Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 226 / 3 / d / Solution

Apply the OSSS inequality to the independent coordinates $(V_x,W_x)$ and to the indicator of the one-arm event $A_R(a)$. Use the randomized OSSS exploration of a one-arm event: choose $k$ uniformly from $\{1,\dots ,R\}$ and reveal the variables needed to explore the superlevel cluster meeting $\partial B_k$. A coordinate can be revealed only if a nearby vertex has an open connection over the relevant distance. Translation invariance, finite-range dependence and the union bound therefore give the revealment estimatefor both kinds of coordinates, after enlarging the explored neighbourhood by a distance depending only on $d$.

For an increasing Gaussian threshold event, the resampling influence of $V_x$ is bounded by a universal constant times $\mathbb{E}[V_x{1}_{A_R(a)}]$. The influence of $W_x$ is bounded by the sum of the corresponding $V$ influences at the $2d$ neighbours of $x$: indeed Gaussian integration by parts givesfirst for smooth increasing approximations and then by a limit. Consequently the OSSS bound becomesPart (c) identifies the final sum with $-{\theta }_R^{\prime }(a)$. Dividing and putting $c=C_d^{-1}>0$ proves

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