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作者：Abert, Miklos; Glasner, Yair; Virag, Balint
作者单位：HUN-REN; HUN-REN Alfred Renyi Institute of Mathematics; Ben-Gurion University of the Negev; University of Toronto; University of Toronto
摘要：We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log vertical bar G vertical bar. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigidity result. If most eigenvalues of a d-regular finite graph G fall in the Al...
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作者：Pekoz, Erol A.; Rollin, Adrian; Ross, Nathan
作者单位：Boston University; National University of Singapore; University of Melbourne
摘要：We study a new class of time inhomogeneous Polya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein's method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations re...
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作者：Angel, Omer; Barlow, Martin T.; Gurel-Gurevich, Ori; Nachmias, Asaf
作者单位：University of British Columbia; Hebrew University of Jerusalem; Tel Aviv University
摘要：We provide a geometric representation of the Poisson and Martin boundaries of a transient, bounded degree triangulation of the plane in terms of its circle packing in the unit disc. (This packing is unique up to Mobius transformations.) More precisely, we show that any bounded harmonic function on the graph is the harmonic extension of some measurable function on the boundary of the disk, and that the space of extremal positive harmonic functions, that is, the Martin boundary, is homeomorphic ...
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作者：Lee, Ji Oon; Schnelli, Kevin; Stetler, Ben; Yau, Horng-Tzer
作者单位：Korea Advanced Institute of Science & Technology (KAIST); Institute of Science & Technology - Austria; Harvard University
摘要：We consider N x N random matrices of the form H = W + V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.
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作者：Decreusefond, Laurent; Schulte, Matthias; Thaele, Christoph
作者单位：IMT - Institut Mines-Telecom; Institut Polytechnique de Paris; Telecom Paris; Helmholtz Association; Karlsruhe Institute of Technology; Ruhr University Bochum
摘要：A Poisson or a binomial process on an abstract state space and a symmetric function f acting on k-tuples of its points are considered. They induce a point process on the target space of f. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamics representation...
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作者：Ortgiese, Marcel; Roberts, Matthew I.
作者单位：University of Munster; University of Bath
摘要：We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that al...
