 Solution

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 $a_c(d)<\infty$.

Solved by gpt-5.6-sol high.