Yukihiro Hashimoto

Notions of independence related to the free group

The central limit problem for algebraic probability spaces associated with the Haagerup states on the free group with countably many generators leads to a new form of statistical independence in which the singleton condition is not satisfied. This circumstance allows us to obtain nonsymmetric distributions from the central limit theorems deduced from this notion of independence. In the particular case of the Haagerup states, the role of the Gaussian law is played by the Ullman distribution. The limit process is explicitly realized on the finite temperature Boltzmannian Fock space. The role of entangled ergodic theorems in the proof of the central limit theorems is discussed.

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

QUANTUM DECOMPOSITION IN DISCRETE GROUPS AND INTERACTING FOCK SPACES

We introduce a kind of quantum decomposition of generators of a discrete group with a suitable length function and construct a sequence of one-mode interacting Fock spaces associated with a filtration of the group. We show stochastic convergence of such a sequence of interacting Fock spaces and obtain a general stochastic limit theorems on discrete groups. As an application to free groups, we see that the Haagerup states give rise to a Gaussian–Poisson transform through its coherent expression.