Related papers: New 1/N expansions in random tensor models
The melonic sector has been proven to be dominant in tensor models at large N. This is true as long as the observables we consider, composites of 2n tensors, are small. That is, if n is much smaller than N. In this paper, I argue that, in…
We use solvable two-dimensional gauge theories to illustrate the issues in relating large N gauge theory to string theory. We also give an introduction to recent mathematical work which allows constructing master fields for higher…
Several recent trends in machine learning theory and practice, from the design of state-of-the-art Gaussian Process to the convergence analysis of deep neural nets (DNNs) under stochastic gradient descent (SGD), have found it fruitful to…
Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erd\H{o}s-Gallai Theorem in random graphs. In particular, we determine, up to a constant…
About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings…
Gauged tensor models are a class of strongly coupled quantum mechanical theories. We present the exact analytic solution of a specific example of such a theory: namely the smallest colored tensor model due to Gurau and Witten that exhibits…
Tensor models are generalizations of matrix models, and are studied as discrete models of quantum gravity for arbitrary dimensions. Among them, the canonical tensor model (CTM for short) is a rank-three tensor model formulated as a totally…
Tensor networks have found a wide use in a variety of applications in physics and computer science, recently leading to both theoretical insights as well as practical algorithms in machine learning. In this work we explore the connection…
We study tensor network varieties associated with the triangular graph, with a focus on the case where one of the physical dimensions is 2. This allows us to interpret the tensors as pencils of matrices. We provide a complete…
The large-n expansion is developed for the study of critical behaviour of d-dimensional systems at m-axial Lifshitz points with an arbitrary number m of modulation axes. The leading non-trivial contributions of O(1/n) are derived for the…
Graphical models have become a very popular tool for representing dependencies within a large set of variables and are key for representing causal structures. We provide results for uniform inference on high-dimensional graphical models…
We consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices and we show that its $L^p$-norm can be upper bounded, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free…
We obtain estimates for the Kolmogorov distance to appropriately chosen gaussians, of linear functions \[ \sum_{i\in [n]^d} \theta_i X_i \] of random tensors $\boldsymbol{X}=\langle X_i:i\in [n]^d\rangle$ which are symmetric and…
Large amount of multidimensional data represented by multiway arrays or tensors are prevalent in modern applications across various fields such as chemometrics, genomics, physics, psychology, and signal processing. The structural complexity…
M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by…
We analyse a class of quantum field theory models illustrating some of the possibilities that have emerged in the general study of the short distance properties of superselection sectors, performed in a previous paper (together with R.…
Many problems in high-dimensional statistics appear to have a statistical-computational gap: a range of values of the signal-to-noise ratio where inference is information-theoretically possible, but (conjecturally) computationally…
Motivated by Erd\H{o}s' ternary conjecture and by recent work of Cui--Ma--Jiang [``Geometric progressions meet Cantor sets'', \textit{Chaos Solitons Fractals} \textbf{163} (2022), 112567.] on intersections between geometric progressions and…
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex…
This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Lapalace…