Stochastic melonic kinetics with random initial conditions

The probability laws associated with random tensors or tensor field theories are traditionally equilibrium distributions. In this paper, we consider a stochastic point of view, and approach the quantization by a Langevin type equation. We especially address the low-temperature behavior of the phase ordering kinetics of a stochastic complex tensor field $T_{i_1\cdots i_d}(x,t)$ of size $N$ and rank $d$ in dimension $D$. The method we propose use the self averaging property of the tensorial invariants in the large $N$ limit. In this regime, the dynamics is governed by the melonic sector, whose behavior is studied in the quenched limit, where the contractions involving $d-1$ indices self-average around a diagonal matrix proportional to the identity. The following work especially focuses on the cyclic (i.e. non-branching) melonic sector, and we study the way that the system returns to the equilibrium regime regarding the temperature and the shape of the potential. In particular, we provide a general formula for the transition temperature between these regimes. The manuscript is accompanied by numerical simulations to support the theoretical analysis, and essentially aims to open towards this new field of investigation.


Introduction
Phase ordering kinetics is a phenomenon classically describing the growth of an ordered phase as a domain coarsening for a quenched system, from the homogeneous phase toward a broken phase [1]. It has been investigated for O(N ) field theory models [2,4] from the methods used to solve spin glass dynamics, and physics exhibits exciting relation with the soft p = 2 spin dynamics [6,7,8,5,9,10]. In this paper, we investigate the growth of the leading order phase for a stochastic random tensor model (RTM), which in contrast with ordinary phase ordering kinetics does not break the underlying U(N ) ×d symmetry.
RTMs were introduced to generalize in higher dimensions the success of random matrix models (RMM) for 2D quantum Einstein gravity [11]. In 2009, it is shown that the colored RTM admits a tractable 1/N expansion involving a generalized degree ≥ 0 such that the Feynman amplitudes A(G) for the graph G scales as A(G) ∼ N d− 2 (d−1)! (G) [12]. The degree can be computed from the set of vertices, edges, and faces building the graph G, which is a 2-simplex rather than an ordinary Feynman graph. The leading order graphs called melons are defined by the condition (G) = 0, and follow a recursive definition reminiscent of branched polymers occurring for large N random vectors [14]. Furthermore, their critical behavior has been investigated as well analytically [15,17], and confirms the branched polymer phase transition from the critical exponent. Recently the so called Sachdev-Ye-Kitaev (SYK) quantum mechanics model which consists of N-Majorana fermions with random interactions are showed to admit this same large N-limit [28]- [30]. Many other works in this direction have also been successful, in particularly: the O(N )-tensor model to derive the fate of the wormholes in a model without quenched disorder with gauge symmetry whose correlation function and thermodynamics in the large N limit are the same as that of the SYK model [31]- [32], see also [33]- [37] for similar works. Generally, what makes the RTM power countable is their global U(N ) invariance, and the statistics of RTM follow an exponential law ρ ∼ e −S , where the classical action S is a sum of U(N ) ×d invariants.
In this paper, we give an introduction to the new field of investigation i.e. the stochastic complex tensors T i 1 ···i d (x, t) on R D+1 , whose dynamics follow a diffusion Langevin type equation, by focusing on the melonic cyclic (i.e. non-branched) sector. Such a stochastic model has been considered recently for a tensorial group field theory (TGFT) [18] where a gravitational field was expected to be out of equilibrium with respect to a weakly coupled scalar field playing the role of a physical "clock " and materialized by the time variable [16]. Our purpose in this manuscript is only to focus on the large time dynamics, where the random tensor is "quenched" around a mean value. In that limit, the closed equations can be solved with elementary analytic methods, and a transition temperature between a power law versus. an exponential relaxation can be computed. However, these conclusions depend on the shape of the confining potential, and in particular on the order of the roots of the polynomial, which we also highlight. This kind of model is studied in the framework of vectors field theory in the large N -limit [2,3,6]. The generalization to the tensor case with a more complicated invariance for a multilinear object is therefore necessary.
Outline. The paper is organized as follows. In section 2 we define the model and conventions. In section 3 we consider the low-temperature regime for D = 0 in the quenched limit, and in section 4 we extend the method for D > 0. In both cases, we investigate the large time closed equation expected from the self-averaging assumption and the transition regime toward this limit. In section 5 the case of a disordered tensor for D = 0, the disorder coupling is materialized by the tensor product of Wigner matrices. We conclude in section 6.

Model and conventions
We consider a time-dependent complex tensor T(x, t) with rank d > 2, t ∈ R and x ∈ R D . We denote by T i 1 ···i d (x, t) ∈ C the components of the tensor, and the indices i 1 , · · · , i d run from 1 to N . We furthermore denote byT(x, t) the complex-conjugate tensor of T(x, t), and byT i 1 ···i d (x, t) its components. The dynamics of the tensor are assumed to follow a Langevin-like equation:Ṫ In this equationṪ i 1 ···i d is the derivative with respect to t, ∂ 2 x denotes the standard Laplacian over R D , η i 1 ···i d (x, t) is a Gaussian tensor field with correlations defined as where is the memory function, and in the white noise limit: the friction coefficient T being interpreted as the temperature for the equilibrium distribution, see below. The UV cut-off Λ provides a UV regularization of the theory, which will be clear in the next sections. Note that, The Hamiltonian H ≡ H[T,T] is assumed to be a sum of tensorial invariant, namely: where we use standard convention in the RTM literature: black and white vertices materialize respectively fields T andT, and solid edges materialize Kronecker Dirac delta between indices of different tensors. Furthermore, all the tensor fields are taken at the same point x and at the same time t for each interaction and the Hamiltonian involves a global integral over R D . We call a bubble such a connected graph and note that we set d = 3 for graphical representations of bubbles in this paper. Note that high valence interactions avoid arbitrary large configurations for the stochastic tensor T . For a fixed initial condition for t = t 0 , one can consider the probability P [T,T, t] that tensor take the values (T(t) = T,T(t) =T) for time t > t 0 . This probability satisfies a Fokker-Planck equation [20], and for a delta-correlated noise in space dimension zero, the equilibrium probability ρ[T,T] := lim t → ∞P [T,T, t] behaves as, which corresponds to a standard RTM at temperature T if the N -scaling of couplings constants µ and g ij labeling each bubble are such that Feynman amplitudes . This can be achieved if coupling constants g b for bubble b scales as where the valence p := n(b) is the number of black vertices in b. Note that for melonic bubbles (b) ≡ 0.
In the next sections we investigate the large-time behavior in the N → ∞ regime. In particular, we compute the transition temperature investigating, in the quenched regime, the effective dynamics for the self-averaging quantity: For our purpose in this paper, we focus on the cyclic sector of the melonic family, and we recall some definitions for self-consistency.
Definition 1 Any melonic bubble b p of valence p may be deduced from the elementary replacing successively (κ − 1)-colored edges (including maybe color "0") by (d − 1)-dipole, the (d − 1)-dipole insertion operator R c being defined as: The cyclic melons are then defined as follows: Definition 2 A non-branching melonic bubble of valence p, b ( ) p is labeled with a single color index c ∈ 1, d , and defined such that: Figure 1 provides the generic structure of melonic non-branching bubbles in rank d = 3.
We then assume that the Hamiltonian H reads as where V c [T ,T ] expands as: where entries of the unitary matrices Φ c are defined as: such that N d r ≡ Tr (Φ c ). 3 The D = 0 model in the quenched limit for a white noise This section is devoted to the dynamic aspects of the zero-dimensional model. Equation (1) can be investigated for a large time, and with some approximations that we will detail, the radius r(t) satisfies a closed equation that can be studied analytically [2]. The way that the system converges toward equilibrium for late time can be investigated as well. The zero-dimensional model has no particular interest in itself, and the aim of this section is essential to introduce the general strategy that we will implement in the next two sections.

Static limit
The equilibrium points of our model are given by the relation: where the deleted nodes correspond to the partial derivatives with respect to the field Φ c . The vacuum solution can be chosen to be U(N ) invariant, namely 4 : which can be easily justified in the large N limit due to our random initial condition (22). We furthermore assume that color permutation symmetry is unbroken, and κ c ≡ κ, ∀ c.
In that way, equation (15) becomes: where I := i 1 · · · i d . Hence, assuming T I = 0, the equilibrium points are defined by the minima of the potential U (κ). For the quartic model, this point is given by: If we assume that the system does not break the U(N ) symmetry dynamically and for large N Φ c self average around a diagonal colored symmetric matrix during the time evolution (see remark 1 at the end of this section), the Langevin equation (1) can be approached as: where κ is assumed to be replaced by its averaged value, such that each component of T decouples from the others. The previous equation can be formally solved as: with: In equation (20), the time t = 0 has to be understood not as a true initial condition but as an arbitrary time, far from the true origin of time. Because we have no information about the true behavior of the tensor at t = 0 for a given draw, we impose it to be randomly distributed accordingly with the centred Gaussian distribution 5 : where |x| is the standard Euclidean norm, X 0 means averaging over initial conditions and Λ is the same UV cutoff as for the noise η(x, t). Hence we get the closed equation: 4 A corollary of the standard Schur Lemma. 5 We define arbitrary D here, despite this section focusing on the case D = 0.
where G(t) := e 2g(t) . For large time, we expect that r(t) reaches one of the equilibrium points of U (κ), i.e., that U (κ(t)) → 0 for t large enough (see Section 4). From this expectation, the previous equation looks like a closed integral equation for G(t), which can be easily solved asymptotically.
• For T = 0, differentiating the closed equation concerning t, ) → (T /2r 0 ), and the system is repelled from the equilibrium point due to thermal fluctuations.

Equilibrium dynamics
The time evolution toward the asymptotic regime can be investigated with a little improvement of the previous argument. Indeed, assuming the validity of the quenching regime, and because from the definition: we have for the quartic model: Deriving the equation with respect to t, we have: This is a linear homogeneous differential equation of second order, that can be easily solved as a combination of exponential: where ω ± is the solution of the characteristic polynomial, namely: explicitly: We have two initial conditions. First G(0) = 1 leading to α + β = 1.
8 (see remark 1 below) and where r 0 := −µ/(ḡ 2 d). Hence, we must have: in other words, and especially for T = 0: For T = 0, we find an exponential law G(t) = αe 2µt + β and the behavior of the system depends on the sign of µ. For µ < 0, G(t) → ∆/r 0 exponentially, as expected. For µ > 0, U (κ(t)) → µ for large time, and then κ(t) → 0. Note that because G(t) is positive, we must have ∆ < r 0 if we expect that the solution holds for all times.
For T = 0, the two solutions ω − and ω + have different signs, and in particular iω − < 0. For small T : and: Hence for µ < 0, we recover the asymptotic behavior expected from the previous analysis. As discussed previously for T = 0, the system fails to reach the equilibrium point of the potential. Nevertheless, it reaches an equilibrium regime that we can characterize. Indeed, from (27), we have for large time: or: To investigate the meaning of this equation, let us focus on the case µ > 0, in which the previous equation can be solved as: Now let us consider the equilibrium distribution . The melonic equilibrium solution can be investigated from the standard methods in RTM literature [12] using Schwinger-Dyson equations, which we recall briefly here in this context for self-consistency. The partition function of the equilibrium state reads: where dT dT := i 1 ,···i d dT i 1 ···i d dT i 1 ···i d is the standard Lebesgue measure, and T is now time independent. Let us consider the Schwinger-Dyson equation: Computing the derivative, and dividing the resulting equation by N d Z eq , we arrive to the equation: where Q is defined as: and used of the large N self averaging condition [13]: The solution of (42) is: where we selected only the solution that goes toward T /2µ asḡ 2 vanishes. This shows that κ(t) → Q for large t, and that the system goes toward the equilibrium regime described by the equilibrium state (6). Note that equilibrium solution (45) is defined below the critical value: at which the free energy f eq := ln Z eq becomes non-analytic, corresponding to the entropy exponent θ = 1/2. Finally, note that the leading order Schwinger Dyson equation means that: hence, G(t) ∼ e (T /κ)t rather than e (T /r 0 )t .

Low temperature regime
The result can be generalized for potentials of the form: where the zeros of the function R(κ) are assumed to be different and far enough to the isolated zero κ = γ. If the system is found initially in the vicinity of the isolated zero, and if we assume that the system converges toward some equilibrium value κ ∞ for late time, close enough to the value κ = γ (see also Section 4, paragraph b), the equation forĠ reads approximately:Ġ which can be solved again as a combination of exponential with: The late time behavior corresponds to the min iω ± , and the equation fixing the value of κ ∞ is: which can be rewritten as: which is nothing but the leading order Schwinger Dyson equation for large N .
It seems that the ability of the system to reach the equilibrium distribution depends strongly on the shape of the potential. As an example, let us investigate the case of a potential with twice degenerate vacua: The equation forĠ reads:Ġ Because G is positive, the right-hand side is positive at all times, and G can only grow over time. For T small enough, one can expand the right-hand side up to linear order in T , and for late time, one can neglect ∆/G with respect to contributions of order G 0 and G 1 . We have:Ġ 11 Assuming that G(t) expands in power series of T : we get for G 0 : In the same way, the equation for G (1) reads: Which can be solved as: The numerical constants C 1 and C 0 will be fixed by the initial condition G(0) = 1, leading to: and G(t) reads: For a later time, the right-hand side can be computed exactly, and we have: where: where Li s (z) is the standard poly-logarithm function. Computing the first derivative with respect to t, a straightforward calculation leads to 6 : Hence, for T small enough, the asymptotic equation fixing κ is: which have two solutions: In contrast, the leading order Schwinger Dyson equation for the equilibrium theory reads: which, for small T solves as: This result shows that, at least for T small enough, equilibrium dynamics selects only one of the three solutions of the equilibrium distribution, the vacuum such that Q = O(T ).
One can expect that this concerns only zeros with even degeneracy, but a simple argument shows that it concerns odd potential as well. To show this, let us consider the potential: In contrast with the previous case, the right-hand side is not positive, and one may expect that the equation converges toward a finite value, say κ ∞ . Hence, assuming the system is closed to this limit: At first order in ε(t), we have: corresponding to the frequencies: For late time, the equation for κ ∞ can be rewritten as:  Figure 2: Shape of the potential V (κ) for some temperature above and below the critical temperature with γ = 0.5 and critical temperature dependency on γ.

Numerical study
The equation forĠ can be investigated numerically. In this section, we focus on the case of purely degenerate vacua, of the form (70). First, let us consider the quadratic case. If equilibrium is reached, the leading contribution to G(t) for a late time has to behave as 7 : where κ ∞ is a zero of the potential V (κ) := κ(κ − γ) 2 − T . There are two regimes, depending on the temperature. For T < T c (γ), there are three real zeros, and for T 1 these zeros display as equation (69) shows, with a zero of order T and two zeros of order γ. For T > T c (γ) in contrast, the potential has only one real zero, identified from continuity with T as the larger zero among the three real zeros occurring from the low-temperature regime. Figure 2 shows the behavior of the potential V (κ) and T c (γ), which is explicitly given by: Let us denote as G num the numerical solution of equation (70) for n = 2. Figure 3 shows the behavior of ln(G num (t))/(iω th t) above and below the critical temperature T c ≈ 0.019 for γ = 0.5 and ∆ = 0.1, the theoretical value iω th being equals to T /κ ∞ , such that V (κ ∞ ) = 0. As we can show on the Figure 3 for T = 0.1, 0.2 and 1, G num (t) ∼ e iω th t for t large enough. The same thing occurs below the critical temperature, and the second diagram in Figure 3 shows the behavior of ln(G num (t))/(iω th t) for T = 0.01 for each of the Hence, if B = T /κ ∞ , the equation reads κ ∞ (κ ∞ − γ) n = T , which is nothing but the leading order Schwinger Dyson equation. After a transition regime, the relaxation toward equilibrium can be numerically investigated, and, indeed: Similar behavior is observed for n > 2. In general, there are a positive and a negative zero for V (κ), and the value toward which ln(G num (t))/(iω th t) converges is compatible with Schwinger-Dyson equation for equilibrium state. Figure 5 shows the behavior of ln(G num (t))/(iω th t) for ∆ = 0.1, γ = 0.5 and T = 0.1. Numerically, the system reaches the equilibrium state and: G num (t) ∼ 0.396 exp (0.103t) . (79)

Remark 1
The assumption (16) can be easily justified along the dynamics from the following argument. Let us computeλ µ , the rate of the µ-th eigenvalue of (Φ c ) ij . Because Φ c is hermitian, it has N real eigenvalues and admits N normalized and orthogonal eigenvectors {u   It is easy to check that: The rateλ µ has then the general structure: where F µ is the projection of the right-hand side of the Langevin equation. Now, let us investigate the large N distribution P 0 ( Φ 0 ) for the initial distribution of Φ 0,c := Φ c (t = 0), where Φ 0 := {Φ 0,c }. Because (22), we have: Computing the 2-point correlation, we have: where the last piece is the connected contribution, which can be easily computed from (22). This leads to: where I/c ∈ 1, N d−1 includes all indices excepts the one of color c. The connected contribution is sub-leading (with a factor 1/N d−1 ) concerning the disconnected piece, and in the large N limit we have: In the same way to find: and the probability P 0 , at leading order in N corresponds to a delta-distribution: where Id is the N × N identity matrix. At the initial time, the distribution is localized around a diagonal matrix with equal entries. Hence, in the rate equation forλ µ , initial conditions are the same for all eigenvalues, and they remain identical for all time. The argument easily generalizes for D > 0.

White noise limit for D > 0: UV regularized theory below critical temperature
For D > 0, the Laplacian contribution in equation (1) modify the effective large N dynamics given by equation (19) as follows: Hence, κ self averages again, but an additional contribution arises because of the Laplacian term. This equation can be formally solved using Fourier transform, with the convention: where k ∈ R D and "u · v" is the standard scalar product. Hence, assuming again that Φ c self averages for large N around a diagonal matrix which is independent of c, and the dynamical equation can be easily solved after they quench as: with: and the initial correlation in the Fourier space reads: From (161) and (93), the equation for r(t) becomes: where: We will consider firstly the quartic case because the closed equation can be solved asymptotically using Laplace transform. This in particular shows the existence of a critical temperature, which is finite in contrast to the zero-dimensional model. We then consider the general cases using a law temperature expansion, where transition temperature looks as the radius of convergence of the series.

The quartic case
Let us investigates the quartic case. If we assume [2,9] that for t large enough, U (κ(t)) → 0, the self-consistent equation for g(t) reads: that can be rewritten as: where and: In the zero temperature limit, the problem is then solved and G(t) ∼ 1/t D/2 , and then U (κ(t)) ∼ 1/t, in agreement with our assumption. Note that G(t) being a positive definite quantity, this makes sense only for µ < 0. Hence, for t τ : for t large enough, and the correlation length grows as √ t, meaning that the system does not reach a thermal regime at a finite time.
For T = 0, one can expect to solve the equation using Laplace transform. Let us denote asf (p) the Laplace transform of the function f (t) as: The standard properties of Laplace transform regarding the convolution product. We get straightforwardly:Ḡ and the Laplace transform of H can be easily computed as: (104) Figure 6 shows the functionH(p). Hence, for large t i.e. for small p, At this stage, we have to clarify a technical point. The closed equation (97) that we considered is only an asymptotic relation. Nerveless, we integrated it for all time taking the Laplace transform, although the closed equation is expected to be wrong for small times. Indeed, we assume the solution of the closed equation provides the true asymptotic behavior for G(t), which can be motivated by the two following observations: • For t large enough, H(t − t ) suppresses low-time contributions provided that G(t) has a finite limit for short times.
• Fluctuations for U (κ(t)) are assumed to have a small standard deviation around the large time-averaged value.
We do the same assumption for higher-order potentials that we will consider in the next section. From table 1, we show that all the computed values forH(0) are non-vanishing, but the sign of this value depends on the dimension and the size of τ . For D = 1, 2, H(0) = ∞, andḠ(0) is non singular. For D = 3,Ḡ(0) is defined for T < T c , with: which vanishes in the limit τ → 0 keeping µ/ḡ 2 fixed. For D > 2, we have: Table 1: Explicit expressions forH(p) in low dimension.
The large time behavior for G(t) can be deduced from the small p expansion oḠ(p). It can be understand from standard theorems on the Laplace transform near the origin [19], and we have for instance the following statement: Theorem 1 Let f (t) be a locally integrable function on [0, ∞) such that f (t) ≈ ∞ m=0 c m t rm as t → ∞ where r m < 0. If the Mellin transformation of this function is defined and if no r m = −1, −2, · · · then the Laplace transformation of f (t) is and for τ = 0, T < T c , we find that G(t) ∼ t −3/2 . For D = 4, the lowest order contribution behaves as p(log(p) + log(τ ) + γ), and G(t) behaves as t −2 log(t). For D > 4, the lowest order in linear with p and G(t) ∼ t −2 . Hence, below the critical dimension, the relaxation time for the zero modes is infinite, and T I (0, t) ∼ t 3/4 for D = 3. The 2-point temporal correlation function for zero initial time C(t) ∼ I T I (0, t)T I (0, 0) behaves as t −3/4 , and the memory of the system follows a power law. For high temperature in contrast, the previous method breaks down, and we expect that G(t) diverges faster than any power law such that Laplace transform for arbitrary small p does not exist, and the system is expected to forget the initial conditions accordingly with an exponential decays.
Taking Laplace to transform (assuming that Laplace transforms for G exists), we get: with:Ḡ assuming µ < 0, the series expansion in T of G 1 (t) leads to: where x := 2dḡ 2 and: Figure 7 shows the behavior of F 0 , F 1 and F 2 for D = 3, 2µ = 1 and τ = 0.1. Asymptotically, F 0 (t) behaves as 1/t 3/2 , precisely: for t large enough. In the second Figure in Figure 7 we show the behavior of F 0 and F 1 and the asymptotic behavior ∼ 0.125/t 3/2 . The functions F n (t), for, n > 0 are also asymptotically uniformly bounded by a/t 3/2 for some a > 0, but is can be checked numerically that F n (t)/t 3/2 → 0 for long time. Indeed, numerically F n (t)/F n−1 (t) → 0 as t → ∞, but remains almost constant for finite but late time, see Figure 8.  For G 0 , we get: where: The typical shape of functions K n is given in Figure 9. Once again we find K 0 ∼ 1/t 3/2 and K n ≥ K n−1 for late time.
where we included an index µ on G refereeing to the selected zero γ µ . This equation looks like the equation of an effective quartic model (see equation (97)), and can be solved using Laplace transform again (assuming the Laplace transform for G(t) exists). We have: Because G(p) is positive definite, this solution exists only below the critical temperature: Hence, in the large N limit, each of the vacua is independent of one from the other, without overlapping, and each of them has its critical temperature looking as the radius of convergence of the low-temperature expansion. During its evolution, the system freezes around one of these vacua with a depth of order N d , and remains around it after an expected short transition period (see Figure 10). Equation (132) shows that a low temperature expansion for G(t): where G (n) (t) satisfies the obvious recursive relation: does make sense only below the critical temperature, which looks like the radius of convergence of the series. Note that our derivation assumes the G(p) exists for all p, and especially for small p (i.e. for large t). Hence, the breakdown of the low-temperature expansion is nothing but the manifestation that this assumption for G(t) becomes wrong, and then that a power time behavior leaves its place to an exponential growth i.e. the system relaxes toward equilibrium exponentially in the high-temperature regime. Finally, one expects that activation effects due to thermal fluctuations could play a role for large but finite N (the famous Kramer's problem) see [26,25,24,21,22,23]. We investigated these aspects in a forthcoming work.
To conclude this subsection, let us investigate the time evolution, for the quartic model. Following (24), we have: The left-hand side can be factorized around the zeros of the polynomial, then: Assuming that u(t) is close to some real zero γ µ , one expects that |u(t) − γ µ | ≡ (t) 1, and expanding the left hand side of the previous equation in power of , we have: where α µ denotes the multiplicity of the zero γ µ and R(γ µ ) is the remaining contribution evaluated at γ µ . If we are only interested by the behavior of the system around γ µ , it is suitable to choose the normalization such that R(γ µ ) = 1/2h m , anḋ For α µ = 1, we recover exactly the analysis of the quartic case, and the system behaves as the other zeros were blinded despite thermal fluctuations. For α µ > 1, the equation can be investigated numerically. Indeed, for α µ = 2 and as for the D = 0 case, G(t) is an increasing function. For T = 0, the function converges indeed exponentially as Figure 12 shows, and we have again: where the index 0 recall that T = 0, and for γ µ = 0.5, we get for instance α ≈ 0.216. Hence for α µ = 2, G(t) behaves exponentially for large t: G(t) ∼ e γ 2 µ t , such that the equilibrium regime reduces to the equation (setting h m = 1/2): and we have two solutions, κ ∞ = 0 and κ ∞ = 2γ µ , meaning that the systems converge towards the zero vacuum due to spatial fluctuations. This obviously contrast with the T = 0 limit for α µ = 1, where G(t) ∼ 1/t 3/2 and κ → γ µ . For T small enough furthermore, the low T expansion for G(t) shows that the exponential law holds for large t. Indeed, because: where Γ (n, z) is the standard incomplete gamma function: The equation can be solved graphically and exhibits generally a single solution, as shown in Figure 11.
Note that for γ µ large enough, and because for large B: we have: The transition temperature can be estimated from the following argument. Assuming G(t) = A(t) e γ 2 * t such that γ 2 * = Q(γ * ), one get an equation forȦ: where δγ 2 µ := γ 2 µ − γ 2 * . Hence as we just discussed, for T small enough in that limit, u(t) is expected to be small for late time, and we can neglect u 2 (t), so that the equation reads: where B(t) := A(t)e −δγ 2 µ t . The equation can be solved recursively, or by Laplace transform, which leads:B whereB(p) denotes the Laplace transform of B(t) and we used B(t = 0) = 1. For small p, this corresponds to an exponential decay: and the assumption κ(t) → 0 for large t requires: .
The same kind of investigation can be performed for other values of α µ . Figure 13 shows the typical behavior of G num,0 (t) (for T = 0) for α µ = 3, γ µ = 0.5 and ∆ = 1. The evolution starts with an exponential phase, G num,0 (t) ∼ e −γ 3 µ t for a small time, and ends with a power law decay G num,0 (t) ∼ t −3/2 for late time. Hence, because of the convolution property, one expect that for t large enough, G num (t) ∼ t −3/2 , as for the case α µ = 1. From this observation, the late time evolution can be investigated from the same method as discussed in section 3.3. In particular, one can estimate the transition temperature from the following argument. During the exponential phase, the system could be close to the zero vacuum, κ ∼ 0. Hence, we find for G(t): that can be solved using Laplace transform as: Hence, the system ends in the non-zero vacuum γ µ as 1/t, provided that T < T c ≈ γ 3 µ /H(0). Note that in this section we provided some estimations for critical temperature, but we do not prove a bound for it, an issue that should be considered in a forthcoming work.

Disordered Langevin dynamics
A way to avoid the difficulty arising from UV divergences without taking care of the definition of the continuum limit is to replace the Laplacian in (1) by a disorder coupling, materialized by a random hermitian Wigner matrix D = D † of size N 2×d : where T I ≡ T I (t) i.e. D = 0 in this section. We assume that the random matrix D decomposes along each colors as a sum of tensorial products: where σ c 's are hermitian Wigner matrices, which can be formally diagonalized as: where {u (λ) i } are orthogonal and normalized. We assume that σ c ∈ GUE are centred gaussian matrices with the same variance µ. Hence, in the large N limit, the empirical distribution for eigenvalues converges toward the Wigner law: and F (t) is still defined as The closed equation is solved as well using Laplace transform and is defined for T < T c , with: For higher order potential furthermore, the previous construction generalizes obviously, and we get T (µ) c = (2σ) d γ µ in replacement of equation (133). As stressed above, the spectrum being bounded, no UV divergence is expected, despite the system exhibiting a non-trivial behavior for a large time. In particular, we recover that the memory of the initial condition does not vanish exponentially, and we have for the 2-point correlation function: and U (κ(t)) → d × 2σ + O(t −1 ) for t large enough.

Discussions and conclusion
In this paper, we investigated the large-time behavior of a stochastic complex tensor in the cyclic melonic regime. We focused on the melonic kinetics and low-temperature regimes for different cases, including white noise limit with Laplacian or tensorial disorder, and memory effects with temporal colored noise. One of the main particularity of the melonic kinetics, which occurs for rank d > 3 regarding kinetics for matrix or vector fields comes essentially from the ability of tensor to self-average without breaking symmetry at leading order (i.e. at the melonic order). Indeed, for vector fields, the kinetics describe a dynamical ordering resulting from the symmetry breaking for U(N ) or O(N ) symmetry. In the melonic case, the vacuum (Φ c ) ij = κδ ij commute with any generator of the Lie algebra associated with this symmetry, which remains unbroken. In the case of the matrice fields, such a tractable vacuum for equilibrium dynamics is unexpected, because of the measured effect in the matrix path integral for equilibrium states, that repelled eigenvalues and leads to a non-trivial spectrum in the large N limit [11]. In contrast for the tensors fields, the measured effect is next to the leading order, and the eigenvalues for the intermediate fields collapse [12]. In this paper, we investigated some new aspects of the low-temperature behavior of the melonic kinetics, which is in connection with our recent contribution in the group field theory context [18]. Some aspects could be addressed in the future, including next to finite N effects and we may implement methods like renormalization group (see [27], in preparation). Furthermore, rigorous bounds for critical temperatures should be proved, rather than estimates provided in this work.