Uniform flat-norm bound for the rescaled cycle-valued random walk (Conjecture)

Establish that for every fixed time horizon T > 0, the rescaled cycle-valued random walk X^{(n)} on the triangulation of the flat torus satisfies a uniform flat-norm bound: lim_{n→∞} sup_{t≤T} ||X^{(n)}_t||_F < ∞. This bound would control the size of cycles under diffusive scaling and underpin tightness and generator convergence arguments.

Background

In analyzing scaling limits on the torus, cycles may develop loops that make their geometric length large, complicating tightness and generator convergence. The authors propose controlling the process using the flat norm (Federer), which measures proximity to boundaries of 2-chains plus area.

They formulate a conjectural uniform bound on the flat norm of the rescaled process over finite horizons. This assumption enables non-exploding bounds in stronger dual norms, facilitates Aldous–Rebolledo tightness criteria, and supports identification of the limit martingale problem. They discuss that certain modifications (e.g., puncturing the torus) would make the conjecture easy to verify, but they prefer the natural complete torus setup.