Deterministic Random Walks on the Two-Dimensional Grid

Random walks with different directions: Drunkards beware !

As an extension of Polya's classical result on random walks on the square grids ($\Z^d$), we consider a random walk where the steps, while still have unit length, point to different directions. We show that in dimensions at least 4, the returning probability after $n$ steps is at most $n^{-d/2 - d/(d-2) +o(1)}$, which is sharp. The real surprise is in dimensions 2 and 3. In dimension 2, where the traditional grid walk is recurrent, our upper bound is $n^{-ω(1)}$, which is much worse than higher dimensions. In dimension 3, we prove an upper bound of order $n^{-4 +o(1)}$. We discover a new conje…

Deterministic Random Walks on Regular Trees

Jim Propp's rotor router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order. Cooper and Spencer (Comb. Probab. Comput. (2006)) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid $\Z^d$ and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is in…

通过确定性差异游走实现谱稀疏化

Spectral Sparsification by Deterministic Discrepancy Walk

本文将确定性差异游走框架从向量差异推广到矩阵差异，统一处理多种谱稀疏化问题，并以确定性分析给出更优结果与更简证明。

A rotor configuration with maximum escape rate

Rotor walk is a deterministic analogue of simple random walk. For any given graph, we construct a rotor configuration for which the escape rate of the corresponding rotor walk is equal to the escape rate of simple random walk, and thus answer a question of Florescu, Ganguly, Levine, and Peres (2014).

Simulating a Random Walk with Constant Error

We analyze Jim Propp's P-machine, a simple deterministic process that simulates a random walk on $Z^d$ to within a constant. The proof of the error bound relies on several estimates in the theory of simple random walks and some careful summing. We mention three intriguing conjectures concerning sign-changes and unimodality of functions in the linear span of $\{p(\cdot,x) : x \in Z^d\}$, where $p(n,x)$ is the probability that a walk beginning from the origin arrives at $x$ at time $n$.

瓷砖棋盘上的路径

Walks on tiled boards

本文结合铺砌与网格路径问题，计算了用正方形和多米诺骨牌覆盖的$(1\ imes n)$和$(2\ imes n)$网格上最短路径的数量，并给出了递归及斐波那契数线性组合的表达式。

Spherical Asymptotics for the Rotor-Router Model in Z^d

The rotor-router model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited aggregation. We prove an isoperimetric inequality for the exit time of simple random walk from a finite region in Z^d, and use this to prove that the shape of the rotor-router aggregation model in Z^d, suitably rescaled, converges to a Euclidean ball in R^d.

扩展枚举组合学中的符号方法。I

Transience of walks in the countable complete digraph with complex edge weights

本文通过扩展生成函数中的有限求和方法至无限级数，推广了Polya定理，研究网格图中行走概率的极限问题，并将网格扩展为带权重的可数完全图。

Escape rates for rotor walk in Z^d

Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z^d for the initial configuration of all rotors aligned. If n particles in turn perform rotor walks starting from the origin, we show that the number that escape (i.e., never return to the origin) is of order n in dimensions d>=3, and of order n/log(n) in dimension 2.

The rotor-router group of directed covers of graphs

A rotor-router walk is a deterministic version of a random walk, in which the walker is routed to each of the neighbouring vertices in some fixed cyclic order. We consider here directed covers of graphs (called also periodic trees) and we study several quantities related to rotor-router walks on directed covers. The quantities under consideration are: order of the rotor-router group, order of the root element in the rotor-router group and the connection with random walks.

Directed random walks on polytopes with few facets

Let $P$ be a simple polytope with $n-d = 2$, where $d$ is the dimension and $n$ is the number of facets. The graph of such a polytope is also called a grid. It is known that the directed random walk along the edges of $P$ terminates after $O(\log^2 n)$ steps, if the edges are oriented in a (pseudo-)linear fashion. We prove that the same bound holds for the more general unique sink orientations.