Higher-order derivatives of first-passage percolation with respect to the environment

We introduce and study derivatives in first-passage percolation with edge weights given by i.i.d. random variables supported on $\{a,b\}$.

We show that the variance of the passage time can be expressed in terms of these derivatives.

We further analyze their structure and establish several fundamental properties and bounds.

Our bounds for the lower Fourier levels on the torus model raise the prospect that, in dimensions 3 and higher, the variance may grow slower than any positive power of $n$.

Such growth would contradict the commonly held belief that the fluctuation exponent is positive.