Dual families of interacting particle systems on graphs
type: Item
instance of: scholarly article
title: Dual families of interacting particle systems on graphs
zbMATH Open document ID: 0968.60096
DOI: 10.1023/A:1007806427774
author: Aidan Sudbury
published in: Journal of Theoretical Probability
publication date: 2001-05-06T00:00:00Z
review text: The author studies the general theory of duality for interacting particle systems (IPS) including many known special cases such as contact process, voter model, annihilating random walk (ARW), branching ARW, annihilating branching process (ABP), and biased ABP, etc. The first rule is about a simple condition for IPS with nearest neighbor interactions to be self-dual. It says: ``Every process with the contact transition, birth rate \(b > 0\) and no spontaneous birth is self-dual with self-duality parameter \(x =\) \(( d - a - c)/b\).'' Here \(a\) (resp. \(c\) or \(d\)) means annihilating (coalescence or death) rate, respectively. The next theorem says: ``There exist families of IPS \(\{ \alpha_i \}\), such that each \(\alpha_i\) is self-dual and dual to every other member of the family.'' Many of the most interesting processes have \(b > 0\). These processes either die out or the number of occupied sites tends to infinity, depending on some critical value \(b_c\) of birth rate \(b\). While, \(s_A\) denotes the survival probability that the set of 1's survives for all time when the initial set \(A\) is finite. When \(b < b_c\), \(s_A =0\), or when \(b > b_c\), \(0 < s_A \leq 1\), it is proved that all processes with \(b > 0\) possess a stationary measure \(\mu_{\infty}\) expressible as a function of \(s_A\), and the self-duality parameter \(x\). If extinction is certain for all finite initial sets \(A\), then \(\mu_{\infty}\) turns out to be \(\delta_{\varphi}\), which is always an equilibrium measure for any IPS without spontaneous birth. The theorem asserts: ``If an IPS on \({\mathbb Z}^k\) has self-duality parameter \(x \geq -1\), then it converges to \(\mu_{\infty}\) from any essentially infinite initial measure.'' The leading ideas and basic concepts such as duality and thinning for proofs of theorems in this paper are greatly due to the author and \textit{P. Lloyd} [Ann. Probab. 23, No. 4, 1816-1830 (1995; Zbl 0853.60079) and ibid 25, No. 1, 96-114 (1997; Zbl 08…
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 1552706
zbMATH Keywords: interacting particle systems
zbMATH Keywords: duality
MaRDI profile type: Publication
full work available at URL: a:1007806427774
OpenAlex ID: W159028949
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description: scientific article; zbMATH DE number 1552706