推演由 AI Bot
Conditional on an exact stopped prefix K , the field has the ordinary strand-conditioned law: whether the rule stopped at this prefix is already decided by that prefix. The independent walk adds no field conditioning. Put U=D(K) . For small fixed δ>0 , except on an event tending uniformly to zero with δ , the following four conditions hold: , d_U(X_T,opposite-sign side)≥2δd,; , diam S[0,τ_T]<δ^ -1 d,; , diam S[t_*,τ_T]<δd/4, t_*=t:d(S_t,∂U)≤δ^2d,; , d(v,K)≥δ^ 1/2 d. The middle two statements are the planar walk hitting estimate uniform in U . The last statement follows from Lemma 52 ([Clearance from the observer]) and K⊂γ . For the observer-separated stopping times needed in the driving argument, the last condition can instead be imposed deterministically. For the first condition delete all crosscuts of diameter at most 4δd joining the two sign sides. The new boundary of the component containing v lies in two balls of radius 8δd . Indeed each deleted crosscut separates v from one of the two sign-change points; any two crosscuts for the same point which touch the retained component must intersect. All such crosscuts are consequently within twice their diameter of a fixed one. Apply the walk hitting estimate to those two balls, using the last condition. This also explains why arbitrary narrowing of U does not invalidate (G10). Fix records satisfying (G10), and suppose the landing is positive. The deterministic clearance construction used in Lemma 50 ([No additional sampled excursions]) gives a compatible crosscut Υ_0 in B(X_T,δd) B(X_T,δd/2) , joining the positive boundary pieces on either side of X_T and separating it from the negative side. If no hexagon centered at a vertex of the stopped walk meets this crosscut, avoidance of this crosscut protects the entire walk. Otherwise add the boundaries of the walk hexagons on its unprotected side and take the boundary facing the negative side. Its diameter is at most 2δ^ -1 d ; its new part is at distance at least δ^2d/2 from ∂U , by the terminal-walk condition. Indeed a contacting walk hexagon occurs before t_* , since the terminal walk piece lies in B(X_T,δd/4) , whereas the crosscut lies outside B(X_T,δd/2) . That contact is at distance at least δ^2d-O(1) from ∂U ; therefore the boundary-attached original crosscut has diameter at least δ^2d-O(1) as well. The old endpoints retain their pure pockets. It is therefore a barrier with parameter cδ^3 and scale 2δ^ -1 d . Proposition 45 ([Uniform barriers]) gives probability at least p_δ>0 that the future interface avoids the resulting barrier. On this event it is adjacent to no vertex of the stopped walk, so Y=X_T . In the first branch Υ_0 can have bounded lattice diameter; this is covered by the all-scale part of that proposition. Reflect signs for a negative landing. Choose δ to make the failure probability of (G10) smaller than , and then take d sufficiently large.
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