Asymptotic Analysis of Discrete Random Structures: Constrained Integer Partitions and Aggregating Particle Systems

The unifying thrust of this thesis is to explore asymptotic properties of discrete random structures of large ``size'', focusing on their limit shapes. The first part is concerned with asymptotic analysis of the so-called Boltzmann distributions over the spaces of strict integer partitions (i.e. with distinct parts) into sums of perfect q-th powers (e.g. squares). The model is calibrated via the hyper-parameters <N> and <M> controlling the expected weight and length of partitions. In this framework, we obtain a variety of limit theorems for ``short'' partitions as <N>⟶ ∞, while <M> is either fixed or grows slower than for unconstrained partitions.
Our results include the asymptotics of the cumulative cardinality in the case of fixed <M> and the derivation of limit shape in the case of slow growth of <M>.

Building on these and other results, we have also designed sampling algorithms for our models, and studied their complexity and performance. Boltzmann sampling is a topical area in computer science research, but we also argue that our algorithms can be used as exploratory tools in additive number theory.

In the second part, we study the limit shape of integer partitions emerging in the classical occupancy problem, i.e. as a result of random allocation of a large number of independent ``balls'' with a given frequency distribution over infinitely many ``boxes''. To clarify the ideas and to streamline calculations, we focus on a specific model based on the Rayleigh frequency distribution (but generalising to a random number of balls).
We also indicate a link with strict partitions, thereby offering an alternative method of sampling.

In the last part of the thesis, we study the mass distribution in a stochastic system comprising particles of integer weight, which can either aggregate via diffusion or fragment by chipping off a single mass unit. For the model of pure aggregation on a one-dimensional cycle, analysed with a combination of computer simulations and analytical techniques. We observe that the Rayleigh distribution represents the limit shape for the spatial mass profile at intermediate times. In a model with linear dependence between the transition rates and the masses, we show that the role of the limit shape is played by the exponentiated Weibull distribution.