The shape fluctuation exponent conjecture for random Riemannian metrics

Let $B_g(t)$ be the random metric ball at time $t$, and let $B(t/\mu)$ be the corresponding rescaled limiting shape, where $\mu$ is the time constant. The fluctuations of these two balls are measured by the scale at which their boundaries typically differ. Shape fluctuation exponent conjecture. The fluctuations of $B_g(t)$ from $B(t/\mu)$ are of order $t^\chi$, for some universal constant $\chi < 1/2$, called the shape fluctuation exponent. In dimension $d=2$, $\chi=1/3$. Such subdiffusive shape fluctuations are expected by analogy with first-passage percolation, but demonstrating these phenomena in the first-passage-percolation setting remains an open question.