1,561 research outputs found
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting
distribution for the number of fixed points for involutions chosen uniformly at
random from that class. In particular we consider monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
Regenerative tree growth: structural results and convergence
We introduce regenerative tree growth processes as consistent families of
random trees with n labelled leaves, n>=1, with a regenerative property at
branch points. This framework includes growth processes for exchangeably
labelled Markov branching trees, as well as non-exchangeable models such as the
alpha-theta model, the alpha-gamma model and all restricted exchangeable models
previously studied. Our main structural result is a representation of the
growth rule by a sigma-finite dislocation measure kappa on the set of
partitions of the natural numbers extending Bertoin's notion of exchangeable
dislocation measures from the setting of homogeneous fragmentations. We use
this representation to establish necessary and sufficient conditions on the
growth rule under which we can apply results by Haas and Miermont for
unlabelled and not necessarily consistent trees to establish self-similar
random trees and residual mass processes as scaling limits. While previous
studies exploited some form of exchangeability, our scaling limit results here
only require a regularity condition on the convergence of asymptotic
frequencies under kappa, in addition to a regular variation condition.Comment: 23 pages, new title, restructured, presentation improve
The free path in a high velocity random flight process associated to a Lorentz gas in an external field
We investigate the asymptotic behavior of the free path of a variable density
random flight model in an external field as the initial velocity of the
particle goes to infinity. The random flight models we study arise naturally as
the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external
field. By analyzing the time duration of the free path, we obtain exact forms
for the asymptotic mean and variance of the free path in terms of the external
field and the density of scatterers. As a consequence, we obtain a diffusion
approximation for the joint process of the particle observed at reflection
times and the amount of time spent in free flight.Comment: 30 page
- …