Convergence criterion for longer-range interaction drift matrices

Prove or disprove that, when the sampling and reinforcement graphs coincide and have a common underlying graph of period d, the stochastic process with drift matrix W^{m_1}Q\widetilde A^{m_2}-I converges to a deterministic limit whenever \gcd(m_1+m_2,d)=1.

Background

The paper studies interacting reinforced stochastic systems in which sampling and reinforcement are represented by network operators W and \widetilde A, respectively. Its main convergence criteria focus primarily on nearest-neighbour interactions and, under common-support assumptions, relate invertibility of the drift matrix WQ\widetilde A-I to graph structure, parity, sampling type, and reinforcement strength.

The authors propose extending the framework to longer-range neighbourhoods, represented by powers W{m_1} and \widetilde A{m_2}. The conjecture predicts that the arithmetic relation between the interaction range m_1+m_2 and the period d of the common graph determines whether the resulting process has a deterministic limit. Establishing this would generalize the paper's graph-theoretic convergence results beyond nearest-neighbour dynamics.

References

Extending the framework to allow sampling and reinforcement mechanisms that depend on longer-range neighbourhoods would provide a natural direction for further investigation. In particular, we have the following conjecture. Suppose the sampling and reinforcement graphs coincide, with common underlying graph G of period d. If the drift matrix is of the form W{m_1}Q\widetilde A{m_2}-I, the process converges to a deterministic limit whenever \gcd(m_1+m_2,d)=1.

— Interacting Stochastic Systems with Agent-Based Sampling and Connection-Based Reinforcement  (2610.13884 - Dahiya et al., 9 Oct 2026) in Section 6, Discussion and Future Work, Conjecture

In the random-limit regime, the distributional properties of the limiting random vector are largely unknown; establishing whether the limit is non-degenerate is an important open problem.

— Interacting Stochastic Systems with Agent-Based Sampling and Connection-Based Reinforcement  (2610.13884 - Dahiya et al., 9 Oct 2026) in Section 6, Discussion and Future Work