There are at least two standard ways to prove this formula in the literature. By the inverse function theorem, is diffeomorphic to for sufficiently close to , and so by further abuse of notation we can identify all such with and now view as an element of and write the preceding integral as. This can be lifted up non-uniquely, and modulo higher order corrections to an infinitesimal -dimensional parallelepiped in ; applying the action of the Hamiltonian vector field for an infinitesimal time , and also letting vary in an infinitesimal interval using some arbitrary smooth connection to identify together different fibres of arising fromthis interval , we obtain modulo higher order corrections a -dimensional parallelepiped in. By 4 , the Hamiltonian vector field, which is tangent to these circles, is symplectically orthogonal to the tangent bundle of.
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Arguing as before, it then suffices to show that varies linearly in for sufficiently close to a regular value. Information References 30 Citations 43 Files Plots. So I guess the summary of itzyksoon situation is that the HCIZ and Brezis-Hikami-Johansson formulae are logically equivalent to each other, and either formula can be proven by solving the relevant heat equation or Brownian motion.
By the Brezin-Hikami-Johansson formula see Theorem 7 of these notesthe eigenvalues of are distributed according to the density. Another way, which I only learned about recently, is to use the formulae for evolution of eigenvalues under Dyson Brownian motion as well as the closely related formulae for the GUE ensemblewhich were derived in this itzykxon blog post. On the other hand, as is preserved by the action, and is the vector field for the infinitesimal generator conjugation with respect to the unitary diagonal matrix for infinitesimalwe see that preserves:.
For this reason we see that we may restrict attention to the case when the are all distinct; as the sum towe conclude that is a permutation of. Terrance Tao remembe… on Jean Bourgain. Email Address never made public. Chiral random matrix models: This is consistent, though, with the fact that the somewhat rarely studied anti-symmetric GOE ensemble has cleaner ityzkson in particular, having a determinantal structure similar to GUE than the much more commonly studied symmetric GOE ensemble.
The Chiral phase transition in a random matrix model with molecular correlations - Wettig, Tilo et al.
Let be the Hamiltonian for some and. By the Ktzykson representation theorem, there must be some density function with the property that. Then, in a sufficiently small neighbourhood of any given regular value ofthe Duistermaat-Heckman measure is a polynomial multiple of Haar measure onwith the polynomial being of degree at itzyksoon.
The Threefold way - Verbaarschot, Jacobus J. The sign conventions are sometimes reversed in the literature; note that and are Hermitian and so this expression is real-valued.
Thanks for the references and early history of this result! We now seek to interpret as a symplectic manifold and as a moment map for a torus action, in order to view as a Duistermmat-Heckman measure.
Di Francesco , Itzykson , Zuber : Classical $W$-algebras
Letand let be given the standard symplectic form. We use exponential coordinateswhere is a small skew-Hermitian matrix, so that. At present, the Hamiltonian can be an arbitrary smooth function, and so the Duistermaat-Heckman measure is also more or less completely arbitrary.
Now consider an infinitesimal -dimensional parallelepiped in. We now use a little bit of de Rham cohomology. If we let be the Hamiltonian. For anywe consider the scalar mapitzkyson can be written as Differentiating this along an arbitrary vector field we see that where is the vector field.
High Energy Physics - Theory
The flow of the hamiltonian vector field is clockwise, while a complex number will rotate anticlockwise when multiplied by a number like exp it. Let denote the space of Hermitian matrices. First, you need to establish that is -invariant. Stationary phase — We now apply the Duistermaat-Heckman theorem to the task of proving the Harish-Chandra-Itzykson-Zuber integral formula.
Thanks a lot for this clarification. Now we express as a moment map and, as a byproduct, conclude the closed nature of. Dates First available in Project Euclid: Following the method of statinonary phase, we now study how the stationary points of.
Recall itzyskon a symplectic manifold is a smooth manifold equipped with a symplectic formthat is to say a smooth anti-symmetric zubed on which is both non-degenerate thus whenever is a vector field that is non-vanishing at and closed thus.