arXiv subjects

Rados Radoicic

Publications and source records attributed to Rados Radoicic.

2 recordsLinked to original sources

arXiv2025

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.

math.PR↗

arXiv2025

Fourier levels and almost sure bounds on higher-order derivatives in first-passage percolation

The variance of first-passage percolation admits a decomposition into Fourier levels indexed by the order of environment derivatives. These Fourier levels capture how local perturbations of different orders contribute to global fluctuations. In this paper, we investigate higher-order environment derivatives and their Fourier-level structure. We prove that derivatives of orders $k\in\{2,3,4\}$ are almost surely bounded below by $-\binom{k-2}{\lceil\frac{k-2}2\rceil}$ and above by $\binom{k-2}{\lceil\frac{k-2}2\rceil}$. We conjecture that these are the correct bounds for all $k$, and we construct explicit environments showing that these extreme values can indeed be attained.

math.PR↗