Unique zero of Iann Sample Clauses

Unique zero of Iann when M < ϵ In [3] we showed that if M < ϵ and α − β < (ϵ − M )/2, then a proof of the LLN can be given that is based on a perturbation argument for the generator of the environment process ζ = (ζt)t≥0, ζt = τXtξt, (4.53) i. e., the random environment as seen relative to the random walk. In particular, it is shown that ζ is uniquely ergodic with equilibrium µe. This leads to a series expansion for v in powers of α − β, with coefficients that are functions of Pµ and α + β and that are computable via a recursive scheme. The speed in the LLN is given by ˜ v = (2ρ − 1)(α − β) (4.54) ˜spends on occupied sites). with ρ = ⟨η(0)⟩µe , where ⟨·⟩µe denotes expectation over µe (ρ˜ is the fraction of time X Proposition 4.9. Let ξ be an attractive spin-flip system with M < ϵ. If α − β < (ϵ − M )/2, then the rate function Xxxx in (4.51) has a unique zero at v.