# Passage time of the frog model has a sublinear variance

## TL;DR

This paper demonstrates that the first passage time in the frog model on multidimensional integer lattices exhibits sublinear variance, challenging the applicability of the standard central limit theorem in this context.

## Contribution

It introduces a novel method combining existing techniques to prove sublinear variance and linearity of optimal path lengths in the frog model.

## Key findings

- First passage time variance is sublinear in the frog model.
- Standard diffusive scaling does not satisfy the central limit theorem.
- Optimal path lengths grow linearly with distance.

## Abstract

In this paper, we show that the first passage time in the frog model on $\Z^d$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not holds at least with the standard diffusive scaling. The proof is based on the method introduced in \cite{BRo, DHS} combining with a control of the maximal weight of paths in locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths..