18,731 research outputs found

### 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 $q$-Painlev\'e System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$

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

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