Ljuben Mutafchiev
We consider procedures of sampling parts from a random integer partition. We determine asymptotically the probabilty distribution of the randomly-selected part whenever the positive integer that is partitioned becomes large.
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For a given sequence of weights (non-negative numbers), we consider partitions of the positive integer n. Each n-partition is selected uniformly at random from the set of all such partitions. Under a classical scheme of assumptions on the weight sequence, which are due to Meinardus (1954), we show that the largest part in a random weighted partition, appropriately normalized, converges weakly, as n tends to infinity, to a random variable having the extreme value (Gumbel's) distribution. This limit theorem extends some known results on particular types of integer partitions and on the Bose-Eins…
On a Class of Partitions with Lower Expected Star Discrepancy and Its Upper Bound than Jittered Sampling
本文提出了一种新分区模型，证明其在预期星偏差上优于经典抖动采样，并推导了更优的上界，解决了相关开放问题。
Let $λ$ be a partition of the positive integer $n$, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of $λ$ at random. They obtained limiting distributions of the multiplicity $μ_n=μ_n(λ)$ of the randomly-chosen part as $n\to\infty$. The asymptotic behavior of the part size $σ_n=σ_n(λ)$, under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partitio…
Growing integer partitions with uniform marginals and the equivalence of partition ensembles
本文构造了整数分拆上具有均匀边际的马尔可夫增长过程，分析其跳跃概率的渐近行为以解释极限形状的形成，并利用该过程给出两个随机分拆系综等价的充分条件，进而证明奇偶部分计数渐近独立且极限分布不同。
Let $\mathcal{P}=\{p_1,p_2,...\}$ be the set of all odd primes arranged in increasing order. A Goldbach partition of the even integer $2k>4$ is a way of writing it as a sum of two primes from $\mathcal{P}$ without regard to order. Let $Q(2k)$ be the number of all Goldbach partitions of the number $2k$. Assume that $2k$ is selected uniformly at random from the interval $(4,2n], n>2$, and let $Y_n=Q(2k)$ with probability $1/(n-2)$. We prove that the random variable $\frac{Y_n}{n/\left(\frac{1}{2}\log{n}\right)^2}$ converges weakly, as $n\to\infty$, to a uniformly distributed random variable in t…
We propose a new method, probabilistic divide-and-conquer, for improving the success probability in rejection sampling. For the example of integer partitions, there is an ideal recursive scheme which improves the rejection cost from asymptotically order $n^{3/4}$ to a constant. We show other examples for which a non--recursive, one--time application of probabilistic divide-and-conquer removes a substantial fraction of the rejection sampling cost. We also present a variation of probabilistic divide-and-conquer for generating i.i.d. samples that exploits features of the coupon collector's proble…
Let $λ$ be a partition of the positive integer $n$ chosen umiformly at random among all such partitions. Let $L_n=L_n(λ)$ and $M_n=M_n(λ)$ be the largest part size and its multiplicity, respectively. For large $n$, we focus on a comparison between the partition statistics $L_n$ and $L_n M_n$. In terms of convergence in distribution, we show that they behave in the same way. However, it turns out that the expectation of $L_n M_n -L_n$ grows as fast as $\frac{1}{2}\log{n}$ We obtain a precise asymptotic expansion for this expectation and conclude with an open problem arising from this study.
We prove a lower bound expansion on the probability that a random $\pm 1$ matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form $e_i \pm e_j$, which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form $e_i \pm e_j \pm e_k \pm e_\ell$, which corresponds to the partition 1111. The fifth…
We develop a new Markov chain on graph partitions that makes relatively global moves yet is computationally feasible to be used as the proposal in the Metropolis-Hastings method. Our resulting algorithm can be made reversible and able to sample from a specified measure on partitions. Both of these properties are critical to some important applications and computational Bayesian statistics in general. Our proposal chain modifies the recently developed method called Recombination (ReCom), which draws spanning trees on joined partitions and then randomly cuts them to repartition. We improve the c…
We introduce a class of convex equivolume partitions. Expected $L_2-$discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected $L_2-$discrepancy than classical jittered sampling for the same sampling number. Second, an explicit expected $L_2-$discrepancy upper bound under this kind of partitions is also given. Further, among these new partitions, there is optimal expected $L_2-$discrepancy upper bound.
The Partition Principle Revisited: Non-Equal Volume Designs Achieve Minimal Expected Star Discrepancy
本文提出非等体积分区方法，通过理论分析证明其在高维数值积分中能实现更低的期望星偏差。
Random sampling of contingency tables and partitions: Two practical examples of the Burnside process
本文利用 Burnside 过程及 lumping 技术构造列联表与整数分拆的高效近似均匀抽样算法，并提出反射 Burnside 过程以显著加速收敛，可抽样大小达 $10^{10}$ 的均匀分拆。
We consider the problem of partitioning $n$ integers into two subsets of given cardinalities such that the discrepancy, the absolute value of the difference of their sums, is minimized. The integers are i.i.d. random variables chosen uniformly from the set $\{1,...,M\}$. We study how the typical behavior of the optimal partition depends on $n,M$ and the bias $s$, the difference between the cardinalities of the two subsets in the partition. In particular, we rigorously establish this typical behavior as a function of the two parameters $κ:=n^{-1}\log_2M$ and $b:=|s|/n$ by proving the existence …
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