Walks with Small Steps in the 4D-Orthant

使用对角线的渐近格路径enumeration

Asymptotic lattice path enumeration using diagonals

本文给出了Weyl室中游动族的新的渐近公式。这里研究的模型定义的步骤集，表现出许多对称性，并限制到第一正交。得到的公式非常简单：每个模型的指数增长性由步数给出，而次指数增长性仅取决于底层晶格的维数和在每个坐标中向前移动的步数。这些表达式是通过分析一个多元有理函数的奇异变化而得到的，该函数的对角线数为所讨论的格路径。此外，我们将展示如何计算这些模型的次显性增长，以及如何确定一阶渐近的偏移。

在四分之一平面上有小台阶的走道中的新台阶

New steps in walks with small steps in the quarter plane

在这篇文章中，我们获得了新的表达式的生成函数计数（非奇异）的走动与小步骤的四分之一平面。这些都是以无穷级数的形式给出的，而在文献中，标准的表达式使用的是边值问题的解。我们用三个例子说明我们的结果（一个代数的情况下，超越D-有限的情况下，和一个无限群模型）。

Bijections Between Łukasiewicz Walks and Generalized Tandem Walks

In this article, we study the enumeration by length of several walk models on the square lattice. We obtain bijections between walks in the upper half-plane returning to the $x$-axis and walks in the quarter plane. A recent work by Bostan, Chyzak, and Mahboubi has given a bijection for models using small north, west, and south-east steps. We adapt and generalize it to a bijection between half-plane walks using those three steps in two colours and a quarter-plane model over the symmetrized step set consisting of north, north-west, west, south, south-east, and east. We then generalize our biject…

四分之一平面上的多步行走

Walks in the Quarter Plane with Multiple Steps

我们扩展的四分之一平面中的最近邻的步行模型，其中多重性附加到每个方向的步骤集的分类。我们的研究导致了少量的无限族，它们完全表征了其相关群为D4，D6或D8的所有模型。这些族涵盖了所有多重数为0、1、2或3的模型，这些模型在实验上被发现是D有限的，但有三个值得注意的例外。

A bijective study of Basketball walks

The Catalan numbers count many classes of combinatorial objects. The most emblematic such objects are probably the Dyck walks and the binary trees, and, whenever another class of combinatorial objects is counted by the Catalan numbers, it is natural to search for an explicit bijection between the latter objects and one of the former objects. In most cases, such a bijection happens to be relatively simple but it might sometimes be more intricate. In this work, we focus on so-called \emph{basketball walks}, which are integer-valued walks with step-set $\{-2,-1,+1,+2\}$. The presence of $-2$ as a…

无共线三点的无限小步$\boldsymbol{\text{Z}^3}$游走

An infinite small-step $\mathbb Z^3$-walk with no collinear triple

本文构造了$\text{Z}^3}$中步长取自16个固定向量的无限游走，无三个顶点共线，解决了由Erdős问题193推广的Gerver与Ramsey问题。

On the enumeration of Hoppy's walks

The enumeration of k-Dyck paths ending at level j after m up-steps, where the last step is an up-step, is given as a sum, improving on a previous formula given by Deng and Mansour.

The vectorial kernel method for walks with longer steps

Asinowski, Bacher, Banderier and Gittenberger (A. Asinowski, A. Bacher, C. Banderier and B. Gittenberger. Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata. Algorithmica, pp. 1-43, 2019.) recently developed the vectorial kernel method - a powerful extension of the classical kernel method that can be used for paths that obey constraints that can be described by finite automata, e.g. avoid a fixed pattern, avoid several patterns at once, stay in a horizontal strip and many others more. However, they only c…

四分之一平面内小步长行走的补偿方法

The compensation approach for walks with small steps in the quarter plane

本文是第一次应用的补偿方法计数问题。我们讨论了如何将这种方法可以应用到一个一般类的四分之一平面$Z_{+}^{2}$的步骤集，是一个子集的$\{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}$在内部的$Z_{+}^{2}$的行走。我们推导出一个显式表达式的计数生成函数，这原来是亚纯和非完整的，可以很容易地被逆转，并可用于获得渐近表达式的计数系数。

象限内大步行走的伽罗瓦结构

Galoisian structure of large steps walks in the quadrant

本文通过赋予大步行走模型的轨道一个伽罗瓦群作用，将小步模型的行走群理论推广到大步情形，并由此建立一般策略，首次证明了带权大步模型的代数性，证实了 Bostan 等人的猜想并发现新的大步代数模型。

大步长象限游走轨道上的伽罗瓦结构

A Galois structure on the orbit of large steps walks in the quadrant

本文为四分之一平面大歩长游走的轨道赋予伽罗瓦群作用，推广了游走群概念，并据此发展伽罗瓦方法证明不变量与解耦的存在性，首次证明大步长加权模型的代数性，验证BBMM猜想并发现新代数模型。

随机游走上海龙树上吸收平均步数

Average Steps until Absorption on Random Walks on Sea Dragon Trees

研究随机游走上海龙树的吸收平均步数，利用马尔可夫链分析几种海龙树类的t(G,v,u)，并探讨一般图上ASUAs的相关方程结果。

On the nature of four models of symmetric walks avoiding a quadrant

We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially al…

小步长三象限游走计数及其他M象限锥上的游走计数

Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones

该研究将四分之一平面小步游走计数的相关结果推广到任意M象限锥情形，推导了刻画生成函数的解析泛函方程，明确了不同M值下生成函数的代数、D有限或D代数性质。

具有$k$独立增量的随机游动

Random walks with $k$-wise independent increments

我们构造了一个几乎必然有界的具有成对独立步数的随机游动的例子，并且对于任何$m$和$k$，构造了一个具有$k$独立步数的随机游动，它没有平稳分布模$m$。

在 $\mathbb{N}^{16}$ 中仅使用单位步长的无限行走，且无三点共线

An infinite walk in $\mathbb{N}^{16}$, using only unit steps, with no three collinear points

该研究证明在16维空间中，仅用单位基向量步长可无限行走且避免三点共线。

Infinite-step stationarity of rotor walk and the wired spanning forest

We study rotor walk, a deterministic counterpart of the simple random walk, on infinite transient graphs. We show that the final rotor configuration of the rotor walk follows the law of the wired uniform spanning forest oriented toward infinity (OWUSF) measure when the initial rotor configuration is sampled from OWUSF. This result holds for all graphs for which each tree in the wired spanning forest has one single end almost surely. This answers a question posed in a previous work of the author (Chan 2018).

Combinatorics of nondeterministic walks of the Dyck and Motzkin type

This paper introduces nondeterministic walks, a new variant of one-dimensional discrete walks. At each step, a nondeterministic walk draws a random set of steps from a predefined set of sets and explores all possible extensions in parallel. We introduce our new model on Dyck steps with the nondeterministic step set {{--1}, {1}, {--1, 1}} and Motzkin steps with the nondeterministic step set {{--1}, {0}, {1}, {--1, 0}, {--1, 1}, {0, 1}, {--1, 0, 1}}. For general lists of step sets and a given length, we express the generating function of nondeterministic walks where at least one of the walks exp…

图上的限制随机游动

Restricted random walks on a graph

利用转移矩阵的方法，研究了图上的限制性随机游动问题。给出了任意图上可走的具有r个反转步数的n步可走数的母函数的封闭表达式。在图的情况下，一个统一的价，我们表明，我们的结果具有概率意义，并推导出明确的表达的生成函数的特征值的邻接矩阵。给出了它在周期格和完全图中的应用。

Counterbalancing steps at random in a random walk

A random walk with counterbalanced steps is a process of partial sums $\check S(n)=\check X_1+ \cdots + \check X_n$ whose steps $\check X_n$ are given recursively as follows. For each $n\geq 2$, with a fixed probability $p$, $\check X_n$ is a new independent sample from some fixed law $μ$, and with complementary probability $1-p$, $\check X_n= -\check X_{v(n)}$ counterbalances a previous step, with $v(n)$ a uniform random pick from $\{1, \ldots, n-1\}$. We determine the asymptotic behavior of $\check S(n)$ in terms of $p$ and the first two moments of $μ$. Our approach relies on a coupling with…

关于四分之一平面内小步数游动的函数

On the functions counting walks with small steps in the quarter plane

模型的空间均匀行走在四分之一平面${\bf Z}_+^{2}$的步骤采取从一个子集$\mathcal{S}$的跳跃到8个最近的邻居。研究了从原点开始到$n$步后结束于$(i,j) \in {\bf Z}_+^{2}$的步数$q(i,j;n)$的生成函数$(x,y,z)\mapsto Q(x,y;z)$.对于所有非奇异游动模型，函数$x \mapsto Q(x,0;z)$和$y\mapsto Q(0,y;z)$在具有无穷多个亚纯分支的${\bf C}$上延拓为多值函数，其中的极点集是确定的.这些功能的性质是来自这个结果：即，对于所有51散步承认一定的无限群的双有理变换的${\bf C}^2$，间隔$]0,1/|\mathcal{S}|[$的变化$z$分裂成两个密集的子集，使功能$x \mapsto Q(x,0;z)$和$y\mapsto Q(0,y;z)$被证明是完整的任何$z$从其中之一和非完整的任何$z$从其他。这需要$(x,y,z)\mapsto Q(x,y;z)$的非完整性，因此证明了Bousquet-Mélou和Mishna的一个猜想。

Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on $\mathbb{Z}^2$ defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or $-1$. We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane $\mathbb{N}^2$, counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna [Contemp. Math., pp. 1--39, Amer. Math. Soc., 2010] identified 19 models of walks that possess a D-finite generating function; linear differential equ…

计算Gessel行走的generating function的显式表达式

Explicit expression for the generating function counting Gessel's walks

Gessel行走是在正象限$\mathbb{Z}_{+}^{2}$内以单位步数在以下任何方向上移动的平面行走：西、东北、东和西南。在本文中，我们找到了一个显式的三元生成函数计数Gessel行走$k\geq 0$步骤，开始于$(0,0)$和结束于一个给定的点$(i,j) \in \mathbb{Z}^2_+$。

非确定性行走的组合学

Combinatorics of nondeterministic walks

本文研究非确定性行走的组合特性，探讨Dyck和Motzkin步集，并证明非确定性桥、游走和meanders等路径类为代数结构，提出可达点概念并扩展至一般步集。