Alon–Puder two-particle spectral-gap conjecture
For every admissible finite weighted graph or hypergraph update structure and every system size , the spectral gap of the corresponding conservative Kipnis–Marchioro–Presutti process satisfies . Equivalently, the first nonconstant spectral mode is represented by an observable involving at most two particles.
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.
One- or two-particle spectral-mode formulation
The dominant nonconstant eigenfunction of the KMP dynamics is represented, under the particle-system intertwining, by either a one-particle observable or a two-particle observable; equivalently, no mode involving more than two particles determines the spectral gap.
source: Aldous' spectral gap phenomena in stochastic exchange models
References
Primary source
Additional references
Progress summary
An unrefereed September claim says the conjecture is solved in a broad setting, but the available preprint record still describes the central case as open.
The conjecture asserts that the first nontrivial spectral mode is always one- or two-particle, equivalently that the spectral gap equals that of the two-particle process .
Known results
A 2026 preprint proves the assertion for several nontrivial hypergraph classes, but states that the equality remains unknown even for graphs.
September 9, 2026 settlement claim
A daily-news item described an unrefereed preprint as resolving the conjecture for broad exchange models, with sharp one-versus-two-particle criteria. The available arXiv analysis, dated September 10, says this claim is unsupported and records the central conjecture as open.
Current status (as of September 2026): The conjecture remains open in the available mathematical record, while a broad-resolution claim is circulating but unverified.
Solutions 0
No solutions have been posted yet.