Sublinear variance in Euclidean first-passage percolation
| Autor: | Michael Damron, Torin Greenwood, Megan Bernstein |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
60K35 60G55 Sublinear function Applied Mathematics 010102 general mathematics Probability (math.PR) First passage percolation 01 natural sciences Upper and lower bounds Combinatorics 010104 statistics & probability Bernoulli's principle Modeling and Simulation Lattice (order) Poisson point process Euclidean geometry FOS: Mathematics 0101 mathematics Invariant (mathematics) Mathematics - Probability Mathematics |
| DOI: | 10.48550/arxiv.1901.10325 |
| Popis: | The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and $ne_1$ obeys a diffusive upper bound: $\mbox{Var}\, T(0,ne_1) \leq Cn$, and in this paper we improve this inequality to $Cn/\log n$. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik-Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the passage time as a function of Bernoulli sequences and uniform sequences, and develop several "greedy lattice animal" arguments. Comment: 40 pages, 1 figure |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|