Flux in Tilted Potential Systems: Negative Resistance and Persistence

The paper’s Theorem 2 precisely states the claim under Morse–Smale and uniqueness-of-minimum-spanning-tree assumptions, proving that for α closed and C1-close to −dU the steady-state [α]-flux is positive and −ε ln Fε([α]) → h* , with the needed setup, definitions of heights h(·), and h* given in §4.1 and the positivity baseline in Prop. 2 . The proof hinges on continuity of minimizing trees and the identification Qv=equality with gains g(e) near −∇U (Lemmas 19–20), leading to the reduction in (128)–(131) and, via the cycle-rooted spanning tree variational formula (Theorem 4), to the limit h* in §7.4 . The candidate solution reaches the same conclusions using a metastable reduction to a finite-state Markov chain on Morse sinks, the Markov chain tree theorem for stationary weights, and a fundamental-cycles decomposition of flux. That argument aligns conceptually with the paper’s Markov-chain-based approach in §6 (but on a different graph) and yields the same exponents. The only substantive difference is that the candidate sketches a slightly stronger claim (uniqueness of the leading off-tree edge e*) that the paper does not require; this is an unnecessary strengthening and can be relaxed without affecting the main asymptotics.