229 research outputs found
Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
For complex Wigner-type matrices, i.e. Hermitian random matrices with
independent, not necessarily identically distributed entries above the
diagonal, we show that at any cusp singularity of the limiting eigenvalue
distribution the local eigenvalue statistics are universal and form a Pearcey
process. Since the density of states typically exhibits only square root or
cubic root cusp singularities, our work complements previous results on the
bulk and edge universality and it thus completes the resolution of the
Wigner-Dyson-Mehta universality conjecture for the last remaining universality
type in the complex Hermitian class. Our analysis holds not only for exact
cusps, but approximate cusps as well, where an extended Pearcey process
emerges. As a main technical ingredient we prove an optimal local law at the
cusp for both symmetry classes. This result is also used in the companion paper
[arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type
matrices is proven.Comment: 58 pages, 2 figures. Updated introduction and reference
Recent trends and future directions in vertex-transitive graphs
A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade
Random subshifts of finite type
Let be an irreducible shift of finite type (SFT) of positive entropy, and
let be its set of words of length . Define a random subset
of by independently choosing each word from with some
probability . Let be the (random) SFT built from the set
. For each and tending to infinity, we compute
the limit of the likelihood that is empty, as well as the limiting
distribution of entropy for . For near 1 and tending
to infinity, we show that the likelihood that contains a unique
irreducible component of positive entropy converges exponentially to 1. These
results are obtained by studying certain sequences of random directed graphs.
This version of "random SFT" differs significantly from a previous notion by
the same name, which has appeared in the context of random dynamical systems
and bundled dynamical systems.Comment: Published in at http://dx.doi.org/10.1214/10-AOP636 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An application of TQFT to modular representation theory
For p>3 a prime, and g>2 an integer, we use Topological Quantum Field Theory
(TQFT) to study a family of p-1 highest weight modules L_p(lambda) for the
symplectic group Sp(2g,K) where K is an algebraically closed field of
characteristic p. This permits explicit formulae for the dimension and the
formal character of L_p(lambda) for these highest weights.Comment: 24 pages, 3 figures. v2: Lemma 3.1 and Appendix A adde
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