Independently at every vertex $x$ of a large core graph $\mathbb G$, we draw a random graph and connect $x$ to a subset of its vertices. This model of a randomly sprinkled graph captures qualitative features of commonly observed graphs; it can also be regarded as a model of quantum disorder, where disorder arises from local perturbations to the graph geometry. We show that it is naturally connected both to the Anderson model and to site percolation on $\mathbb G$ by deriving an Anderson-percolation representation for its spectrum. We characterize the support of the spectrum and derive quantitative bounds on the integrated density of states. We also analyse in detail the regime of sparse sprinkling, in particular investigating the behaviour of the density of states in the vicinity of atoms.
A random geometric graph is a random graph model whose vertex set is a binomial point process, where two vertices are connected by an edge if each lies within the ball of radius $r$ centered at the other. In this paper, we investigate the asymptotic fluctuations of the threshold radius required for the maximum degree o...
We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\geq3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the $\mathrm{Sine}_1$ process with intensity $1/\pi$. We also establish universality of consecutive gaps...
We develop the thinning--sprinkling technique, a general method for proving robustness of graph properties under random vertex sampling. Using it, we show that random induced subgraphs of tough graphs, high-degree connected vertex-transitive graphs, and nearly regular sublinear expanders retain strong connectivity or e...
We study the limit shape of the range of the random walk excited to the center on $\mathbb Z^d$, with $d\geq2$. A cookie is placed at each nonzero vertex. On its first visit to such a vertex, the walker consumes the cookie and takes its next nearest-neighbor step with a bias toward the origin. On a visit to the origin,...
Higher-order networks take group interactions as given. I ask what happens when a group interaction is made by pairwise dynamics on a graph, and follow it through the transition that decides whether it matters at large scale, Anderson localisation. An emergent hyperedge is a $k$-body bound state of a Hamiltonian whose...
We study the graphon variational problem that governs lower tail large deviations of the number of copies of a fixed nonbipartite graph $H$ in a sparse Erd\H{o}s--R\'enyi random graph. Our main new insight is that any non-constant minimizer of the variational problem exhibits a discontinuous jump away from the constant...
Byron Chin· 0 citations
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