Electronic Communications in Probability

Noninvadability implies noncoexistence for a class of cancellative systems

Jan Swart

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There exist a number of results proving that for certain classes of interacting particle systems in population genetics, mutual invadability of types implies coexistence. In this paper we prove a sort of converse statement for a class of one-dimensional cancellative systems that are used to model balancing selection. We say that a model exhibits strong interface tightness if started from a configuration where to the left of the origin all sites are of one type and to the right of the origin all sites are of the other type, the configuration as seen from the interface has an invariant law in which the number of sites where both types meet has finite expectation. We prove that this implies noncoexistence, i.e., all invariant laws of the process are concentrated on the constant configurations. The proof is based on special relations between dual and interface models that hold for a large class of one-dimensional cancellative systems and that are proved here for the first time.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 38, 12 pp.

Accepted: 23 May 2013
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C22: Interacting particle systems [See also 60K35]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92D25: Population dynamics (general) 82C24: Interface problems; diffusion-limited aggregation

Cancellative system interface tightness duality coexistence Neuhauser-Pacala model affine voter model rebellious voter model balancing selection branching annihilation parity preservation

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Swart, Jan. Noninvadability implies noncoexistence for a class of cancellative systems. Electron. Commun. Probab. 18 (2013), paper no. 38, 12 pp. doi:10.1214/ECP.v18-2471. https://projecteuclid.org/euclid.ecp/1465315577

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