The Watts-Strogatz random graph model on $n$ vertices with parameters $K$ (a positive even integer) and $p \in [0, 1]$ is constructed in two steps. First, one starts with a ring lattice on $n$ vertices, where each vertex is connected to its $K/2$ nearest neighbors on each side. Each edge in turn is then independently rewired with probability $p$ by replacing one endpoint with a uniformly chosen vertex not already adjacent to it. We study the empirical eigenvalue distribution of the adjacency matrix for this model, whose entries are highly dependent due to the rewiring construction. In the regime where both $K$ and $pK$ grow to infinity with the vertex size $n$, we show that, after appropriate scaling, the empirical eigenvalue distribution converges to the semicircle law. The proof is based on a novel coupling argument that approximates the adjacency matrix by a sum of two independent random matrices, one a sparse Wigner matrix and the other a random band matrix. In the case where $K$ and $p$ remain fixed, we propose conjectural formulas for the first five moments of the limiting eigenvalue distribution. These conjectures are supported by a convergence result relating the Watts-Strogatz model to another random graph model, together with numerical simulations.
For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we provide an alternative technique to establish that $q(G) \leq 4$ for any threshold graph $G$ as presented in [L. Emilio Allem, C. Hoppen, J. Lazzarin, L. Siviero Sibemberg, F. Colman Tura, The minimum number of distinct eigenvalues of a threshold graph is at most 4, Linear Algebra and its Applications, 726 (2025) 32 to 53]. In addition, we show that all connected threshold graphs admit a matrix having any four distinct eigenvalues. Further
Jane Breen, Shaun M. Fallat, Johnna Parenteau· 0 citations
A sequence of sparse weighted graphs $(G_N:N\ge 1)$ indexed by the number of vertices $N$ is said to be left-convergent if all (suitably weighted) subgraph counts converge to a limit as $N\to\infty$. This notion generalizes in a natural way Benjamini-Schramm's definition of local weak convergence. A broad research agenda aims at determining which `global'graph properties are determined by left or local weak convergence (in other words, which of these properties are in fact local). We prove that, under a spectral condition on the weighted adjacency matrix ${\boldsymbol A}_N$ of graph $G_N$, the free energy density of the Ising model on this weighted graph is continuous in the left convergence topology (and hence is a local function). Our proof uses the decomposition of the Ising measure as a log-concave combination of product measures, and of the rapid mixing of Langevin dynamics for log-concave measures. As applications, we derive new limit theorems for the free energy density of spin glasses, antiferromagnets, and magnetization constrained ferromagnetic models, on locally tree-like graphs.
We consider the voter model on the giant component of a hyperbolic random graph, which is a spatial scale-free network, in the sparse and linear-giant regime $\alpha\in(1/2,1)$. We find that the quenched expected consensus time has order $n^{2-1/\alpha}$, as the number of vertices $n\to\infty$, with probability arbitrarily close to one. This is generalised to the voter model where each vertex changes its opinion at rates $q(v)={\rm d}(v)^\varphi$, where we also establish the consensus time orders for all $\varphi\geq 0$. These orders have 3 regimes, with a phase transition at $\varphi=2-2\alpha$. For the upper bounds, our main proof idea is to connect the meeting set to some fixed target vertex of appropriate height in the product chain electrical network, to make rigorous an argument due to Durrett.
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.
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sawhney confirmed this conjecture for every constant $p$, leaving the regime where $p=o(1)$ open. In this regime, the only case addressed in the literature is when $H=K_3$, where, in a recent paper, Ara\'ujo and Mattos confirmed the conjecture for $p \in (4n^{-1/2}, 1/2)$. This, together with a general result of R\"ollin and Ross, essentially settles the conjecture for the triangle. We generalise these results by showing that an LCLT holds for $H = K_r$ (for any fixed $r \ge 3$) in the regime $n^{-1/m(H)}\ll p\leq 1/2$, essentially settling the conjecture for cliques.
Asaf Cohen Antonir, Ilay Hoshen, Maksim Zhukovskii· 0 citations
Brouwer conjectured that the sum of the $k$ largest Laplacian eigenvalues of an $n$-vertex graph is less than or equal to the number of its edges plus $\binom{k+1}{2}$ for every $k\in \{1,2,\dots,n\}$, which has been confirmed by Kothari and Tudose (2026) recently. In this note, we characterize the equality case in this inequality. Our main result is that for every $n$-vertex graph $G=(V,E)$ and for every $k\in \{1,2,\dots,n-1\}$, the equality $\sum_{i=1}^k\mu_i(G)=|E(G)|+\binom{k+1}{2}$ holds if and only if $G$ is a threshold graph with clique number $k+1$, where $\mu_1(G)\geq \mu_2(G)\geq \cdots\geq \mu_{n}(G)$ are the Laplacian eigenvalues of $G$. This, together with the confirmed Brouwer's conjecture, would yield a complete solution to the full Brouwer's conjecture posed by Li and Guo (2022). Our proof relies on the projection method of Kothari and Tudose and shows directly that the equality case can occur only for threshold graphs.
Yuhang Cui, Xiao-Dan Chen· 1 citation
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