2( \beta + \epsilon (D-1))}} C_n\Bigl ( \frac{N^{D}}{N^{(D-1)(\epsilon (D-2) + 2 \beta )}}{\textrm{Tr}}(AUBU^* )\Bigr ) \\{} & {} \quad = \sum _{ \begin{array}{c} {{{\varvec{\sigma }}}\,\in { {\varvec{S}} } _n \text { connected} }\\ { (D+1)\text {-melonic}} \end{array}} {\textrm{tr}}_{{{\varvec{\sigma }}}}({a}) \, {\textrm{tr}}_{{{\varvec{\sigma }}}^{-1}} ({b}) . \end{aligned}$$

Proof

Using $\omega ({{\varvec{\tau }}})$ from (4.5), $\Box$ from Proposition 4.8, $\Box _{{{\varvec{\tau }}}}$ from (4.9), and $\Omega _{D+1}({{\varvec{\tau }}})$ from (4.6), we rewrite (5.1) as:

$$\begin{aligned}&2\beta + 2\epsilon (D-1)+ n\bigl ({\gamma _{{}_{\beta \epsilon } }}- D + \epsilon D (D-1) + 2(\beta - \epsilon )(D-1) \bigr ) -2\sum _c g(\sigma _c, \tau _c) \\ {}&\qquad - \epsilon (\omega ({{\varvec{\tau }}}) + \omega ({{\varvec{\sigma }}})) - (\beta - \epsilon )(\Omega _{D+1}({{\varvec{\sigma }}}) + \Omega _{D+1}({{\varvec{\tau }}}) ) - 2\, \Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \\&\qquad \qquad -(1+\epsilon - \beta - \epsilon D)(|\Pi ({{\varvec{\tau }}})|+ |\Pi ({{\varvec{\sigma }}})|- 2) \; . \end{aligned}$$

Since $1+\epsilon - \beta - \epsilon D>0$, at leading order we have $|\Pi ({{\varvec{\tau }}})|=|\Pi ({{\varvec{\sigma }}})|=1$, so that:

$$\begin{aligned}{} & {} \Box _{{{\varvec{\tau }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) = \sum _c\bigl [\#(\tau _c) - |\Pi (\sigma _c, \tau _c)|\bigr ] =0, \qquad \textrm{and} \qquad \\{} & {} \quad \Box _{{{\varvec{\sigma }}}}({{\varvec{\sigma }}}, {{\varvec{\tau }}}) = \sum _c\bigl [\#(\sigma _c) - |\Pi (\sigma _c, \tau _c)|\bigr ] =0 \;, \end{aligned}$$

which together with $g(\sigma _c,\tau _c)=0$ imposes that ${{\varvec{\sigma }}}= {{\varvec{\tau }}}$ (Proposition 4.4). As (see Proposition 4.2) $\Omega _{D+1}({{\varvec{\tau }}})=0$ implies $\omega ({{\varvec{\tau }}})=0$, we find that at leading order ${{\varvec{\sigma }}}$ is connected $(D+1)$-melonic, and ${{\varvec{\tau }}}= {{\varvec{\sigma }}}$. $\square$

1.4 Regime S-V

Lemma C.4

For $\epsilon > \beta$, $\epsilon >0$ and $\beta < 1/D$, the leading order graphs are such that all the $\tau _c,\sigma _c$ are the same cycle, and:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^{2\beta D} } C_n\Bigl ( N^{D(1- \epsilon (D-1)) }{\textrm{Tr}}(AUBU^* )\Bigr ) = (n-1)! \ {\textrm{tr}}({a}^n) \, {\textrm{tr}}({b}^n) \;. \end{aligned}$$

Proof

For $\epsilon > \beta$ and $\beta < 1/D$ we rewrite (5.1) as:

$$\begin{aligned}&2\beta D \!+\! n\bigl [ {\gamma _{{}_{\beta \epsilon } }}\!-\!D \!+\! \epsilon D(D-1)\bigr ] \! -\!(\epsilon \!-\! \beta )\bigl [\sum _c \bigl (\#(\sigma _c) \!+\! \#(\tau _c)\bigr )\! -\! D\bigl (|\Pi ({{\varvec{\sigma }}})|\! + \!|\Pi ({{\varvec{\tau }}})|\bigr )\bigr ] \\&\qquad \!-\sum _c 2g(\sigma _c, \!\tau _c)\!-\! \epsilon (\omega ({{\varvec{\sigma }}}) \!+\! \omega ({{\varvec{\tau }}})) \!-\! 2\!\,\Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \! -\! (1\! -\! \beta D) (|\Pi ({{\varvec{\sigma }}})|\!+\!|\Pi ({{\varvec{\tau }}})|\!-\!2) \; . \end{aligned}$$

The leading order graphs are such that $|\Pi ({{\varvec{\tau }}})|=1$, which implies $\sum _c\#(\tau _c) - D|\Pi ({{\varvec{\tau }}})|=\sum _c\#(\tau _c) - D\ge 0$, which in turn vanishes if and only if $\#(\tau _c)=1$. They also have $\omega ({{\varvec{\tau }}})=0$, which imposes that all the $\tau _c$ are the same cycle. The same goes for ${{\varvec{\sigma }}}$. For the genera $g(\sigma _c, \tau _c)$ to vanish, $\sigma _c$ and $\tau _c$ must be the same cycle. $\Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}})$ indeed vanishes for such contributions. $\square$

1.5 Regime S-VII

Lemma C.5

For $\epsilon > \beta = 1/D$, the leading order graphs are such that for all c, $\tau _c = \sigma _c=\sigma$, not necessarily connected, and:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^2} C_n\Bigl ( N^{D(1- \epsilon (D-1) }{\textrm{Tr}}(AUBU^* )\Bigr ) = \sum _{ \sigma \in S_n} \ \prod _{{\eta \text { cycle }}{\text {of} \sigma }} {\textrm{tr}}\bigl (A^{l(\eta )}\bigr ){\textrm{tr}}\bigl (B^{l(\eta )}\bigr ) f[{{\varvec{\sigma }}}, {{\varvec{\sigma }}}] , \end{aligned}$$

with $l(\eta )$ the length of the cycle $\eta$.

We can give a close expression for $f[{{\varvec{\sigma }}}, {{\varvec{\sigma }}}]$ using (2.5):

$$\begin{aligned} f[{{\varvec{\sigma }}}, {{\varvec{\sigma }}}] = 2^{nD} \sum _{\begin{array}{c} {\pi _1,\ \ldots \,\ \pi _D}\\ {|\Pi (\sigma )\vee \pi _1\vee \ldots \vee \pi _D | = 1 }\\ {\sum _c |\Pi (\pi _c)|= 1 + nD - |\Pi (\sigma )|} \end{array}} \ \prod _{c=1}^D \prod _{B\in \pi _c} \frac{(3|B |- 3)!}{(2|B |)!} . \end{aligned}$$

Proof

For $\epsilon > \beta = 1/D$, (5.1) becomes:

$$\begin{aligned}&2 + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon D(D-1)\bigr ] -\sum _c 2g(\sigma _c, \tau _c) - \epsilon ( \omega ({{\varvec{\sigma }}}) + \omega ({{\varvec{\tau }}})) - 2\,\Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \\&\qquad -(\epsilon - \frac{1}{D})\bigg [\sum _c \bigl (\#(\sigma _c) + \#(\tau _c)\bigr ) - D\bigl (|\Pi ({{\varvec{\sigma }}})|+ |\Pi ({{\varvec{\tau }}})|\bigr )\bigg ] \; . \end{aligned}$$

At leading order, ${{\varvec{\sigma }}}$ and ${{\varvec{\tau }}}$ must be melonic, and since $\sum _c \#(\sigma _c) =D|\Pi ({{\varvec{\sigma }}})|$ and $\sum _c \#(\tau _c) = D |\Pi ({{\varvec{\tau }}})|$, from (4.5) it follows that $\sigma _c =\sigma$ and $\tau _c = \tau$ for all c. With these constraints, $\Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}})=0$ is equivalent to $\#(\sigma )=\#(\tau ) = |\Pi (\sigma , \tau )|$ and since $g(\sigma , \tau )=0$, $\sigma =\tau$ (Proposition 4.4). $\square$

1.6 Regime S-VIII

Lemma C.6

In the regions $\epsilon \ge 1/D$, $\beta > 1/D$ and $0\le \epsilon < 1/D$, $\beta > 1- \epsilon (D-1)$, the leading order graphs are such that $\sigma _c=\tau _c = {\textrm{id}}$ for all c:

$$\begin{aligned} \lim _{N\rightarrow +\infty } \frac{1}{N^2 } C_n\Biggl ( \frac{N^{D+2}}{N^{\epsilon D(D-1) + 2\beta D}} {\textrm{Tr}}(AUBU^* )\Biggr ) = \bigl ( {\textrm{tr}}({a}) \cdot {\textrm{tr}}({b})\bigr )^n. \end{aligned}$$

Proof

We divide the proof for this regime in two regions: $\epsilon \ge 1/D$ and $\beta > 1/D$ on one hand, and $0\le \epsilon < 1/D$ and $\beta > 1- \epsilon (D-1)$ on the other.

For $\epsilon \ge 1/D$ and $\beta > 1/D$, we rewrite (5.1) as:

$$\begin{aligned}&2 + n\bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon D(D-1) + 2\beta D- 2\bigr ] -2\sum _c g(\sigma _c, \tau _c) - \epsilon (\omega ({{\varvec{\sigma }}}) + \omega ({{\varvec{\tau }}})) - 2\, \Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \\ {}&- (\beta - \frac{1}{D} )\sum _c \bigl [n-\#(\sigma _c) +n- \#(\tau _c)\bigr ] \\&\quad -(\epsilon - \frac{1}{D})\bigg [\sum _c \bigl (\#(\sigma _c) + \#(\tau _c)\bigr ) - D\bigl (|\Pi ({{\varvec{\sigma }}})|+ |\Pi ({{\varvec{\tau }}})|\bigr )\bigg ] \; , \end{aligned}$$

which at leading order, due to the first term in the second line, restricts to $\sigma _c=\tau _c={\textrm{id}}$ for all c. For $0\le \epsilon < 1/D$ and $\beta > 1- \epsilon (D-1)$, we reach the same conclusion by writing (5.1) as:

$$\begin{aligned}&2+ n\Bigl [ {\gamma _{{}_{\beta \epsilon } }}-D + \epsilon D(D-1) + 2(1-\epsilon D)(D- 1) + 2(\beta - \epsilon + \epsilon D - 1)D\Bigr ] \\&\qquad - \sum _c 2g(\sigma _c, \tau _c) - (\beta - \epsilon + \epsilon D - 1)\sum _c \bigl [n-\#(\sigma _c) +n- \#(\tau _c)\bigr ]\\&\qquad \qquad - \epsilon (\omega ({{\varvec{\sigma }}}) + \omega ({{\varvec{\tau }}})) - 2\,\epsilon D\, \Box ({{\varvec{\sigma }}}, {{\varvec{\tau }}}) - 2(1 - \epsilon D ){\Delta }({{\varvec{\sigma }}}, {{\varvec{\tau }}}) \; . \end{aligned}$$

$\square$