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

The family $(Y_x)$ is a Gaussian random field. For distinct $x,x^{\prime }$ its covariance isTwo distinct vertices of ${\mathbb{Z}}^d$ have a common neighbour exactly when their graph distance, equivalently their ${\ell }^1$ distance, is two. Since jointly Gaussian variables are independent exactly when they are uncorrelated,Thus the field has finite-range dependence, even though nearest-neighbour values are independent.

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Write $p(a)=P(Y_0\ge a)$; because $Y_0\sim N(0,1+1/(2d))$, $p(a)\to 0$ as $a\to \infty$. Every path in a graph of length $n$ contains, by a greedy selection, at least $n/M_d$ vertices at mutual graph distance greater than two, where $M_d=|B_2(0)|$. The corresponding field values are jointly independent by part (a). Hence the probability that a fixed path lies in the superlevel set is at mostThere are at most $(2d{)}^n$ length-$n$ paths from the origin. The union bound therefore givesChoose a finite $a$ for which $(2d)p(a{)}^{1/M_d}<1$ and let $n\to \infty$. There is then no unbounded component through the origin, and translation invariance rules out an unbounded component anywhere almost surely. Thus the critical threshold for level-set percolation satisfies $a_c(d)<\infty$.

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The one-arm event $A_R(a)$ depends on the field values in the finite ball $B_R$. Replace $V_x$ by $U_x+a$ for $x\in B_R$. In the new variables the event has threshold zero, while the product normal distribution density is ${\prod }_{x\in B_R}\phi (U_x+a)$. Differentiating this finite-dimensional integral under the integral sign gives the Gaussian shift identityThis is also an instance of Gaussian integration by parts. In particular, the asserted inequality holds, in fact with equality.

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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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