In the past two years, several points of view have been proposed to address the question of the generalization of the theory of free probability to random tensors with different invariances, and it is unclear at this point whether they lead to the same notions of tensorial free cumulants and freeness. One way to approach this problem, developed by Collins, Gurau and the second named author for local unitary invariant random tensors, relies on finite size quantities involving averages over the invariance group, and whose asymptotics naturally possess the properties expected for tensorial generalizations of free cumulants of arbitrary orders. At this point, this approach has only been carried out for certain distributions, and for a subset of the moments that define such theories, and a more systematic and exhaustive study is lacking. This is the program initiated in this paper: we link this approach to the one proposed by Nechita and Park; extend a number of their results as well as those of the aforementioned paper to arbitrary orders of fluctuations, thereby generalizing higher order free cumulants; push further the study of distributions with larger invariance groups; detail the link with the asymptotics of the free-energies of the tensor HCIZ and BGW integrals; and provide formulae for tensorial free cumulants of products of tensors. Another important question is that of the definition of concrete distributions whose tensorial free-cumulants take non-trivial values. We compute the tensorial free cumulants for Gaussian random tensors with non-trivial covariances, and show that they provide such examples.
It was recently proven that, in contrast to their matrix analogues, the moments of a real Gaussian tensor of size N do not in general factorize over their connected components in the asymptotic large N limit. While the original proof of this rather surprising result was not constructive, explicit examples of non-factorizing moments, which are expectation values of trace-invariants, have since then been discovered. We explore further aspects of this problem, with a focus on Haar-distributed (or Gaussian) complex random tensors, which are more directly relevant to quantum information. We start out by exhibiting an explicit example of non-factorizing trace-invariant, thereby filling a gap in the recent literature. We then turn to the opposite question: that of finding interesting families of trace-invariants that do in fact factorize at large N. We establish three main theorems in this regard. The first one provides a sufficient combinatorial bound ensuring large N factorization, that is also simple enough to be applicable to various cases of practical relevance. Our second main result shows that the expectation value of any compatible trace-invariant is dominated by certain tree-like combinatorial structures at large N, which we refer to as tree-like dominant pairings. Our third main theorem establishes that any trace-invariant admitting tree-like dominant pairings does actually factorize at large N. In this way, we are able to prove that various families of trace-invariants that have been previously studied in the literature do factorize at large N. We apply our findings to the theory of multipartite quantum entanglement: to any trace-invariant is associated a multipartite generalization of Rényi entanglement entropy, whose typical expectation value in the uniform random quantum state can be explicitly computed assuming large N factorization.
Abstract In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.
We construct a measure that exhibits two aspects of a new type of universality and dramatically simplifies the integration of tensors
Ta1,a2,…,aD∈C (
a1,…,aD=1,…,N) at large
N. In contrast to matrix integration, in which matrix traces canonically yield the integrand, tensors need additional information (equivalent to a
D-coloured graph
B) to contract their indices and form a tensor trace
B(T). We show that, whenever each
B1,…,Bn can be obtained by a recursive construction known as melonicity, then the leading order in
N of the integral of $ {B_1}(T) {B_2}(T) \cdots {B_n}(T) $ is independent of the -- often intricate -- combinatorics of the traces
Bi, but also, to our surprise, independent of
D as far as
D≥3. Instead, at large
N, these integrals are some functions (indexed by
n) of the number of vertices
2pi of
Bi which we call melonic polynomials. Melonic traces cumulants with respect to any ('interacting') measure
exp{−ND−1∑i=1mgiBi(T)}dμ0(T)(g1,…,gm∈R,dμ0(T)=the tensor Gaussian) with each
Bi melonic, can be computed with our universal measure that replaces each
Bi by a canonical trace depending only on
pi. We prove that any two melonic tensor models are indistinguishable at large-
N, independently of the number of tensor indices (first universality aspect), and of the fine-grainedness of their interactions (second universality), being a sufficient condition that the couplings (the parameters
gi above) agree and their respective traces are monomials with the same degree in
T.
Gaussian-process limits are powerful in describing overparameterized machine learning models, yet their validity in structured tensor-network architectures remains unclear. Here we analytically present a moment-based approach that identifies precise conditions for the emergence and breakdown of Gaussian universality in tensor-network learning models, with a focus on matrix product states. We prove that in the large bond dimension limit, the learning models with both local and global observables converge to Gaussian processes, with explicit finite-size bounds on higher-order moment deviations. Whereas in the large physical dimension limit, the Gaussian universality no longer persists: while the models with local observables retain Gaussian-process behavior, those global cases exhibit persistent non-Gaussian corrections. Our results reveal that Gaussian-process behavior in tensor-network learning is controlled not only by parameter number, but also by architectural scaling, observable locality, and the spectral properties.
We introduce a new method, based on the recently developed random tensor theory, to study the p-spin glass model with non-Gaussian, correlated disorder. Using a suitable generalization of Gurau's theorem on the universality of the large N limit of the p-unitary ensemble of random tensors, we exhibit an infinite family of such non-Gaussian distributions which leads to same low temperature phase as the Gaussian distribution. While this result is easy to show (and well known) for uncorrelated disorder, its robustness with respect to strong quenched correlations is surprising. We show in detail how the critical temperature is renormalized by these correlations. We close with a speculation on possible applications of random tensor theory to finite-range spin glass models.