Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts

We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form $\mathcal{F}(z)=\prod_{\ell=1}^\infty \mathcal{F}_0(z^\ell)$ (which entails equal weighting among possible parts $\ell\in\mathbb{N}$). Under mild technical assumptions on the function $H_0(u)=\ln(\mathcal{F}_0(u))$, we show that the limit shape $\omega^*(x)$ exists and is given by the equation $y=\gamma^{-1}H_0(\mathrm{e}^{-\gamma x})$, where $\gamma^2=\int_0^1 u^{-1}H_0(u)\,\mathrm{d}u$. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta-types of decomposable combinatorial structures --- assemblies, multisets and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.Comment: Minor editorial corrections. Published in "Random Structures and Algorithms" (18 Apr 2014, Early View, online), http://onlinelibrary.wiley.com/doi/10.1002/rsa.20540/abstrac

The limit shape of random permutations with polynomially growing cycle weights

In this work we are considering the behavior of the limit shape of Young diagrams associated to random permutations on the set $\{1,\dots,n\}$ under a particular class of multiplicative measures. Our method is based on generating functions and complex analysis (saddle point method). We show that fluctuations near a point behave like a normal random variable and that the joint fluctuations at different points of the limiting shape have an unexpected dependence structure. We will also compare our approach with the so-called randomization of the cycle counts of permutations and we will study the convergence of the limit shape to a continuous stochastic process.Comment: 36 pages, 3 figures. The paper was subject to a major revision (compared to v1): 1) we considered more general weights, i. e. $\theta_m= (\log m)^j m^\alpha$, 2) title replaced, 3) improvements of the presentation, 4) correction of typos and minor mathematical error

Limit Shape of Minimal Difference Partitions and Fractional Statistics

The class of minimal difference partitionsMDP(q) (with gap q) is defined by the condition that successive parts in an integer partition differ from one another by at least q≥0. In a recent series of papers by A. Comtet and collaborators, the MDP(q) ensemble with uniform measure was interpreted as a combinatorial model for quantum systems with fractional statistics, that is, interpolating between the classical Bose–Einstein (q=0) and Fermi–Dirac (q=1) cases. This was done by formally allowing values q∈(0,1) using an analytic continuation of the limit shape of the corresponding Young diagrams calculated for integer q. To justify this “replica-trick”, we introduce a more general model based on a variable MDP-type condition encoded by an integer sequence q=(qi), whereby the (limiting) gap q is naturally interpreted as the Cesàro mean of q. In this model, we find the family of limit shapes parameterized by q∈[0,∞) confirming the earlier answer, and also obtain the asymptotics of the number of parts

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 and controlling the expected weight and length of partitions. In this framework, we obtain a variety of limit theorems for ``short'' partitions as ⟶ ∞, while is either fixed or grows slower than for unconstrained partitions. Our results include the asymptotics of the cumulative cardinality in the case of fixed and the derivation of limit shape in the case of slow growth of . 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