We investigate heavy-Wigner tensors: symmetric random tensors whose independent entries, up to the tensor symmetries, are centered and have moments of order $N^{-(p-1)}$, where $N$ is the tensor dimension. This framework includes normalized adjacency tensors of sparse Erd\H{o}s-R\'enyi hypergraphs and truncated heavy-tailed tensor models. We study trace invariants, a complete family of polynomial invariants under permutations of the tensor indices. We prove that, after the natural normalization, the only non-vanishing asymptotic contributions are those associated with fat hypertrees, and we derive a central limit theorem for these injective trace invariants. As applications, we first analyze Erd\H{o}s-R\'enyi $p$-uniform hypergraphs with edge probability $\alpha_N=c/N^{p-1}$. We prove local weak convergence to a uniform Galton-Watson hypertree with Poisson offspring distribution. We also prove convergence of the empirical spectral distribution of the matrix obtained by contracting the adjacency tensor; in the sparse regime the limiting law depends on the sparsity parameter $c$ and has unbounded support, while in the regime $N^{p-1}\alpha_N\to\infty$ with $N^{p-1}(1-\alpha_N)\to\infty$, the limiting spectral distribution is the semicircle law. This result generalizes for matrices obtained by contracting an arbitrary heavy-Wigner tensor and we derive an explicit formula for the moments of the limiting spectral measure.
We develop a common framework for random width laws, spectral populations, and geometric reconstruction. For a $d$-dimensional orthotope, we prove an exact parity law for the maximal $\pi^{-1}$-grade of every spherical width cumulant, including noncancellation and sign in all dimensions and orders. The first $d$ scalar width moments recover the unordered side vector, and $d-1$ moments are generically insufficient. Each Laplace mode generates an auxiliary width law whose upper endpoint satisfies $M_{n,a} = \lambda_n(a)^{1/2}/\pi$. At high energy the modal coordinate partitions converge to a universal Dirichlet law, while an unsmoothed measure-valued cutoff expansion retains the first geometric memory at face scale. Its simplex moment determines, up to an explicit nonzero factor and a separate off-diagonal argument, a basis-independent projector-gradient Weyl tensor that reconstructs the orthotope. Genuine edge-scale jumps obstruct a third coefficient for the total raw cutoff; exact mixed-boundary Mobius inversion isolates every coordinate stratum and restores a recursive bulk-boundary expansion with a smaller remainder. Beyond orthotopes, we prove direction-labelled identifiability for a canonical linear-quadratic class and finite recovery from direction-sensitive ridge moments under a generator bound. In dimension three, a global great-circle incidence calculus gives the exact step, fold, endpoint-fold, and corner coefficients of reduced zonotopal width densities, including an explicit non-simple corner cancellation. The results distinguish universal aggregation, recoverable geometric memory, and the remaining scalar inverse problem.
It is proved that sketching dimension m = O(k^{3/2}/\epsilon^2) suffices for subspace embedding with a Khatri-Rao sketching matrix with any fixed order $d$.
In this paper, we study the large deviation principle (LDP) for a tensor-weighted functional of i.i.d. random variables, when the sequence of tensors converges under a variant of the"bad"cut norm. Using the LDP, we analyze a Gibbs measure with a tensor-valued Hamiltonian, and characterize the optimizers of the limiting variational problem in terms of a functional fixed point equation. As applications, we focus on several concrete examples, which include monochromatic subgraph counts in sparse random graphs, Erd\H{o}s-R\'enyi hypergraphs, and a generalized Potts statistic of order $v\ge 2$. Studying the optimization problem, we give sufficient conditions for uniqueness of the optimizer, as well as for existence of constant optimizers (replica symmetry). Our results demonstrate universal weak laws for a large class of tensor Gibbs models with approximately regular tensors.
Reihaneh Malekian, Sohom Bhattacharya, Nabarun Deb et al.· 0 citations
We prove universality of limiting local eigenvalue statistics for random matrices over $\mathbb{Z}_p$. In previous work of the author and Van Peski (arXiv:2601.06283), the limiting eigenvalue correlation functions of additive Haar random matrices were studied in arbitrary finite extensions of $\mathbb{Q}_p$. The same Haar random matrix model plays a central role in the Ellenberg-Jain-Venkatesh heuristic for zeros of $p$-adic $L$-functions. We show that its limiting local eigenvalue statistics are unchanged for a broad class of random matrices with independent entries satisfying a mild non-concentration condition. Thus the random matrix predictions underlying the Ellenberg-Jain-Venkatesh heuristic are not artifacts of the particular Haar ensemble, but instead reflect universal limiting eigenvalue statistics. In this sense, our results provide additional theoretical support for the robustness of their random matrix heuristic. Our proof is based on a new framework, which we call the resultant distribution method. The method recovers limiting laws and root statistics of $p$-adic polynomials from the distributions of their resultant valuations against fixed test polynomials, together with suitable degree estimates. As a second application, we consider random $p$-adic polynomials with independent coefficients satisfying a mild non-concentration condition. Caruso (arXiv:2110.03942) determined the joint root correlation functions of the Haar coefficient model over finite extensions of $\mathbb{Q}_p$. We prove that, for roots of absolute value one, these limiting correlation functions are universal and persist for a broad class of independent coefficient distributions.
We study robust estimation of simple random tensors of arbitrary order $q\in\mathbb{N}$ under finite-moment assumptions and adversarial contamination. We propose the first robust estimator achieving near-optimal dimension-free statistical rates in this setting. The estimator attains the near-optimal corruption rate whenever $p\ge2q$ moments are finite and continues to provide nontrivial guarantees throughout the weak-moment regime $q\le p\le2q$. Being based on directional trimmed means and minimax aggregation, our estimator is adaptive to $p$ and upper bounds on hypercontractive constants without resorting to interval-intersection procedures. Our analysis extends the trimmed-mean framework underlying recent advances in robust mean and covariance estimation to arbitrary tensor order. In particular, we establish concentration inequalities for higher-order counting and truncated empirical multi-vector product processes. We believe these inequalities could be of independent interest beyond the present application, including algorithmic robust estimation.
R. Oliveira, Zoraida F. Rico, Philip Thompson· 0 citations
We study finite-rank multiplicative deformations of rotationally invariant non-Hermitian random matrices. More precisely, we consider models of the form $\mathbf{A}(\mathbf{I}+\mathbf{T})$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix, $\mathbf{T}$ is a finite-rank normal perturbation, and $\mathbf{I}$ denotes the identity matrix. We characterize the emergence of outlier eigenvalues, their fluctuations, and the associated eigenvector overlaps. Our results provide a multiplicative non-Hermitian counterpart to the classical Baik--Ben Arous--P\'ech\'e framework.
Unknown authors· 0 citations
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