Dimension-uniform control of noisy finite-dimensional interactions

Establish dimension-uniform control of the finite-dimensional interaction in discrete-time mean-field coupled dynamical systems with noise, extending beyond the uniformly expanding setting treated by Tanzi (2022).

Background

The paper contrasts the technical setting it resolves—conditioned dynamics for arbitrary products of independent copies of a one-dimensional mixing Markov chain—with the more difficult setting of discrete-time mean-field coupled systems. In the latter, one must simultaneously control particle interactions and the conditioning induced by a hole. The authors state that, outside the uniformly expanding case, dimension-uniform control of the finite-dimensional interaction is not yet available, particularly for systems with discrete-time noise. Their product-system analysis is presented as an intermediate step that isolates the conditioning difficulties while retaining a connection to the motivating mean-field problem.

References

Beyond the uniformly expanding setting treated in , however, dimension-uniform control of the finite-dimensional interaction remains largely open, especially for discrete-time systems with noise.

— Stability of quasi-stationary measures in high-dimensional products of mixing Markov chains  (2609.10178 - Tanzi et al., 9 Sep 2026) in Introduction, paragraph beginning “Establishing the correspondence in...”