[Paper Review] Self-organized criticality in a discrete model for Smoluchowski's equation

Mathieu Merle, Raoul Normand|arXiv (Cornell University)|Oct 30, 2014

Stochastic processes and statistical mechanics24 references7 citations

TL;DR

This paper studies a discrete coagulation model with a dynamic threshold α(N) for cluster size, showing that under the condition N²/³ log⁴(N) ≤ α(N), the system exhibits self-organized criticality: the distribution of typical finite clusters converges to a critical Galton-Watson tree. The model's small clusters form a subcritical Erdős-Rényi random graph before time 1 and remain critical after, verifying Aldous' conjecture on local weak convergence of cluster components.

ABSTRACT

We study a discrete model of coagulation, involving a large number $N$ of particles. Pairs of particles are given i.i.d exponential clocks with parameter $1/N$. When a clock rings, a link between the corresponding pair of particles is created only if its two ends belong to small clusters, i.e. of size less than $α(N)$, with $1 \ll α(N) \ll N$. The concentrations of clusters of size $m$ in this model are known to converge as $N o \infty$ to the solution to Smoluchowski's equation with a multiplicative kernel. Under the additional assumption $N^{2/3} \log^γ(N) \le α(N)$, for some $γ> 1/3$, we study finer asymptotic properties of this model, namely the combinatorial structure of the graph consisting of small clusters. We prove that this graph is essentially an Erdos-Renyi random graph, which is subcritical before time 1, and remains critical after time 1. In particular, we show that our model exhibits self-organized criticality at a microscopic level: the limiting distribution of a typical finite cluster is that of a critical Galton-Watson tree. Our approach allows in particular to verify, under our additional assumption, a conjecture of Aldous.

Motivation & Objective

• To investigate the emergence of self-organized criticality in a discrete coagulation model with a finite-size threshold α(N).
• To analyze the combinatorial structure of small clusters as a random graph and determine its criticality regime.
• To verify Aldous' conjecture on the local weak convergence of the component containing a typical particle to a critical Galton-Watson tree.
• To examine the dependence between the gelation time Z and the component size C(Z) in the limit.

Proposed method

• Model particles with i.i.d. exponential clocks of rate 1/N, allowing coagulation only between clusters of size < α(N).
• Use a hydrodynamic limit to show convergence of cluster concentrations to solutions of Smoluchowski’s equation with a multiplicative kernel.
• Apply random graph techniques to analyze the small-cluster graph as an Erdős-Rényi model, showing subcriticality before time 1 and criticality after.
• Use the exploration process of ER graphs to compute the expected number of surplus edges in the component containing a typical particle.
• Condition on the absence of large excursions in a Brownian motion approximation to model the threshold constraint.
• Leverage results from random graph theory and branching process convergence to establish local weak convergence of cluster components.

Experimental results

Research questions

• RQ1 Does the discrete coagulation model with a dynamic threshold α(N) exhibit self-organized criticality at the microscopic level?
• RQ2 How does the structure of the small-cluster graph evolve over time, and is it subcritical or critical?
• RQ3 Does the component containing a typical particle converge locally weakly to a critical Galton-Watson tree, as conjectured by Aldous?
• RQ4 Are the gelation time Z and the component size C(Z) asymptotically independent in the limit?
• RQ5 What happens to the model when α(N) is smaller than N²/³, particularly in the critical window?

Key findings

• Under the assumption N²/³ log⁴(N) ≤ α(N), the graph of small clusters is asymptotically an Erdős-Rényi random graph that is subcritical before time 1 and critical after.
• The limiting distribution of a typical finite cluster is that of a critical Galton-Watson tree, confirming self-organized criticality at the microscopic scale.
• The component containing a typical particle, C(Z), converges locally weakly to a critical Galton-Watson tree, verifying Aldous' conjecture.
• The expected number of surplus edges in C(Z) converges to Z²/12 as N → ∞, implying that Z and C(Z) are not asymptotically independent.
• The model remains robust to threshold reduction down to α(N) ≍ N²/³, though conditioning becomes non-trivial in the critical window.