Transversal fluctuations for a first passage percolation model

@article{Bakhtin2019TransversalFF,
  title={Transversal fluctuations for a first passage percolation model},
  author={Yuri Bakhtin and Wei Wu},
  journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
  year={2019}
}
  • Yuri Bakhtin, Wei Wu
  • Published 19 May 2016
  • Mathematics
  • Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
We introduce a new first passage percolation model in a Poissonian environment on $\mathbb{R}^{2}$. In this model, the action of a path depends on the geometry of the path and the travel time. We prove that the transversal fluctuation exponent for point-to-line action minimizers is at least $3/5$. 

Figures from this paper

Shape of shortest paths in random spatial networks.
TLDR
The results shed some light on the Euclidean first-passage process but also raise some theoretical questions about the scaling laws and the derivation of the exponent values and also whether a model can be constructed with maximal wandering, or non-Gaussian travel fluctuations, while embedded in space.

References

SHOWING 1-10 OF 22 REFERENCES
Euclidean models of first-passage percolation
Summary. We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝd. Compared to standard FPP on ℤd, these models have some technical
Geodesics in two-dimensional first-passage percolation
We consider standard first-passage percolation on Z 2 . Geodesics are nearest-neighbor paths in Z 2 , each of whose segments is time-minimizing. We prove part of the conjecture that doubly infinite
First-Passage Percolation, Subadditive Processes, Stochastic Networks, and Generalized Renewal Theory
In 1957, Broadbent and Hammersley gave a mathematical formulation of percolation theory. Since then much work has been done in this field and has now led to first-passage percolation problems. In the
A Surface View of First-Passage Percolation
Let \(\tilde B\)(t) be the set of sites reached from the origin by time t in standard first-passage percolation on Z d , and let B0 (roughly lim \(\tilde B\) (t)/t) be its deterministic asymptotic
SUPERDIFFUSIVE BEHAVIOR OF TWO-DIMENSIONAL BROWNIAN MOTION IN A POISSONIAN POTENTIAL
We consider d-dimensional Brownian motion in a truncated Poissonian potential conditioned to reach a remote location. If Brownian motion starts at the origin and ends in an hyperplane at distance L
A simplified proof of the relation between scaling exponents in first-passage percolation
In a recent breakthrough work, Chatterjee (5) proved a long standing conjecture that relates the transversal exponentand the fluctuation exponentin first-passage percolation on Z d . The purpose of
The universal relation between scaling exponents in first-passage percolation
It has been conjectured in numerous physics papers that in ordinary rst-passage percolation on integer lattices, the uctuation exponent and the wandering exponent are related through the universal
Superdiffusivity in first-passage percolation
Summary.In standard first-passage percolation on ${\Bbb Z}^d$ (with $d\geq 2$), the time-minimizing paths from a point to a plane at distance $L$ are expected to have transverse fluctuations of order
Divergence of Shape Fluctuations in Two Dimensions
We consider stochastic growth models, such as standard first-passage percolation on Z d , where to leading order there is a linearly growing deterministic shape. Under natural hypotheses, we prove
Bigeodesics in First-Passage Percolation
In first-passage percolation, we place i.i.d. continuous weights at the edges of $${\mathbb{Z}^2}$$Z2 and consider the weighted graph metric. A distance-minimizing path between points x and y is
...
...