ABSTRACT We study the matrix discrepancy problem in the average‐case setting. Given a sequence of symmetric matrices , its discrepancy is defined as the minimal spectral norm over all signed sums with . Our contributions are twofold. First, we study the asymptotic discrepancy of random matrices. When the matrices belong to the Gaussian orthogonal ensemble, we provide a sharp characterization of the asymptotic discrepancy and show that the limiting distribution is concentrated around , under the assumption . We show that a broad class of random matrices has discrepancy when . For , we give efficiently achievable complementary upper bounds of and , which match the lower bound up to constants when . Second, we analyse the matrix hyperbolic cosine algorithm, an online algorithm for matrix discrepancy minimization due to Zouzias (2011), in the average‐case setting. We show that the algorithm achieves with high probability a discrepancy of for a broad class of random matrices, including Wigner matrices with entries satisfying a hypercontractive inequality and Gaussian Wishart matrices.
Let be the largest size of an induced -free subgraph that every -vertex -free graph is guaranteed to contain. We prove that for any triangle-free graph , Along the way we give a slight improvement of a construction of Erd & odblac;s-Frankl-R & ouml;dl for the Brown-Erd & odblac;s-S & oacute;s -problem when is large. In contrast to our result for , for any -free graph containing a cycle, we prove there exists such that For every graph , we prove that there exists such that whenever is a non-empty graph such that is not contained in any blowup of , then . On the other hand, for graph that is not a clique, and every , we exhibit a -free graph such that .
ABSTRACT We study the chromatic number of typical triangle‐free graphs with edges and establish the width of the scaling window for the transitions from to and from to . The transition from 3‐ to 4‐colorability has scaling window of width . To prove this, we show a high probability equivalence of the 3‐colorability of a random triangle‐free graph at this density and the satisfiability of an instance of bipartite random 2‐SAT, for which we establish the width of the scaling window following the techniques of Bollobás, Borgs, Chayes, Kim, and Wilson. The transition from 4‐ to 5‐colorability has scaling window of width . To prove this, we show a high probability equivalence of the 4‐colorability of a random triangle‐free graph at this density and the simultaneous 2‐colorability of two independent Erdős–Rényi random graphs. For this transition, we also establish the limiting probability of 4‐colorability inside the scaling window.
We study the inference of the root for a random nearest neighbor tree generated by sequentially embedding vertices uniformly at random in the d-dimensional torus and connecting each new vertex to the nearest existing vertex. Given an error parameter and the unlabeled tree, we want to efficiently find a small "confidence set" of epsilon > 0 candidate vertices containing the root with probability at least 1-epsilon. We define several problem variations-such as embedded and metric root finding-which differ based on the available metric information provided in addition to the graph structure (torus embedding, edge lengths, or none). For embedded and metric root finding, we construct efficient algorithms and derive bounds on the confidence set size. For embedded root finding, the upper bound is subpolynomial in 1/epsilon and the information-theoretic lower bound is polylogarithmic in 1/epsilon . For embedded root finding in d=1 , we obtain matching upper and lower bounds for a confidence set of size Theta (log(1/epsilon)/log log(1/epsilon)) .
We provide new exact formulas for the distribution of massive spanning forests on the complete graph, which give also a new outlook on the celebrated special case of the uniform spanning tree. As a corollary we identify their local limit. This generalizes a well-known theorem of Grimmett on the local limit of uniform spanning trees on the complete graph.
