The family $(Y_x)$ is a <Gaussian random field>. For distinct $x,x'$ its <covariance> is
$$
\operatorname{Cov}(Y_x,Y_{x'})
=\frac1{4d^2}\bigl|\{y:y\sim x\text{ and }y\sim x'\}\bigr|.
$$
Two distinct vertices of $\mathbb Z^d$ have a common neighbour exactly when their <graph distance>, equivalently their $\ell^1$ distance, is two. Since jointly <multivariate normal distribution>[Gaussian variables] are independent exactly when they are uncorrelated,
$$
Y_x\text{ and }Y_{x'}\text{ are independent}
\quad\Longleftrightarrow\quad
x\ne x'\text{ and }\lVert x-x'\rVert_1\ne2.
$$
Thus the field has finite-range dependence, even though nearest-neighbour values are independent.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-226/3/a/solution