Meinardus加权分拆定理的推广及概率证明

A Probabilistic Characterization of the Dominance Order on Partitions

A probabilistic characterization of the dominance partial order on the set of partitions is presented. This extends work in "Symmetric polynomials and symmetric mean inequalities". Electron. J. Combin., 20(3): Paper 34, 2013. Let $n$ be a positive integer and let $ν$ be a partition of $n$. Let $F$ be the Ferrers diagram of $ν$. Let $m$ be a positive integer and let $p \in (0,1)$. Fill each cell of $F$ with balls, the number of which is independently drawn from the random variable $X = Bin(m,p)$. Given non-negative integers $j$ and $t$, let $P(ν,j,t)$ be the probability that the total number of…

Partitions into distinct parts with bounded largest part

We prove an asymptotic formula for the number of partitions of $n$ into distinct parts where the largest part is at most $t\sqrt{n}$ for fixed $t \in \mathbb{R}$. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres' asymptotic formula for distinct parts partitions when instead the number of parts is bounded by $t\sqrt{n}$. Although equivalent to a circle method/saddle-point method calculation, the probabilistic approach predicts the shape of the asymptotic formula, to some degree.

通用参数化分布族的运行数

Universal parameterized family of distributions of runs

本文提出了一种通用参数化分布族的运行数概率公式，推广了二元值随机变量中μ重叠概率，并给出了计算该概率的高效算法。

Sampling Parts of Random Integer Partitions: A Probabilistic and Asymptotic Analysis

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…

Macdonald对称函数的概率测度及其算法

Probabilistic measures and algorithms arising from the Macdonald symmetric functions

Macdonald对称函数用于定义所有整数的所有分区的集合上的度量。根据这些措施的概率算法的增长分区。将Hall-Littlewood多项式的情形与有限典型群的情形联系起来，从而简化了相应的算法。Schur函数的情况下，导致一个$q$ -模拟的Plancharel措施，和相应的算法的条件版本产生的组合的钩子行走的推广。

A probabilistic analysis of a continuous-time evolution in recombination

We study the continuous-time evolution of the recombination equation of population genetics. This evolution is given by a differential equation that acts on a product probability space, and its solution can be described by a Markov chain on a set of partitions that converges to the finest partition. We study an explicit form of the law of this process by using a family of trees. We also describe the geometric decay rate to the finest partition and the quasi-stationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit.

A probabilistic interpretation of the Macdonald polynomials

The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new two-parameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Prop…

关于Hardy-Ramanujan分拆函数渐近式

About the Hardy-Ramanujan partition function asymptotics

本研究以拉普拉斯变换将分拆函数的代数生成函数转为指数形式为核心视角，沿用Fristedt-Romik概率方法推导Hardy-Ramanujan分拆函数的首阶渐近式，并补充了该方法的更多应用示例。

The probabilistic vs the quantization approach to Kähler-Einstein geometry

In the probabilistic construction of Kähler-Einstein metrics on a complex projective algebraic manifold X - involving random point processes on X - a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kähler-Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler-Einstein geometry.

Muttalib--Borodin plane partitions and the hard edge of random matrix ensembles

We study probabilistic and combinatorial aspects of natural volume-and-trace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hard- and soft-edge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy--Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP):…

多 pivoting 快速排序的概率分析

Probabilistic analysis of optimal multi-pivot QuickSort

本文研究了使用 K 个 pivots 的多 pivoting 快速排序算法，分析其在随机模型下的期望关键比较次数，并推导了 K 的渐进行为及收敛率。

Plancherel型随机分拆的渐近性

Asymptotics of Plancherel-type random partitions

本文给出了Philippe Biane提出的一个问题的一个解决方案：我们证明了当划分变大时，划分上的某种Plancherel型概率分布收敛于集合{0，1，2，.}上的一个新的行列式随机点过程.非负整数。这可以被看作是一个边缘极限转换。极限过程由{0，1，2，.}上的相关核决定。它通过Hermite多项式表示，我们称之为离散Hermite核。证明是基于一个简单的参数，从无界自伴差分算子的收敛性导出相关核的收敛性。我们的方法也可以应用到其他一些概率模型。作为一个例子，我们讨论了一个随机分区的Plancherel型模型的体积极限。