二维和三维简单随机游走的相交指数

中文摘要 AI 辅助

多个独立随机游走彼此回避的概率在二维和三维中随步数 $N\to\infty$ 代数衰减，$P_N\sim N^{-\xi/2}$，其中相交指数 $\xi$ 取决于随机游走的数量。更一般地，可以考虑若干组独立随机游走，组内允许相交，组间禁止相交。我们首先研究二维中的相交指数，其结果与已知的精确公式一致，并检验了数值方法。在三维中，由于没有已知的一般精确表达式，我们确定了一系列情况下的 $\xi$。对于分别包含 $k$ 和 $m$ 个随机游走的两组，我们获得了 $k=1,2$ 和 $m=1,2,\ldots,7$ 的指数 $\xi(k,m)$。我们进一步将 $m$ 扩展为连续参数 $\lambda$，从而获得 $k=1,2,3$ 和 $0.25\leq\lambda\leq3$ 的 $\xi(k,\lambda)$ 数值。在最小矩阶数之外，这些函数随 $\lambda$ 增加而斜率递减，这与布朗相交指数的严格凹性预期一致。我们还研究了若干代表性情况下的三组配置。所得的整数和连续指数为非相交随机路径的解析研究提供了数值。

英文摘要

The probability that several independent random walks avoid one another decays algebraically with the number of steps $N\to\infty$ in two and three dimensions, $P_N\sim N^{-ξ/2}$, where the intersection exponent $ξ$ depends on the number of random walks. More generally, one may consider several groups of independent random walks, with intersections allowed within each group and forbidden between different groups. We first study intersection exponents in two dimensions, where our results agree with the known exact formulas and test the numerical approach. In three dimensions, where no general exact expression is known, we determine $ξ$ for a range of cases. For two groups containing $k$ and $m$ random walks, respectively, we obtain the exponents $ξ(k,m)$ for $k=1,2$ and $m=1,2,\ldots,7$. We further extend $m$ to a continuous parameter $λ$, allowing us to obtain numerical values for $ξ(k,λ)$ for $k=1,2,3$ and $0.25\leqλ\leq3$. Away from the smallest moment orders, these functions increase with $λ$ with decreasing slopes, as expected from the strict concavity of Brownian intersection exponents. We also investigate three-group configurations for several representative cases. The resulting integer and continuous exponents supply numerical values for analytical studies of non-intersecting random paths.

发表机构

• University of Science and Technology of China（中国科学技术大学）
• Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）
• Peking University（北京大学）
• Beijing International Center for Mathematical Research, Peking University（北京国际数学研究中心，北京大学）
• Hefei University of Technology（合肥工业大学）
• Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China（微尺度物质科学国家研究中心（合肥），中国科学技术大学）

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