Analysis on Graphs in Sendai 2011
Date: 2011 Feb. 21st,Place: Graduate School of Science, Tohoku University, 合同棟 508 B・C

Program
10:00-10:50

Tomoyuki Shirai

Title: Random analytic functions and their zeros
Abstract: After recalling the notions of Gaussian analytic functions and determinantal(or fermion) point processes, we discuss a central limit theorem for a certain class ofrandom analytic functions including random power series with i.i.d. random coefficients.
Also we discuss a limit theorem for the zeros of random analytic functions.
11:00-11:50

Akihiro Munemasa

Title: Accumulation points of the smallest eigenvalues of graphs
Abstract: The method of Alan Hoffman developed in 1970's was formalized by Woo andNeumaier in 1995. In this talk, we proceed further to introduce decompositions ofHoffman graphs, and reprove Hoffman's theorem on the smallest eigenvalues of graphs
in the new formalism.
14:00-14:50

Nobuaki Obata

Title: Asymptotic Spectral Analysis of Large Graphs~~ A Quantum Probabilistic Approach
Abstract: Quantum probability theory provides a framework of extending themeasure-theoretical (Kolmogorovian) probability theory. The idea traces back to von
Neumann (1932), who, aiming at the mathematical foundation for the statisticalquestions in quantum mechanics, initiated a parallel theory by making a selfadjoint
operator and a trace play the roles of a random variable and a probability measure,respectively. During the last 25 years quantum probability theory has developed
considerably with wide applications. Our theme is relatively new and further
development seems promising.
In this talk I will review how quantum probabilistic ideas are applied to spectral graphtheory, in particular, to the study of asymptotic spectral distributions of large graphs (orgrowing graphs) . We have so far developed three tools: (i) quantum decomposition andinteracting Fock spaces; (ii) various concepts of independence and associated central
limit theorems; (iii) partition statistics and moment--cumulant formulae. The method of

quantum decomposition has been proved to be effective when the graph structure is
reduced in a sence to a one-sided (finite or infinite) path graph because the theory oforthogonal polynomials of one-variable is fully applicable. It is apparently important toweaken this restriction and to go further. I will illustrate the basic idea of quantum
decomposition and report some recent attempts in this line.
15:00-15:50

Fumihiko Nakano

Title: Domino tilings with impurities
Abstract: We consider the dimer problem on a planar non-bipartite graph, where there
are two types of dimers one of which we regard as impurities. Computer simulations
imply a tendency that impurities are attracted to the boundary, which is the motivation
to study this particular graph. We show that (1) the local move connectedness yielding
an ergodic Markov chain on all possible dimer coverings, and (2) a bound of the numberof dimer coverings and that of the probability of finding an impurity at a given edge.
16:00-16:50

Pavel Exner

Title: Vertex approximations in quantum graphs
Abstract: It is a longstanding problem how to understand the coupling in vertices of aquantum graph using approximations, either by a family of appropriate "fat graphs" orby operators on the graph itself. In particular, within an approximation by Neumann
Laplacians on a tube network the squeezing limit yields only the free (or Kirchhoff)boundary conditions. In this talk I will report first a recent result coming from acommon work with Olaf Post: it will be shown that adding families of suitably scaled
potentials to those Laplacians one can get spectrally nontrivial vertex couplings,
including those with wave functions discontinuous at the vertices. Furthermore, I will
describe a fresh result obtained together with Taksu Cheon and Ondřej Turek onapproximations by Schrödinger operators on graphs which shows a way how theproblem can be solved in full generality.

東北大学大学院
理学研究科
合同棟 508 B・C