  This article investigates the behavior of the continuous-time simple random
walk on $\mathbb{Z}^d$, $d \geq 3$. We derive an asymptotic lower bound on the
principal exponential rate of decay for the probability that the average value
over a large box of some non-decreasing local function of the field of
occupation times of the walk exceeds a given positive value. This bound matches
at leading order the corresponding upper bound derived by Sznitman in
arXiv:1906.05809, and is given in terms of a certain constrained minimum of the
Dirichlet energy of functions on $\mathbb{R}^d$ decaying at infinity. Our proof
utilizes a version of tilted random walks, a model originally constructed by Li
in arXiv:1412.3959 to derive lower bounds on the probability of the event that
the trace of a simple random walk disconnects a macroscopic set from an
enclosing box.
