[Paper Review] Pemantle's min-plus binary tree

Antonio Auffinger, Dylan Cable|arXiv (Cornell University)|Sep 22, 2017

Stochastic processes and statistical mechanics8 references3 citations

TL;DR

This paper studies a stochastic min-plus binary tree process where particles at leaves merge via addition or annihilate via minimum selection with equal probability. It establishes that in the critical case $ p = 1/2 $, the logarithm of the final mass $ X_N $, scaled by $ rac{1}{ ilde{c}N} $ with $ c = rac{ ilde{ au}^2}{3} $, converges in distribution to a non-degenerate limit, revealing a $ ilde{c}N $-scaling law for the typical size of $ X_N $ in the large $ N $ limit.

ABSTRACT

We consider a stochastic process that describes several particles interacting by either merging or annihilation. When two particles merge, they combine their masses; when annihilation occurs, only the particle of smallest mass survives. Particles start at the bottom of a binary tree of depth N and move towards the root. Assuming that merging or annihilation happens independently at random, we determine the limit law of the final mass of the system in the large N limit.

Motivation & Objective

• To resolve Robin Pemantle’s open problem on the asymptotic behavior of a stochastic min-plus binary tree process at the critical threshold $ p = 1/2 $.
• To characterize the limit law of the final mass $ X_N $ in the large $ N $ limit when operations (addition or minimum) are chosen independently at random with equal probability.
• To establish the precise scaling of $ X_N $, showing it grows as $ \exp(\sqrt{cN}) $ with $ c = \pi^2/3 $, and to derive the limiting distribution of $ \frac{1}{\sqrt{cN}} \log X_N $.
• To extend the analysis to subcritical ($ p < 1/2 $) and supercritical ($ p > 1/2 $) regimes, proving tightness and exponential growth, respectively.
• To provide a rigorous framework for modeling particle systems or network flows with merging and selection operations via recursive random processes on trees.

Proposed method

• Formalizes the process as a recursive random function on a binary tree of depth $ N $, where each internal node applies either addition or minimum operation based on i.i.d. Bernoulli variables with $ p = 1/2 $.
• Defines $ X_N $ as the value at the root, computed recursively from leaf values $ X_v = 1 $, using $ X_w = X_{v_1} + X_{v_2} $ if $ \eta_w = 1 $, or $ \min(X_{v_1}, X_{v_2}) $ if $ \eta_w = -1 $.
• Derives a recurrence for the tail probability $ p_{N,k} = \mathbb{P}(X_N \geq k) $, given by $ p_{N+1,k} - p_{N,k} = \frac{1}{2} \sum_{\ell=1}^{k-1} (p_{N,\ell} - p_{N,\ell+1})(p_{N,k-\ell} - p_{N,k}) $, which captures the stochastic evolution.
• Uses the non-negativity of partial derivatives in the recurrence to derive upper and lower bounds on $ p_{N,k} $, enabling asymptotic analysis.
• Applies a continuous approximation to the recurrence, leading to a differential equation-like analysis that supports the conjectured scaling $ \mathbb{P}(X_N \geq k) \sim 1 - \frac{\log^2 k}{cN} $.
• Employs a limiting argument via fixed-point analysis of the recurrence in the limit $ N \to \infty $, showing convergence to a non-trivial distribution with explicit cumulative form.

Experimental results

Research questions

• RQ1 What is the asymptotic distribution of the final mass $ X_N $ in the critical case $ p = 1/2 $ as $ N \to \infty $?
• RQ2 How does the typical size of $ X_N $ scale with $ N $, and what is the precise constant in the exponent of the growth rate?
• RQ3 What happens to the distribution of $ X_N $ when $ p < 1/2 $ or $ p > 1/2 $, and how does the system behavior differ across these regimes?
• RQ4 Can the recurrence for $ p_{N,k} = \mathbb{P}(X_N \geq k) $ be used to derive tight bounds and establish convergence to a limiting law?
• RQ5 What is the behavior of $ \mathbb{E}[\log X_N] $ in the critical case, and how does it scale with $ N $?

Key findings

• For $ p = 1/2 $, the limit law of $ \frac{1}{\sqrt{cN}} \log X_N $ with $ c = \pi^2/3 $ converges to a non-degenerate distribution with cumulative distribution function $ F(t) = 0 $ for $ t \leq 0 $, $ t^2 $ for $ 0 \leq t \leq 1 $, and $ 1 $ for $ t \geq 1 $.
• The final mass $ X_N $ grows as $ \exp(\sqrt{cN}) $ with $ c = \pi^2/3 $, confirming Pemantle’s conjecture on the critical scaling.
• The expectation of $ \log X_N $ satisfies $ \lim_{N \to \infty} \frac{1}{\sqrt{N}} \mathbb{E}[\log X_N] = \frac{2\pi}{3\sqrt{3}} $, providing a precise asymptotic for the logarithmic moment.
• For $ p < 1/2 $, the sequence $ X_N $ is tight, meaning $ \mathbb{P}(X_N \geq k) \to 0 $ as $ k \to \infty $ uniformly in $ N $, implying convergence in distribution to a non-degenerate limit.
• For $ p > 1/2 $, the expectation of $ X_N $ grows exponentially as $ \mathbb{E}[X_N] \geq (2p)^N $, indicating explosive growth due to dominance of merging operations.
• The recurrence for $ p_{N,k} $ has non-negative partial derivatives, enabling the use of monotonicity and comparison arguments to derive upper and lower bounds for $ p_{N,k} $, which are key to the asymptotic analysis.