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I'm trying to get my head around the "replica trick" and it's mathematically rigorous formulations (due to Talagrand, Parchenko, etc.). I was wondering to myself that a solution or insight ...

Let $n$ and $d$ be large positive integers such that $d/n = a \in (0,1)$, fixed. Let $x_1,\ldots,x_n$ be iid random vectors from $N(0,I_d)$. Fix $b \in (0,1]$ and a unit-vector $v \in \mathbb R^d$, ...