Finite-depth scaling and an exact Bernoulli-leaf identity for the min-plus process on the binary tree

Abstract

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.

AMSart style, 20 pages, 8 figures, 26 refs

References in corpus (1)

• A numerical investigation into the scaling behavior of the longest increasing subsequences of the symmetric ultra-fat tailed random walk