A random walk in random conductances moves on a lattice where the jump rates between neighboring sites are themselves random — drawn from some distribution and then frozen. The classical question: does the rescaled walk converge to Brownian motion? For many environments, yes — this is the invariance principle, and it holds under progressively weaker conditions on the conductances.
The upgrade here is topological. Classical convergence is in the uniform topology: the rescaled path approaches Brownian motion as a continuous function. Rough path convergence is stronger — the rescaled path, together with its iterated integrals (the “area process”), converges to enhanced Brownian motion in the p-variation rough path topology. This stronger convergence preserves nonlinear information that uniform convergence discards.
Why this matters: many quantities of interest — solutions to stochastic differential equations driven by the random walk, for instance — depend continuously on the driving signal only in the rough path topology, not in the uniform one. Without rough path convergence, you can't pass to the limit in these equations. With it, the entire machinery of rough differential equations applies.
The result holds in both annealed (averaged over environments) and quenched (for almost every environment) settings, with degenerate conductances and long-range jumps allowed. The quenched case replaces the usual reversibility assumption with a softer condition: existence of a stationary potential for the corrector.
The structural point: the invariance principle is not one theorem but a family of theorems at different levels of regularity. Each level unlocks different applications. Rough path convergence is the version that makes stochastic analysis work.
(arXiv:2603.18748)