Simply generated non-crossing partitions

环带上的高斯自由场：BPZ方程与能级线的穿越概率

Gaussian free field in annulus: BPZ equations and crossing probabilities for level lines

该研究针对带交替边界条件的环带高斯自由场，构造配分函数并证明其满足环带BPZ方程，通过控制配分函数推导出能级线穿越环带的概率。

Annular non-crossing permutations and partitions, and second-order asymptotics for random matrices

We study the set $S_{ann-nc}$ of permutations of $\{1, ..., p+q \}$ which are non-crossing in an annulus with $p$ points marked on its external circle and $q$ points marked on its internal circle. The algebraic approach to $S_{ann-nc}$ goes by identifying three possible crossing patterns in an annulus, and by defining a permutation to be annular non-crossing when it does not display any of these patterns. We prove the annular counterpart for a ``geodesic condition'' shown by Biane to characterize non-crossing permutations in a disc. We point out that, as a consequence, annular non-crossing per…

Central Limit Theorems for some Set Partition Statistics

We prove the conjectured limiting normality for the number of crossings of a uniformly chosen set partition of [n] = {1,2,...,n}. The arguments use a novel stochastic representation and are also used to prove central limit theorems for the dimension index and the number of levels.

非交叉划分上的乘性与半乘性函数及其与累积量的关系

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

本文引入非交叉划分上的半乘性函数群，统一解释自由随机变量乘法与布尔及t-布尔累积量的关系，并证明其与对称函数Hopf代数的对偶联系。

The splitting process in free probability theory

Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equa…

Noncommutative Brownian motions with Kesten distributions and related Poisson processes

We introduce and study a noncommutative two-parameter family of noncommutative Brownian motions in the free Fock space. They are associated with Kesten laws and give a continuous interpolation between Brownian motions in free probability and monotone probability. The combinatorics of our model is based on ordered non-crossing partitions, in which to each such partition $P$ we assign a weight depending on the numbers of disorders and orders in $P$ related to the natural partial order on the set of blocks of $P$ implemented by the relation of being inner or outer. In particular, we obtain a simp…

Higher order infinitesimal freeness

We define higher order infinitesimal noncommutative probability space and infinitesimal non-crossing cumulant functionals. In this framework, we generalize to higher order the notion of infinitesimal freeness, via a vanishing of mixed cumulants condition. We also introduce and study some non-crossing partitions related to this notion. Finally, as an application, we show how to compute the successive derivatives of the free convolution of two time-indexed families of distributions from their individual derivatives.

分数布朗运动的水平穿越

Level crossings of fractional Brownian motion

本文将布朗运动局部时由水平穿越极限刻画的经典结果推广到非 Markov、非鞅的分数布朗运动，利用次可加遍历思想与平移随机缝合引理构造其逐路径局部时，并证明 Lebesgue 划分上第 1/H 次变差的几乎必然收敛。

Non-crossing Annular Pairings and The Infinitesimal Distribution of the GOE

We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (GOE). In particular we show how the infinitesimal moments are described by non-crossing partitions, but not of type B. We demonstrate the asymptotic infinitesimal freeness of independent complex Wishart matrices. With the combinatorial picture we can easily compute the infinitesimal cumulants of the GOE and demonstrate the lack of asymptotic infinitesimal freeness of independent GOE ensembles. I have added a new figure and some additional explanatory remarks. This update corrects a number…

Poisson type operators on the Fock space of type B and in the Blitvi{ć} model

In \cite{Bia97} Biane proposed a new statistic on set partitions which he called \emph{restricted crossings}. In a series of papers \cite{Ans01,Ans04,Ans04b,Ans05} Anshelevich showed that this statistic is an essential tool to investigate stochastic processes on $q$-Fock space. In particular, Anshelevich constructed operators whose moments count restricted crossings and used these operators to develop a beautiful theory of noncommutative $q$-Lévy processes. In the present paper following Anshelevich we define gauge operators on $(α,q)$-Fock and cumulants which are governed by statistics on par…