Theodoros Assiotis
We obtain a combinatorial formula for the positive integer moments of the partition function of the $CβE_{N}$ field, or equivalently the moments of the moments of the characteristic polynomial of the $CβE_{N}$ ensemble. We then use this formula to establish the large $N$ asymptotics of these moments in the "moment-supercritical" regime. A key role is played by Jack polynomials.
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A note on the polynomial moments of the partition function in the SK model
证明了SK模型中分拆函数$Z_N(\cdot)$的$k$阶矩与分拆函数$Z_k(\cdot)$的$N$阶矩之间的一个简单恒等式.作为一个推论，我们发现一个表征的极限$\lim_{N \to \infty} \frac{1}{N} \log \mathbb{E} Z_N(β)^k$替代米歇尔Talagrand以前发现的\cite{4}。
Moments of C$β$E field partition function, $\mathsf{Sine}_β$ correlations and stochastic zeta
本文通过Hua-Pickrell随机zet函数证明了C$β$E场分区函数的超临界矩猜想，并给出了$\mathsf{Sine}_β$点过程的相关函数表达式。
Negative moments of the CREM partition function in the high temperature regime
本研究聚焦于连续随机能量模型（CREM）的高温区间，证明了在$β<β_c$时，CREM配分函数的负$s$矩与其期望的$-s$次幂可比，且常数与$N$无关。
Closed expressions for averages of set partition statistics
在研究有限域上上三角形矩阵群的超特征标计数理论时，我们发现关于集合划分的新统计量的矩（均值、方差和高阶矩）具有简单的封闭表达式，即移位Bell数的线性组合.这里表明，其他统计量的家庭有类似的时刻。线性组合中的系数是$n$中的多项式。这允许精确枚举小$n$的矩，以确定所有$n$的精确公式。
We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of Möbius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by inse…
Asymptotic normality through factorial cumulants and partitions identities
本文利用阶乘累积量给出了一种求渐近正态性的方法。阶乘累积量以同样的方式从阶乘矩中产生，就像（普通）累积量从（普通）矩中产生一样。另一个工具，我们利用的是一个新的身份的“时刻”的分区数。然后利用一般极限结果重新推导了几个模型的渐近正态性，包括经典离散分布、广义分配方案中的占用问题以及与负多项分布相关的两个模型.
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter N. Then we emulate the theory of random matrices in a combinatorial framework: for any parameter N, we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory. A renormalization of the partition algebras allows us to c…
Evaluating moments of length of Pitman partition
本文研究了Pitman分割中随机分割长度K的r阶矩的渐近行为，探讨了参数n和θ趋近于无穷时的近似方法。
The generating function for the number of purely crossing partitions of {1,...,n} is found in terms of the generating function for Bell numbers. Further results about generating functions for asymptotic moments of certain random Vandermonde matrices are derived.
The sequence of so-called signature moments describes the laws of many stochastic processes in analogy with how the sequence of moments describes the laws of vector-valued random variables. However, even for vector-valued random variables, the sequence of cumulants is much better suited for many tasks than the sequence of moments. This motivates us to study so-called signature cumulants. To do so, we develop an elementary combinatorial approach and show that in the same way that cumulants relate to the lattice of partitions, signature cumulants relate to the lattice of so-called "ordered parti…
The partition function of the directed polymer model on Z^{2+1} undergoes a phase transition in a suitable continuum and weak disorder limit. In this paper, we focus on a window around the critical point. Exploiting local renewal theorems, we compute the limiting third moment of the space-averaged partition function, showing that it is uniformly bounded. This implies that the rescaled partition functions, viewed as a generalized random field on R^2, have non-trivial subsequential limits, and each such limit has the same explicit covariance structure. We obtain analogous results for the stochas…
The calculation of multi-fractal properties of directed random walks on hierarchic trees with continuous branching
考虑具有连续分支的层次树随机能量模型，计算了相应的配分函数的矩。我们建立了这些时刻的多重分形性质。我们推导出随机变量正态分布的公式，以及一般情况下。我们比较我们的结果的时刻的配分函数与相应的结果的对数1-D REM和猜想一个特定的幂律尾的配分函数分布在高温阶段。我们的结果建立了反应扩散方程和多尺度之间的联系。
Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice with open boundaries. It has been known for awhile that there is a tight connection between the partition function of the ASEP and moments of Askey-Wilson polynomials, a family of orthogonal polynomials which are at the top of the hierarchy of classical orthogonal polynomials in one variable. On the other hand, Askey-Wilson polynomials can be viewed as a specialization of the…
Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- II
本文研究二维高斯聚合物在亚临界窗口中的配分函数矩，证明了下界与已有上界匹配，方法基于解耦和泊松收敛。
We deal with a sequence of integer-valued random variables $\{Z_N\}_{N=1}^{\infty}$ which is related to restricted partitions of positive integers. We observe that $Z_N=X_1+ \ldots + X_N$ for independent and bounded random variables $X_j$'s, so $Z_N$ has finite mean ${\bf E}Z_N$ and variance ${\bf Var}Z_N$. We want to find the limit distribution of ${\hat Z}_N=\left(Z_N-{\bf E}Z_N\right)/{\sqrt{{\bf Var}Z_N}}$ as $N \to \infty.$ While in many cases the limit distribution is normal, the main results established in this paper are that ${\hat Z}_N \overset{d}{\to} Z_{*},$ where $Z_{*}$ is a bound…
This article offers a simplified approach to the distribution theory of randomly weighted averages or $P$-means $M_P(X):= \sum_{j} X_j P_j$, for a sequence of i.i.d.random variables $X, X_1, X_2, \ldots$, and independent random weights $P:= (P_j)$ with $P_j \ge 0$ and $\sum_{j} P_j = 1$. The collection of distributions of $M_P(X)$, indexed by distributions of $X$, is shown to encode Kingman's partition structure derived from $P$. For instance, if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}$, the $n$th moment of $M_P(X_p)$ is a polynomial function of $p$ which equals the probability gener…
Continuous-time directed polymers to the Critical SHF via moments
本文针对关联布朗环境中的二维连续时间定向随机聚合物，利用矩方法证明其配分函数收敛到临界随机热流（SHF），为临界SHF的离散近似提供了收敛性理论支撑。
Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -- I
本文研究二维有向高斯高分子在弱无序次临界窗口下配分函数的矩，通过排除三重相交的方法分析q为根号log N量级时的矩性质，补充了该模型配分函数渐近行为的相关结果。
High temperature limits for $(1+1)$-dimensional directed polymer with heavy-tailed disorder
Alberts-Khanin-Quastel~ \cite{AKQ12}提出了中间无序区的定向聚合物模型。证明了在温度$βn^{-γ}$与$γ=1/4$成反比时，适当中心化的分拆函数按分布收敛，并给出了其极限.这一结果是在无序变量服从指数矩的假设下得到的，但在六阶矩的假设下也证明了它的普适性。我们证明了这一猜想是有效的，我们进一步扩展了它的展示类不同的普遍限制行为的情况下，小于六个时刻。我们还解释了在不同矩假设和$γ$值下对数分拆函数标度指数的行为。
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