The Bethe-Hessian is a symmetric matrix for which the negative spectrum has been observed to encode the informative structure of sparse stochastic block models. We prove that, in the stochastic block model where all vertices have expected degree $d>1$, the number of negative eigenvalues of the Bethe-Hessian is exactly the number predicted by the eigenvalues of the planted model lying outside the bulk spectrum. The condition $d>1$ is optimal, and matches a regime in which existing spectral approaches based on larger non-Hermitian matrices apply. Our result extends a theorem of Stephan and Zhu, who established the same conclusion under the assumption $d\geq 2$. Our improvement relies on two main ideas. First, we construct test vectors on the $2$-core, where degree fluctuations are substantially smaller, and then extend them to the entire graph while controlling the quadratic form. Second, we construct the test vectors using an isotropic basis of the underlying Markov random field, with coefficients adapted to each relevant planted eigenvalue. This allows us to control the fluctuations of the test vectors throughout the sparse regime.
We study the eigenvalues of the random Laplacian matrix $D - A$, where $A$ is a Wigner matrix with sub-Gaussian entries and the diagonal matrix $D$ contains the row sums of $A$. Our main results show that the extreme eigenvalues of this model exhibit Poisson statistics; in particular, after the appropriate centering an...
We investigate the eigenvalues of Hermitian Toeplitz matrices generated by the symmetric Fisher-Hartwig symbol $\hat a(z) = (1-z)^{\alpha/2}(1-z^{-1})^{\alpha/2}$ for $\alpha \in (0,2)$. Using Dirichlet-Neumann bracketing of the discrete Laplacian, we establish explicit, non-asymptotic bounds for the individual eigenva...
We develop a novel approach to matrix discrepancy based on matrix small-ball estimates. Specifically, we use a determinantal weight (obtained from the log-barrier) to scale the small-ball probability into a partition function of a tilt of the Gaussian measure. We then employ matrix-weighted Poincar\'e inequalities to c...
We derive, in closed form, the complete spectrum of the combinatorial Laplacian L(L) on the level- L Berker–Ostlund diamond hierarchical lattice DL(b) with branching parameter b⩾2. Because the vertex degree grows exponentially under inflation, the renormalized eigenvalue equation is not a Laplacian at the coarser scale...
Andrea Ladiana· Journal of Physics A: Mathem...· 0 citations
We establish the large-$n$ asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order $2k+\frac{1}{2}$ for $k \in \mathbb{N}...
The spectral sample and pivotal set of a percolation crossing have the same one- and two-coordinate inclusion probabilities under the uniform product measure. We prove that both Shannon entropies are comparable to total influence for critical square-lattice bond and triangular-lattice site crossings, with matching esti...
Yitzchak Shmalo· 1 citation
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