It is shown that r-uniform Erd\H{o}s-R\'enyi hypergraphs on n vertices exhibit a spectral gap as soon as their expected number of hyperedges satisfies $m \gg n^{r/2}$.
Abstract
Friedman and Wigderson (1995) introduced a notion of second eigenvalue for hypergraphs that generalizes the second eigenvalue of the adjacency matrix of a graph. We show that $r$-uniform Erd\H{o}s-R\'enyi hypergraphs on $n$ vertices exhibit a spectral gap as soon as their expected number of hyperedges $m$ satisfies $m \gg n^{r/2}$. Prior work identified this scale only up to logarithmic factors; removing these factors is the main technical challenge. Our proof overcomes this obstacle through an explicit decomposition of an associated selector process, inspired by a generic decomposition theorem of Talagrand (2021). As a consequence of our techniques, we obtain improved norm bounds for sparse random tensors with independent entries. Finally, under a mild moment equivalence assumption, we extend to tensors a seminal result of Seginer (2000) for random matrices with i.i.d. entries.
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 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 establish a tensor spectral stability theorem for uniform hypergraphs with bounded matching number. More precisely, for fixed integers $k\geq 3$ and $\beta\geq2$, and sufficiently large $n$, we prove that every $n$-vertex $k$-uniform hypergraph $H$ with matching number at most $\beta$ and tensor spectral radius close to the maximum possible value among all such hypergraphs must be structurally close to the extremal hypergraph $S_{n,k,\beta}$, whose edges consist of all $k$-sets intersecting a fixed set of $\beta$ vertices. Furthermore, we show that every edge of $H$ intersects this distinguished vertex set and that $H$ contains all but a small proportion of the edges of $S_{n,k,\beta}$. As an application, we obtain a new proof of the spectral version of the Erd\H{o}s matching conjecture for sufficiently large $n$.
The Erd\H{o}s Matching Conjecture concerns the maximum number of hyperedges in an $r$-uniform hypergraph with bounded matching number. In this paper, we study a spectral counterpart of this conjecture. For sufficiently large $n$, we determine the maximum spectral radius over all $n$-vertex $r$-uniform hypergraphs whose matching number is less than $s$, and characterize the unique extremal hypergraph. To establish the main theorem, we first apply the shifting method to reduce the problem to shifted hypergraphs. We then derive several spectral upper bounds through hypergraph decomposition and related variational estimates for tensor spectral radii. With these estimates, we analyze the structural properties of shifted-saturated hypergraphs and prove the spectral extremal theorem for shifted hypergraphs with bounded matching numbers. Finally, we drop the shifted condition and extend our spectral bound to general $r$-uniform hypergraphs. Our main theorem states that for any $n$-vertex $r$-uniform hypergraph $H$ with matching number $\nu(H)<s$, the inequality $\rho(H)\leq \rho(\mathcal{F}_{s-1}(n))$ holds whenever $n$ is sufficiently large. Here $\mathcal{F}_{a}(n)$ denotes the family of all $r$-subsets of $[n]$ intersecting the vertex set $[a]$, and equality is attained if and only if $H$ is isomorphic to $\mathcal{F}_{s-1}(n)$. As an immediate corollary, we derive a spectral counterpart of the classical Erd\H{o}s-Ko-Rado theorem for intersecting hypergraph families.
Liying Kang, Yongchun Lu, Xiying Yuan et al.· 2 citations
We study the problem of recovering latent inner products from a random geometric graph with anisotropic Gaussian latent points. More precisely, for an i.i.d. sample $x_1, \dots, x_n \sim N(0,\Sigma)$ where $\Sigma \in \mathbb{R}^{d \times d}$, an edge $(i,j)$ is present in the graph if and only if $\langle x_i, x_j \rangle \ge \zeta$ for a threshold $\zeta$. We assume the threshold $\zeta$ to be chosen such that the average edge density of the graph is of constant order. To address the undesired degree fluctuations amplified by the anisotropy of the latent points, we consider the doubly centered adjacency matrix of the graph, and estimate the latent inner products using a rank-$d$ spectral approximation of the doubly centered matrix. The estimator obtains a mean squared error with a rate involving the stable rank of the covariance matrix $\Sigma$. Notably, the rate of estimation matches the state of the art for the isotropic case $\Sigma = I_d$, and permits an ill-conditioned covariance matrix with a diverging condition number. The analysis of the spectral method proceeds via the entrywise Hermite expansion of the doubly centered adjacency matrix with respect to the latent inner products. Instead of the standard trace method, it uses a decoupling argument recently introduced by Kaushik, Romberg, and Muthukumar (2025) to control nonlinear error terms.
We study the spectrum of the adjacency matrix $A_n$ of directed inhomogeneous random graphs on $n$ vertices. We assume that $A_n$ has independent entries and diverging average degree scale $s_n$. This framework includes, as special cases, the directed Chung--Lu random graph and directed stochastic block models. Assuming boundedness of the variance profile and that $s_n$ diverges faster than a suitable logarithmic function of $n$, we show that the rank-one Chung--Lu model satisfies a non-homogeneous version of the circular law, which in some situations allows for an explicit expression. Moreover, under mild conditions, we identify the asymptotic singular value distribution using tools from free probability. Finally, for finite-rank directed models, we prove the existence of eigenvalues outside the bulk and establish their joint Gaussian fluctuations at the scale $\sqrt{s_n/n}$, with an explicit covariance matrix. These results extend the theory of spectral outliers and their fluctuations to directed inhomogeneous random graphs.