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3. Fluctuations and scaling relations

The best current bounds on fluctuations of the first-passage metric says that for some constant $c>0$ we have $$ c\log n\,\le\,{\rm Var}\big(T(0,n{\bf e}_1)\big)\,\le\,\frac{n}{c\log n}, $$ where the lower bound holds for $d=2$, see [MR1283187], [MR1349159], and the upper bound for $d\ge2$, see [MR2016607], [MR3405617].


    1. Add remark

      Problem 3.1.

      [Jack Hanson] Let $\Gamma_n$ denote the geodesic of winding number 1 on the $d$-dimensional torus of side length $n$. Prove that ${\rm Var}(T(\Gamma_n))$ is divergent.


        • Add remark

          Problem 3.2.

          [Michael Damron] Assume finite exponential moment of the weight distribution. Does there exists a constant $c>0$ such that for $\lambda>0$ $$ \mathbb{P}\bigg(\big|T(0,z)-\mathbb{E}[T(0,z)]\big|>\lambda\sqrt{\frac{|z|}{\log|z|}}\bigg)\,<\,e^{-c\lambda^2}? $$

            •     Recall that $\chi$ and $\xi$ denote the fluctuation and wandering exponents respectively; see section on main open problems.

              Add remark

              Problem 3.3.

              [Kenneth Alexander] For $d=2$, prove anything about the correlation between $T(0,x)$ and $T(0,y)$, for $x$ and $y$ that are close, but far from the origin, and use that to say something about the ratio $2\chi/\xi$.


                • Add remark

                  Problem 3.4.

                  [Partha Dey] Can one obtain a comparison principle for ${\rm Var}(T(0,n{\bf e}_1))$?

                    •     An inhomogeneous model for first-passage percolation was introduced in [MR3485346] in which edges in the left and right half-planes of $\mathbb{Z}^2$ are assigned weights according to different distributions $F_-$ and $F_+$. It was further proven that a shape theorem holds in this setting, and that the asymptotic shape equals the convex hull of the shapes associated with the distributions $F_-$ and $F_+$, intersected with respective half-plane, and a potential additional line segment along the vertical axis. In the presence of a line segment this would produce a ‘pyramid’ on the shape in the vertical direction, and it is known that there exists pairs $(F_-,F_+)$ for which such a pyramid is produced.

                      Add remark

                      Problem 3.5.

                      [org.aimpl.user:ahlberg.daniel@gmail.com] In the inhomogeneous model for which the pair $(F_-,F_+)$ produces a ‘pyramid’ on the asymptotic shape, does that imply that ${\rm Var}(T(0,n{\bf e}_2))$ is order $n$, and that $T(0,n{\bf e}_2)$, properly rescaled, satisfies a Gaussian central limit theorem?
                          This question was answered affirmatively by Ahlberg, Bhatnagar, Dey and Ganguly during the workshop.

                          Cite this as: AimPL: First passage percolation, available at http://aimpl.org/firstpercolation.