Melonic Turbulence

Abstract

We propose a new application of random tensor theory to studies of non-linear random flows in many variables. Our focus is on non-linear resonant systems which often emerge as weakly non-linear approximations to problems whose linearized perturbations possess highly resonant spectra of frequencies (non-linear Schrödinger equations for Bose–Einstein condensates in harmonic traps, dynamics in Anti-de Sitter spacetimes, etc). We perform Gaussian averaging both for the tensor coupling between modes and for the initial conditions. In the limit when the initial configuration has many modes excited, we prove that there is a leading regime of perturbation theory governed by the melonic graphs of random tensor theory. Restricting the flow equation to the corresponding melonic approximation, we show that at least during a finite time interval, the initial excitation spreads over more modes, as expected in a turbulent cascade. We call this phenomenon melonic turbulence.

Proof of Lemma 3

The graph amplitudes can be expressed as

$$\begin{aligned} {\mathcal {A}}_{r}(G)=\frac{1}{N^n} \Bigl (\prod _{v\in {\tilde{{\mathcal {V}}}}} \sum _{S_v}\sum _{j_v, k_v \le S_v}\Bigr ) \Bigl (\prod _{\begin{array}{c} e \text { dashed } \\ \text {or solid} \end{array}} \delta _{j_e j'_e} \Bigr ) \prod _{e\text { dashed} } p^{j_e}. \end{aligned}$$

(137)

We know that for every corner, the corner momentum $i_c$ (which is $j_v, S_v-j_v, k_v, S_v - k_v$ for the corresponding node v) is equal to all of the indices of the other corners in the face it belongs to. Indeed, a face consists of edges linking corners, and the constraint on a dashed or solid edge e between two corners c and $c'$ identifies the corner momenta $j_e=i_c$ and $j_e' = i_{c'}$. So if we introduce a new set of indices $\{i_f\}_{f}$, we can rewrite the constraints on the edges as

$$\begin{aligned} \prod _{\begin{array}{c} e \text { dashed } \\ \text {or solid} \end{array}} \delta _{j_e j'_e} = \Bigl (\prod _{f} \sum _{i_f\ge 0}\Bigr )\Bigl (\prod _f \prod _{c\in f} \delta _{i_f}^{i_c}\Bigr ). \end{aligned}$$

(138)

Indeed, to recover the left hand side, we perform the sums over the $\{i_f\}$ on the right hand side. For each face, it identifies all of the indices of the visited corners and gives back the constraints on the dashed and solid edges that the face visits. We rewrite (137) as

$$\begin{aligned} {\mathcal {A}}_{r}(G)= & {} \frac{1}{N^n} \Bigl (\prod _{f} \sum _{i_f\ge 0}\Bigr ) \Bigl (\prod _{v\in {\tilde{{\mathcal {V}}}}} \sum _{S_v}\sum _{j_v, k_v \le S_v}\Bigr ) \Bigl (\prod _f \prod _{c\in f} \delta _{i_f}^{i_c}\Bigr ) \prod _{e\text { dashed} } p^{j_e} \end{aligned}$$

(139)

$$\begin{aligned}= & {} \frac{1}{N^n} \Bigl (\prod _f \sum _{i_f}p^{i_f L^\alpha _f}\Bigr )\Biggl [ \Bigl (\prod _{v\in {\tilde{{\mathcal {V}}}}} \sum _{S_v}\sum _{j_v, k_v \le S_v}\Bigr ) \Bigl (\prod _f \prod _{c\in f} \delta _{i_f}^{i_c}\Bigr )\Biggr ], \end{aligned}$$

(140)

where $L^\alpha _f$ is the number of dashed edges in the face f. We would now like to perform the sums over the $\{S,j,k\}$, to reduce the term between brackets into a product of Kronecker deltas. This would succeed if we use one delta for each one of the 3n/2 sums. There is exactly one delta per corner. For a given vertex $v_0$, these four deltas depend only on the three indices $j_v, k_v$ and $S_v$, and on some face momenta which are fixed. Therefore, we should be able to reduce the node constraints to a product of deltas.

More precisely, we consider a node $v_0$ and perform the sums for the two corners $c_1$ and $c_3$ for which the corner momenta are $i_{c_1} = j_{v_0}$ and $i_{c_3}=k_{v_0}$. This uses one delta each, $\delta _{i_{f_1}}^{i_{c_1}}$ and $\delta _{i_{f_3}}^{i_{c_3}}$ (where we shortened the notation $i^{(a)}_{f,v_0} = i_{f_a}$), and it means that for the two remaining corners $c_2$ and $c_4$ of the node, respectively corresponding to the face momenta $S_{v_0} - j_{v_0}$ and $S_{v_0} - k_{v_0}$, the remaining deltas are $\delta _{i_{f_2}}^{S_{v_0} - i_{f_1}}$ and $\delta _{i_{f_4}}^{S_{v_0} - i_{f_3}}$. We stress that the condition $S_{v_0}\ge i_{f_1}, i_{f_3}$ is implemented in the deltas because $i_{f_2}$ and $i_{f_4}$ are non-negative. We perform the sum over $S_{v_0}$, leaving us with a $\delta _{i_{f_2}}^{i_{f_3} + i_{f_4} - i_{f_1}}$ which we re-arrange as a $\delta _{i_{f_1} + i_{f_2}}^{i_{f_3} + i_{f_4}}$. We have to be sure that the constraint $S_{v_0}\ge \max (i_{f_1},i_{f_3})$ is implemented, and this is the case, as $i_{f_3} + i_{f_4} - i_{f_1}$ is non-negative since $i_{f_2}\ge 0$ and $i_{f_3} + i_{f_4}\ge i_{f_3}$ so $i_{f_3} + i_{f_4}\ge \max (i_{f_1},i_{f_3})$ and similarly for $i_{f_1} + i_{f_2}$. The sums corresponding to the vertex $v_0$ in (140) have been taken care of and we proceed with another vertex. $\quad \square$