12 research outputs found
Staircase polygons: moments of diagonal lengths and column heights
We consider staircase polygons, counted by perimeter and sums of k-th powers
of their diagonal lengths, k being a positive integer. We derive limit
distributions for these parameters in the limit of large perimeter and compare
the results to Monte-Carlo simulations of self-avoiding polygons. We also
analyse staircase polygons, counted by width and sums of powers of their column
heights, and we apply our methods to related models of directed walks.Comment: 24 pages, 7 figures; to appear in proceedings of Counting Complexity:
An International Workshop On Statistical Mechanics And Combinatorics, 10-15
July 2005, Queensland, Australi
Area limit laws for symmetry classes of staircase polygons
We derive area limit laws for the various symmetry classes of staircase
polygons on the square lattice, in a uniform ensemble where, for fixed
perimeter, each polygon occurs with the same probability. This complements a
previous study by Leroux and Rassart, where explicit expressions for the area
and perimeter generating functions of these classes have been derived.Comment: 18 pages, 3 figure
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
Limit laws for discrete excursions and meanders and linear functional equations with a catalytic variable
We study limit distributions for random variables defined in terms of
coefficients of a power series which is determined by a certain linear
functional equation. Our technique combines the method of moments with the
kernel method of algebraic combinatorics. As limiting distributions the area
distributions of the Brownian excursion and meander occur. As combinatorial
applications we compute the area laws for discrete excursions and meanders with
an arbitrary finite set of steps and the area distribution of column convex
polyominoes. As a by-product of our approach we find the joint distribution of
area and final altitude for meanders with an arbitrary step set, and for
unconstrained Bernoulli walks (and hence for Brownian Motion) the joint
distribution of signed areas and final altitude. We give these distributions in
terms of their moments.Comment: 33 pages, 1 figur
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
Asymptotics and scaling analysis of 2-dimensional lattice models of vesicles and polymers
PhDThe subject of this thesis is the asymptotic behaviour of generating functions
of different combinatorial models of two-dimensional lattice walks
and polygons, enumerated with respect to different parameters, such as
perimeter, number of steps and area. These models occur in various applications
in physics, computer science and biology. In particular, they
can be seen as simple models of biological vesicles or polymers. Of particular
interest is the singular behaviour of the generating functions around
special, so-called multicritical points in their parameter space, which correspond
physically to phase transitions. The singular behaviour around
the multicritical point is described by a scaling function, alongside a small
set of critical exponents.
Apart from some non-rigorous heuristics, our asymptotic analysis mainly
consists in applying the method of steepest descents to a suitable integral
expression for the exact solution for the generating function of a given
model. The similar mathematical structure of the exact solutions of the
different models allows for a unified treatment. In the saddle point analysis,
the multicritical points correspond to points in the parameter space at
which several saddle points of the integral kernels coalesce. Generically,
two saddle points coalesce, in which case the scaling function is expressible
in terms of the Airy function. As we will see, this is the case for Dyck and
Schr枚der paths, directed column-convex polygons and partially directed
self-avoiding walks. The result for Dyck paths also allows for the scaling
analysis of Bernoulli meanders (also known as ballot paths).
We then construct the model of deformed Dyck paths, where three saddle
points coalesce in the corresponding integral kernel, thereby leading to an
asymptotic expression in terms of a bivariate, generalised Airy integral.Universit盲t Erlangen-N眉rnberg
Queen Mary Postgraduate Research Fun