Asymptotics and scaling analysis of 2-dimensional lattice models of vesicles and polymers
Abstract
The 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
Schröder 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.
Authors
Haug, Nils AdrianCollections
- Theses [4122]