Multidimensional-velocity scaling limit on the constrained hypercube

Prove a scaling-limit theorem for the discrete Hamiltonian dynamics with multidimensional velocity on the constrained hypercube target defined on \(\{0,1\}^{3m}\), showing that accelerating the process by an additional factor of \(O(\sqrt{m})\) yields a limit analogous to the one-dimensional-velocity scaling limit in Theorem \(\ref{theo:scaling_limits_constrained_hypercube}\).

Background

The paper proves a scaling limit for discrete Hamiltonian dynamics on the constrained $3m$-dimensional hypercube when the momentum is one-dimensional and the direction records whether a bit is added or removed. Under time acceleration by a factor of $m$, the normalized number of unconstrained active bits converges to a randomized Hamiltonian process on $[0,1]\times\mathbb{R}$.

For the multidimensional-velocity version, the momentum lies in $\mathbb{R}^{3m}$ and the transition directions are the coordinatewise vectors $\sigma_{x,y}=y-x$. Numerical experiments suggest that averaging among the velocity components produces ballistic behavior with an additional $O(\sqrt{m})$ slowdown, corresponding to an overall mixing-time scale of $O(m^{3/2})$. The authors explicitly conjecture that an analogous scaling limit holds in this regime but state that they could not prove it; they suggest stochastic homogenization as a possible method for analyzing the velocity averaging.