Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges

For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion determinant. For the parameter dependent Gaussian and Laguerre ensembles the matrix elements of the determinant are expressed in terms of corresponding skew-orthogonal polynomials, and their limiting value for infinite matrix dimension are computed in the vicinity of the soft and hard edges respectively. A connection formula relating the distributions at the hard and soft edge is obtained, and a universal asymptotic behaviour of the two point correlation is identified.Comment: 37 pgs., 1fi

Pfaffian Expressions for Random Matrix Correlation Functions

It is well known that Pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices. Moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of Pfaffians. In this article, we review the formulations and applications of Pfaffian formulas. For that purpose, we first present the general Pfaffian expressions in terms of the corresponding skew orthogonal polynomials. Then we clarify the relation to Eynard and Mehta's determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials. Applications of Pfaffian formulas to random matrix theory and other fields are also mentioned.Comment: 28 page

A Variation of the qq-Painlev\'e System with Affine Weyl Group Symmetry of Type E7(1)E_7^{(1)}

Recently a certain $q$-Painlev\'e type system has been obtained from a reduction of the $q$-Garnier system. In this paper it is shown that the $q$-Painlev\'e type system is associated with another realization of the affine Weyl group symmetry of type $E_7^{(1)}$ and is different from the well-known $q$-Painlev\'e system of type $E_7^{(1)}$ from the point of view of evolution directions. We also study a connection between the $q$-Painlev\'e type system and the $q$-Painlev\'e system of type $E_7^{(1)}$. Furthermore determinant formulas of particular solutions for the $q$-Painlev\'e type system are constructed in terms of the terminating $q$-hypergeometric function

Ginsparg-Wilson Relation and Admissibility Condition in Noncommutative Geometry

Ginsparg-Wilson relation and admissibility condition have the key role to construct lattice chiral gauge theories. They are also useful to define the chiral structure in finite noncommutative geometries or matrix models. We discuss their usefulness briefly.Comment: Latex 4 pages, uses ptptex.cls. Talk given at Nishinomiya-Yukawa Memorial Symposium on Theoretical Physics ``Noncommutative Geometry and Quantum Spacetime in Physics", Japan, Nov.11-15, 2006. (To be published in the Proceedings

Donaldson-Thomas theory and cluster algebras

We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we give an alternative proof of Fomin-Zelevinsky's conjectures on $F$-polynomials and $g$-vectors.Comment: 39 pages, 8 figures, mostly rewritte

Non-commutative Donaldson-Thomas theory and vertex operators

In arXiv:0907.3784, we introduced a variant of non-commutative Donaldson-Thomas theory in a combinatorial way, which is related with topological vertex by a wall-crossing phenomenon. In this paper, we (1) provide an alternative definition in a geometric way, (2) show that the two definitions agree with each other and (3) compute the invariants using the vertex operator method, following Okounkov-Reshetikhin-Vafa and Young. The stability parameter in the geometric definition determines the order of the vertex operators and hence we can understand the wall-crossing formula in non-commutative Donaldson-Thomas theory as the commutator relation of the vertex operators.Comment: 29 pages, 4 figures, some minor changes, descriptions about symmetric obstruction theory (section 5.2 and 6.1) are improve