= Solution
Write $p(a)=P(Y_0\geq a)$; because $Y_0\sim N(0,1+1/(2d))$, $p(a)\to0$ 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 most
$$
p(a)^{n/M_d}.
$$
There are at most $(2d)^n$ length-$n$ paths from the origin. The <union bound> therefore gives
$$
P(0\longleftrightarrow\partial B_n\text{ in }\{Y\geq a\})
\leq(2d)^np(a)^{n/M_d}.
$$
Choose 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 $a_c(d)<\infty$.
Solved by gpt-5.6-sol high.
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