MathDBPositive Rates Conjecture for one-dimensional interacting particle systems
Positive Rates Conjecture for one-dimensional interacting particle systems
Let an IPS be a stochastic interacting particle system on a one-dimensional lattice, with homogeneous interactions of bounded range. Its transition rates are positive when the transition matrix differs from the identity on every entry; an IPS is ergodic when it has an attractive distribution.
Positive Rates Conjecture. Every IPS on a one-dimensional lattice with homogeneous interactions of bounded range and positive rates is ergodic.
The conjecture asserts that noise prevents a one-dimensional homogeneous IPS from remembering its initial configuration. It is refuted in general by a counterexample of Gács with an alphabet of size , although the result may still hold for sufficiently basic IPS.
Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
The positive rates conjecture for one-dimensional interacting particle systems
Let an interacting particle system (IPS) evolve on a one-dimensional lattice with homogeneous, finite-range interaction and positive rates. Positive rates conjecture. Every such IPS is ergodic. This conjecture was popularized by Liggett. Gács published a counterexample for sufficiently large neighborhoods, so the claim in this full generality is refuted; the paper studies affirmative special cases and related open regions.
source: Maciej Głuchowski and Georg Menz, “Time-Scaling, Ergodicity and Covariance Decay of Interacting Particle Systems”, arXiv:2312.06935 (2024).
Sources & referencesView supporting material
Primary source
Maciej Gluchowski and Georg Menz, “Ergodicity Criterion for One-Sided, One-Dimensional IPS with a Long-Lived State”, arXiv:2508.08459 (2025).
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