Mixing times for coagulation-fragmentation processes

Definitions and mathematical statement

A standard finite coagulation--fragmentation model has states describing partitions of a fixed finite mass or set, with transitions that merge blocks or split a block. The rate kernel and splitting rule are model parameters. The source does not specify labeled versus size partitions or a particular family of rates.

For an explicit continuous-time formulation, let $Q=(q(x,y))$ be the generator of an irreducible chain on a finite state space $\Omega$, with stationary probability $\pi$, and assume reversibility, \[ \pi(x)q(x,y)=\pi(y)q(y,x). \] Let $P_t=e^{tQ}$, let $\lambda_1>0$ be the smallest nonzero eigenvalue of $-Q$, and define \[ t_{\mathrm{rel}}=\lambda_1^{-1},\qquad t_{\mathrm{mix}}(\varepsilon)=\inf\left\{t\ge0: \max_x\frac12\sum_{y\in\Omega}|P_t(x,y)-\pi(y)|\le\varepsilon\right\}, \quad0<\varepsilon<1. \] Obtain useful bounds on these quantities in terms of the intended coagulation--fragmentation parameters. These definitions are a conventional candidate; the source does not choose continuous versus discrete time, irreducibility/aperiodicity conventions, or a mixing threshold.

The actual rate family, reversible measure and parameters are missing. A rate normalization is essential because rescaling all rates rescales the time unit. A meaningful bound must also specify its permitted dependence on mass/particle count, rate variation and splitting structure. No universal polynomial-time law is asserted for arbitrary reversible chains, and generic finite-chain bounds are not confused with a model-specific sharp analysis.

The supplied summary reports special-case literature but no systematic general parameter-dependent bounds verified. Its cited coalescence review and double-coset work are expressly related background rather than solutions. The source's open presentation and later search assessment remain attributions pending primary-literature review.

Short English statement

Bound spectral relaxation and total-variation mixing for reversible finite coagulation–fragmentation chains using their actual rate and size parameters. The chain family, time normalization and target bound are not specified in the excerpt.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $9700004$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• David Aldous, Open problems, item coag_frag.html, maintained index: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html. Supplied HTML text with linked-primary-PDF text accessed 2026-07-29; the index is reported to have 23 active problem pages.
• Aldous, "Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists", Bernoulli (1999), supplied as background rather than a mixing-time result.
• "Double Coset Markov Chains", arXiv (2021), supplied as related work without authors, identifier or a resolution of this question.