Tree-like properties of cycle factorizations

We provide a bijection between the set of factorizations, that is, ordered (n-1)-tuples of transpositions in ${\mathcal S}_{n}$ whose product is (12...n), and labelled trees on $n$ vertices. We prove a refinement of a theorem of D\'{e}nes that establishes new tree-like properties of factorizations. In particular, we show that a certain class of transpositions of a factorization correspond naturally under our bijection to leaf edges of a tree. Moreover, we give a generalization of this fact.Comment: 10 pages, 3 figure

Minimal factorizations of permutations into star transpositions

We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form (excluded due to format error) source. This generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored. Our result exhibits an unexpected and simple symmetry of star factorizations that has yet to be explained in a satisfactory manner

Universality in chaotic quantum transport: The concordance between random matrix and semiclassical theories

Electronic transport through chaotic quantum dots exhibits universal, system independent, properties, consistent with random matrix theory. The quantum transport can also be rooted, via the semiclassical approximation, in sums over the classical scattering trajectories. Correlations between such trajectories can be organized diagrammatically and have been shown to yield universal answers for some observables. Here, we develop the general combinatorial treatment of the semiclassical diagrams, through a connection to factorizations of permutations. We show agreement between the semiclassical and random matrix approaches to the moments of the transmission eigenvalues. The result is valid for all moments to all orders of the expansion in inverse channel number for all three main symmetry classes (with and without time reversal symmetry and spin-orbit interaction) and extends to nonlinear statistics. This finally explains the applicability of random matrix theory to chaotic quantum transport in terms of the underlying dynamics as well as providing semiclassical access to the probability density of the transmission eigenvalues.Comment: Refereed version. 5 pages, 4 figure

Fock factorizations, and decompositions of the L2L^2 spaces over general Levy processes

We explicitly construct and study an isometry between the spaces of square integrable functionals of an arbitrary Levy process and a vector-valued Gaussian white noise. In particular, we obtain explicit formulas for this isometry at the level of multiplicative functionals and at the level of orthogonal decompositions, as well as find its kernel. We consider in detail the central special case: the isometry between the $L^2$ spaces over a Poisson process and the corresponding white noise. The key role in our considerations is played by the notion of measure and Hilbert factorizations and related notions of multiplicative and additive functionals and logarithm. The obtained results allow us to introduce a canonical Fock structure (an analogue of the Wiener--Ito decomposition) in the $L^2$ space over an arbitrary Levy process. An application to the representation theory of current groups is considered. An example of a non-Fock factorization is given.Comment: 35 pages; LaTeX; to appear in Russian Math. Survey