返回题库

三维随机游走回原点的概率

Three-Dimensional Random Walk

专题
Algorithmic Programming / 算法编程
难度
L4

题目详情

As in the two- dimensional walk, a particle starts at an origin OO in threespace. Think of the origin as centered in a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight corners of the cube. Thus, at every move the particle has a 50- 50 chance of moving one unit up or down, one unit east or west, and one unit north or south. If the walk continues forever, find the fraction of particles that return to the origin.

解析

题目中的三维游走每步以等概率走向立方体 8 个角点之一（每个坐标同时以 ±1\pm 1 变化）。该随机游走是暂态（transient），回到原点的概率小于 1。

计算“在第 2n2n 步回到原点”的概率约为

[(2nn)2−2n]3≈1(πn)3/2,\left[\binom{2n}{n}2^{-2n}\right]^3\approx \frac{1}{(\pi n)^{3/2}},

其和收敛，从而总回归次数期望有限。文中给出的数值近似：平均回归次数约 0.3150.315，于是

P≈0.3151+0.315≈0.239.P\approx \frac{0.315}{1+0.315}\approx 0.239.

所以

P(返回原点)≈0.239.\boxed{\mathbb{P}(\text{返回原点})\approx 0.239}.