Akihito Hora

Central limit theorems for large graphs: Method of quantum decomposition

A new method is proposed for investigating spectral distribution of the combinatorial Laplacian (adjacency matrix) of a large regular graph on the basis of quantum decomposition and quantum central limit theorem. General results are proved for Cayley graphs of discrete groups and for distance-regular graphs. The Coxeter groups and the Johnson graphs are discussed in detail by way of illustration. In particular, the limit distributions obtained from the Johnson graphs are characterized by the Meixner polynomials which form a one-parameter deformation of the Laguerre polynomials

Central Limit Theorems and Asymptotic Spectral Analysis on Large Graphs

Regarding the adjacency matrix of a graph as a random variable in the framework of algebraic or noncommutative probability, we discuss a central limit theorem in which the size of a graph grows in several patterns. Various limit distributions are observed for some Cayley graphs and some distance-regular graphs. To obtain the central limit theorem of this type, we make combinatorial analysis of mixed moments of noncommutative random variables on one hand, and asymptotic analysis of spectral structure of the graph on the other hand.

Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians

Abstract On the adjacency algebra of a distance-regular graph we introduce an analogue of the Gibbs state depending on a parameter related to temperature of the graph. We discuss a scaling limit of the spectral distribution of the Laplacian on the graph with respect to the Gibbs state in the manner of central limit theorem in algebraic probability, where the volume of the graph goes to ∞ while the temperature tends to 0. In the model we discuss here (the Laplacian on the Johnson graph), the resulting limit distributions form a one parameter family beginning with an exponential distribution (which corresponds to the case of the vacuum state) and consisting of its deformations by a Bessel function.

A Noncommutative Version of Kerov’s Gaussian Limit for the Plancherel Measure of the Symmetric Group

We give a noncommutative extension of Kerov’s central limit theorem for irreducible characters of the symmetric group with respect to the Plancherel measure [S.Kerov:C.R.Acad.Sci.Paris 316 (1993)]in the framework of algebraic probability theory.For adjacency operators associated with the cycle classes, we consider their decomposition according to the length function on the Cayley graph of the symmetric group.We develop a certain noncommutative central limit theorem for them, in which the limit picture is described by creation and annihilation operators on an analogue of the Fock space equipped with an orthonormal basis labelled by Young diagrams.The limit Gaussian measure in Kerov’s theorem appears as the spectral distribution of the field operators in our setting.

SCALING LIMIT FOR GIBBS STATES OF JOHNSON GRAPHS AND RESULTING MEIXNER CLASSES

Asymptotic behavior of spectral distribution of the adjacency operator on the Johnson graph with respect to the Gibbs state is discussed in infinite volume and zero temperature limit. The limit picture is drawn on the one-mode interacting Fock space associated with Meixner polynomials.

Asymptotic spectral analysis of growing regular graphs

We propose the quantum probabilistic techniques to obtain the asymptotic spectral distribution of the adjacency matrix of a growing regular graph. We prove the quantum central limit theorem for the adjacency matrix of a growing regular graph in the vacuum and deformed vacuum states. The condition for the growth is described in terms of simple statistics arising from the stratification of the graph. The asymptotic spectral distribution of the adjacency matrix is obtained from the classical reduction.

Projective Representations and Spin Characters of Complex Reflection Groups

This paper is a continuation of two previous papers in MSJ Memoirs, Vol.\,29 (Math. Soc. Japan, 2013) with the same title and numbered as I and II. Based on the hereditary property given there, from mother groups

G(m,p,n)

G(m,1,n)

and

, the generalized symmetric groups, to child groups

G(m,p,\infty)

G(m,p,n)