  In first-passage percolation (FPP), one assigns i.i.d.~weights to the edges
of the cubic lattice $\mathbb{Z}^d$ and analyzes the induced weighted graph
metric. If $T(x,y)$ is the distance between vertices $x$ and $y$, then a
primary question in the model is: what is the order of the fluctuations of
$T(0,x)$? It is expected that the variance of $T(0,x)$ grows like the norm of
$x$ to a power strictly less than 1, but the best lower bounds available are
(only in two dimensions) of order $\log \|x\|$. This result was found in the
'90s and there has not been any improvement since. In this paper, we address
the problem of getting stronger fluctuation bounds: to show that $T(0,x)$ is
with high probability not contained in an interval of size $o(\log
\|x\|)^{1/2}$, and similar statements for FPP in thin cylinders. Such
statements have been proved for special edge-weight distributions, and here we
obtain such bounds for general edge-weight distributions. The methods involve
inducing a fluctuation in the number of edges in a box whose weights are of
"hi-mode" (large).
