497 research outputs found
Patterns in random permutations avoiding the pattern 132
We consider a random permutation drawn from the set of 132-avoiding
permutations of length and show that the number of occurrences of another
pattern has a limit distribution, after scaling by
where is the length of plus
the number of descents. The limit is not normal, and can be expressed as a
functional of a Brownian excursion. Moments can be found by recursion.Comment: 32 page
Asymptotic bias of some election methods
Consider an election where N seats are distributed among parties with
proportions p_1,...,p_m of the votes. We study, for the common divisor and
quota methods, the asymptotic distribution, and in particular the mean, of the
seat excess of a party, i.e. the difference between the number of seats given
to the party and the (real) number Np_i that yields exact proportionality. Our
approach is to keep p_1,...,p_m fixed and let N tend to infinity, with N random
in a suitable way.
In particular, we give formulas showing the bias favouring large or small
parties for the different election methods.Comment: 54 page
The largest component in a subcritical random graph with a power law degree distribution
It is shown that in a subcritical random graph with given vertex degrees
satisfying a power law degree distribution with exponent , the
largest component is of order . More precisely, the order of
the largest component is approximatively given by a simple constant times the
largest vertex degree. These results are extended to several other random graph
models with power law degree distributions. This proves a conjecture by
Durrett.Comment: Published in at http://dx.doi.org/10.1214/07-AAP490 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Conditioned Galton-Watson trees do not grow
An example is given which shows that, in general, conditioned Galton-Watson
trees cannot be obtained by adding vertices one by one, as has been shown in a
special case by Luczak and Winkler.Comment: 5 pages, 2 figure
Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
This survey is a collection of various results and formulas by different
authors on the areas (integrals) of five related processes, viz.\spacefactor
=1000 Brownian motion, bridge, excursion, meander and double meander; for the
Brownian motion and bridge, which take both positive and negative values, we
consider both the integral of the absolute value and the integral of the
positive (or negative) part. This gives us seven related positive random
variables, for which we study, in particular, formulas for moments and Laplace
transforms; we also give (in many cases) series representations and asymptotics
for density functions and distribution functions. We further study Wright's
constants arising in the asymptotic enumeration of connected graphs; these are
known to be closely connected to the moments of the Brownian excursion area.
The main purpose is to compare the results for these seven Brownian areas by
stating the results in parallel forms; thus emphasizing both the similarities
and the differences. A recurring theme is the Airy function which appears in
slightly different ways in formulas for all seven random variables. We further
want to give explicit relations between the many different similar notations
and definitions that have been used by various authors. There are also some new
results, mainly to fill in gaps left in the literature. Some short proofs are
given, but most proofs are omitted and the reader is instead referred to the
original sources.Comment: Published at http://dx.doi.org/10.1214/07-PS104 in the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Congruence properties of depths in some random trees
Consider a random recusive tree with n vertices. We show that the number of
vertices with even depth is asymptotically normal as n tends to infinty. The
same is true for the number of vertices of depth divisible by m for m=3, 4 or
5; in all four cases the variance grows linearly. On the other hand, for m at
least 7, the number is not asymptotically normal, and the variance grows faster
than linear in n. The case m=6 is intermediate: the number is asymptotically
normal but the variance is of order n log n.
This is a simple and striking example of a type of phase transition that has
been observed by other authors in several cases. We prove, and perhaps explain,
this non-intuitive behavious using a translation to a generalized Polya urn.
Similar results hold for a random binary search tree; now the number of
vertices of depth divisible by m is asymptotically normal for m at most 8 but
not for m at least 9, and the variance grows linearly in the first case both
faster in the second. (There is no intermediate case.)
In contrast, we show that for conditioned Galton-Watson trees, including
random labelled trees and random binary trees, there is no such phase
transition: the number is asymptotically normal for every m.Comment: 23 page
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