A new type of quantum walks based on decomposing quantum states

Absence of wave operators for one-dimensional quantum walks

We show that there exist pairs of two time evolution operators which do not have wave operators in a context of one-dimensional discrete time quantum walks. As a consequence, the borderline between short range type and long range type is decided.

周期量子游走的瓶颈效应与谐波型速度界

Bottleneck Effects and Harmonic-Type Velocity Bounds for Periodic Quantum Walks

本文证明了具有任意周期硬币的一维量子游走传播速度的显式上界，包括微扰区域中瓶颈效应导致的线性速度界和一般情况下的谐波平均速度界，并给出了下界。

Type transition of simple random walks on randomly directed regular lattices

Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function, the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of the simple random walk, i.e.\ its being recurrent or transient, and show that there exists a critical value of the decay power, ab…

关于正定矩阵上随机游走的Matsumoto-Yor和Dufresne型定理

Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices

本文研究了正定矩阵上乘法随机游走的Matsumoto-Yor和Dufresne型定理，揭示了类型II Beta分布增量下的随机矩阵递归结构及其显式解。

Random walks on randomly oriented lattices

Simple random walks on various types of partially horizontally oriented regular lattices are considered. The horizontal orientations of the lattices can be of various types (deterministic or random) and depending on the nature of the orientation the asymptotic behaviour of the random walk is shown to be recurrent or transient. In particular, for randomly horizontally oriented lattices the random walk is almost surely transient.

Super-Walk Formulae for Even and Odd Laplacians in Finite Graphs

The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix. In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph. By defining two types of walks and giving orientation to a finite graph, one can easily count the number of the total signs of each kind of walk from one element to another of a fixed length.

To fixate or not to fixate in two-type annihilating branching random walks

We study a model of competition between two types evolving as branching random walks on $\mathbb{Z}^d$. The two types are represented by red and blue balls respectively, with the rule that balls of different colour annihilate upon contact. We consider initial configurations in which the sites of $\mathbb{Z}^d$ contain one ball each, which are independently coloured red with probability $p$ and blue otherwise. We address the question of \emph{fixation}, referring to the sites eventually settling for a given colour, or not. Under a mild moment condition on the branching rule, we prove that the p…

离散时间量子游走的费曼公式

Feynman Formula for Discrete-time Quantum Walks

本文通过费曼公式将离散时间量子游走与四状态马尔可夫加性过程关联，推导谱分解与极限密度的关系，并利用概率表示提出高效的蒙特卡洛计算方法。

关于立方格点游走的代数面积

On the algebraic area of cubic lattice walks

本研究提出立方格点上指定长度与3D代数面积的闭合随机游走的显式枚举公式，该公式可映射到三类量子排斥统计粒子的集团系数，满足两类g=1费米子数量相等的约束。