AI 中文总结
针对二叉树上的min-plus过程，通过蒙特卡洛模拟验证其极限与闭式解，确定伯努利初值变体的恒等式，提出的分层递归与FFT叶表可优化大型树相关递归方程的模拟。
AI 中文摘要
min-plus过程是二叉树上的一种随机凝聚-湮灭型过程，作为max型递归分布方程的可处理实例，受到数学、物理和计算机科学领域的关注。我们开展了有效树深度达N=60的大型蒙特卡洛模拟，为Chen、Duquesne、Shi及Morfe近期提出的随机齐次系统分类中，p=1/2时根节点值X_N的Beta(2,1) stretched-exponential极限提供了有限深度的佐证，该极限位于非对称√N侧。在非临界状态下，我们的模拟在蒙特卡洛误差范围内验证了亚临界闭式解P(X_∞=1)=(1-2p)/(1-p)，并记录了在我们达到的深度上，超临界平均增长超出了基本下界(2p)^N。对于伯努利(q)初值变体，我们在p=1/2时确定了一个基本闭式恒等式，该恒等式精确确定了序参量P(X_N=0)=q，在算子混合概率而非初始零密度中定位了吸收态相变点p_c=1/2，并表明正节点上的条件分布随q发生显著变形。我们的模拟采用分层递归和基于FFT的预计算叶表，在保留递归树律的同时降低了有效模拟深度，可能对大型树上相关递归方程的模拟有用。
英文摘要
The min-plus process is a stochastic coagulation-annihilation-type process on the binary tree, of interest in mathematics, physics, and computer science as a tractable instance of max-type recursive distributional equations. We carry out large Monte Carlo simulations at effective tree depths up to $N=60$ that provide finite-depth corroboration of the Beta(2,1) stretched-exponential limit for its root value $X_{N}$ at $p=1/2$, on the asymmetric $\sqrt{N}$ side of the random-homogeneous-systems classification recently introduced by Chen, Duquesne, and Shi and by Morfe. Off criticality, our simulations confirm the sub-critical closed form $\mathbb{P}(X_{\infty}=1)=(1-2p)/(1-p)$ within Monte Carlo error and document a super-critical mean growth exceeding the elementary $(2p)^{N}$ lower bound at the depths we reach. For a Bernoulli($q$)-initial-condition variant, we identify an elementary closed-form identity at $p=1/2$ that pins down the order parameter $\mathbb{P}(X_{N}=0)=q$ exactly, locates the absorbing-state phase transition at $p_{c}=1/2$ in the operator-mixing probability rather than in the initial-zero density, and shows that the conditional law on positives deforms substantially with $q$. Our simulations use a level-wise recursion and an FFT-based precomputed leaf table which reduce the effective simulation depth while preserving the recursive tree law and may be useful for the simulation of related recursive equations on large trees.
CommentsAMSart style, 20 pages, 8 figures, 26 refs
Journal refJ. Phys. A: Math. Theor. 59 (32), 325202 (2026)
