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 $2^{293}$ , although the result may still hold for sufficiently basic IPS.

Equivalent formulations 1Other wordings

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).

References

Primary source

Maciej Gluchowski and Georg Menz, “Ergodicity Criterion for One-Sided, One-Dimensional IPS with a Long-Lived State”, arXiv:2508.08459 (2025).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The full conjecture is claimed false because of a large-alphabet counterexample, while the simplest two-state cases remain unresolved.

The conjecture asserts that every noisy, homogeneous one-dimensional system with bounded-range interactions eventually forgets its starting state. Gács constructed a claimed counterexample with alphabet size $2^{293}$ , so the unrestricted conjecture is false if that construction is accepted.

Known results

Gray proved ergodicity for periodic, weakly monotone, nearest-neighbor systems with positive rates and alphabet size two.
Two-state one-sided nearest-neighbor systems are exponentially ergodic when $p_{1\mid 11}\le p_{1\mid 10}<1$ and $0<p_{1\mid 01}\le p_{1\mid 00}$ .

August 2025 ergodicity criterion

An August 2025 preprint reports ergodicity for one-sided nearest-neighbor systems whenever $\delta<\sqrt{2}\,\beta$ , covering much of the previously unresolved parameter region. It identifies noisy versions of the one-dimensional East model as the remaining unresolved cases, rather than proving the full conjecture.

Current status (as of September 2026): the general conjecture is claimed refuted by Gács’s counterexample, while the two-state one-sided nearest-neighbor case and noisy East models remain open.

Sources

No solutions have been posted yet.

The positive rates conjecture for one-dimensional interacting particle systems

Primary source

Progress summary