[Paper Review] Balls-in-boxes duality for coalescing random walks and coalescing Brownian motions

Steven N. Evans, Xiaowen Zhou|ArXiv.org|Jun 17, 2004

Stochastic processes and statistical mechanicsMathematics27 references20 citations

TL;DR

This paper establishes a novel duality between coalescing random walks and coalescing Brownian motions by relating the distribution of particles in space to the occupancy of spatial intervals, showing that the joint distribution of particle locations in one system matches the interval containment indicators in a dual system. The key contribution is a time-inhomogeneous duality that extends to systems with Poisson immigration, enabling explicit computation of the limiting distribution of particle clusters via Airy functions.

ABSTRACT

We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration.

Motivation & Objective

• To establish a duality between coalescing random walks and coalescing Brownian motions, extending known duality results in stochastic processes.
• To generalize the duality to systems where particles and spatial boxes originate at different times, not just at time zero.
• To apply the duality to analyze a system of coalescing Brownian motions with Poisson immigration, deriving the asymptotic distribution of particle clusters.
• To characterize the limiting distribution of particle locations in the infinite-time regime using random intensity measures and Cox processes.
• To derive explicit expressions for the probability of no particles in an interval and the expected number of particles in a bounded interval, using Airy functions.

Proposed method

• The duality is derived from a combinatorial identity in discrete-time coalescing random walks, which is then lifted to continuous-time coalescing Brownian motions via weak convergence.
• The authors define two systems: one with particles coalescing upon meeting, and another with fixed balls and evolving boxes; their indicator arrays are shown to have identical joint distributions.
• The duality is extended to allow particles and boxes to start at different times by introducing a time-inhomogeneous coupling of the two systems.
• The method uses the avoidance function of the particle system, which is linked to the integral of the distance between coalescing pairs, and shown to match the Laplace functional of a Cox process.
• The limiting distribution of particle clusters is characterized via a random Radon measure $ M_{ au} $, which arises from the time integral of the distance between coalescing pairs.
• Explicit formulas for the probability of no particles in an interval $]a,b]$, and the expected number of particles in such an interval, are derived using the Airy function and its derivative.

Experimental results

Research questions

• RQ1 Can a duality be established between coalescing random walks and coalescing Brownian motions that preserves the joint distribution of particle locations and interval occupancies?
• RQ2 How can the duality be extended to systems where particles and spatial boxes are not synchronized in time?
• RQ3 What is the limiting distribution of particle clusters in a system with coalescing Brownian motions and Poisson immigration as time goes to infinity?
• RQ4 How can the probability of no particles in a given interval be computed explicitly in such a system?
• RQ5 What is the expected number of particles in a bounded interval in the stationary regime of the system?

Key findings

• The joint distribution of particle locations in a system of coalescing Brownian motions is dual to the joint distribution of interval occupancies in a system of coalescing Brownian motions with fixed balls, under a time-inhomogeneous coupling.
• For a system with Poisson immigration at rate $ \lambda $, the limiting distribution of particle clusters is characterized by a Cox process with random intensity measure $ \lambda M_{\infty} $, where $ M_{\infty} $ is the time integral of the distance between coalescing pairs.
• The probability that no particles lie in the interval $]a,b] $ in the infinite-time limit is $ \frac{\mathrm{Ai}(\lambda^{1/3}(b-a))}{\mathrm{Ai}(0)} $, where $ \mathrm{Ai} $ is the Airy function.
• The expected number of particles in the interval $]a,b] $ in the stationary regime is $ (3\lambda)^{1/3} \frac{\Gamma(2/3)}{\Gamma(1/3)} (b-a) $.
• The limiting measure $ M_{\infty} $ is almost surely finite and has atoms, so the resulting Cox process is not a simple point process, meaning particle clusters can have multiplicity greater than one.
• An almost sure construction of the limiting particle configuration $ S_{\infty} $ is given via a Poisson random measure on $ \mathbb{R}_{-} \times \mathbb{R} $, where particles are located at $ \phi(s,0,x) $ for $ s \leq 0 $.