271 research outputs found
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Fractional Langevin equation
We investigate fractional Brownian motion with a microscopic random-matrix
model and introduce a fractional Langevin equation. We use the latter to study
both sub- and superdiffusion of a free particle coupled to a fractal heat bath.
We further compare fractional Brownian motion with the fractal time process.
The respective mean-square displacements of these two forms of anomalous
diffusion exhibit the same power-law behavior. Here we show that their lowest
moments are actually all identical, except the second moment of the velocity.
This provides a simple criterion which enables to distinguish these two
non-Markovian processes.Comment: 4 page
Star Formation and Dynamics in the Galactic Centre
The centre of our Galaxy is one of the most studied and yet enigmatic places
in the Universe. At a distance of about 8 kpc from our Sun, the Galactic centre
(GC) is the ideal environment to study the extreme processes that take place in
the vicinity of a supermassive black hole (SMBH). Despite the hostile
environment, several tens of early-type stars populate the central parsec of
our Galaxy. A fraction of them lie in a thin ring with mild eccentricity and
inner radius ~0.04 pc, while the S-stars, i.e. the ~30 stars closest to the
SMBH (<0.04 pc), have randomly oriented and highly eccentric orbits. The
formation of such early-type stars has been a puzzle for a long time: molecular
clouds should be tidally disrupted by the SMBH before they can fragment into
stars. We review the main scenarios proposed to explain the formation and the
dynamical evolution of the early-type stars in the GC. In particular, we
discuss the most popular in situ scenarios (accretion disc fragmentation and
molecular cloud disruption) and migration scenarios (star cluster inspiral and
Hills mechanism). We focus on the most pressing challenges that must be faced
to shed light on the process of star formation in the vicinity of a SMBH.Comment: 68 pages, 35 figures; invited review chapter, to be published in
expanded form in Haardt, F., Gorini, V., Moschella, U. and Treves, A.,
'Astrophysical Black Holes'. Lecture Notes in Physics. Springer 201
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