Skip to Main content Skip to Navigation
Theses

Percolation and first passage percolation : isoperimetric, time and flow constants

Abstract : In this thesis, we study the models of percolation and first passage percolation on the graph Zd, d≥2. In a first part, we study isoperimetric properties of the infinite cluster Cp of percolation of parameter p>pc. Conditioning on the event that 0 belongs to Cp, the anchored isoperimetric constant φp(n) corresponds to the infimum over all connected subgraph of Cp containing 0 of size at most nd, of the boundary size to volume ratio. We prove that n φp (n) converges when n goes to infinity towards a deterministic constant φp, which is the solution of an anisotropic isoperimetric problem in the continuous setting. We also study the behavior of the anchored isoperimetric constant at pc, and the regularity of the φp in p for p>pc. In a second part, we study a first interpretation of the first passage percolation model where to each edge of the graph, we assign independently a random passage time distributed according to a given law G. This interpretation of first passage percolation models propagation phenomenon such as the propagation of water in a porous medium. A law of large numbers is known: for any given direction x, we can define a time constant µG(x) that corresponds to the inverse of the asymptotic propagation speed in the direction x. We study the regularity properties of the µG in G. In particular, we study how the graph distance in Cp evolves with p. In a third part, we consider a second interpretation of the first passage percolation model where to each edge we assign independently a random capacity distributed according to a given law G. The capacity of G edge is the maximal amount of water that can cross the edge per second. For a given vector v of unit norm, a law of large numbers is known: we can define the flow constant in the direction v as the asymptotic maximal amount of water that can flow per second in the direction v per unit of surface. We prove a law of large numbers for the maximal flow from a compact convex source to infinity. The problem of maximal flow is dual to the problem of finding minimal cutset. A minimal cutset is a set of edges separating the sinks from the sources that limits the flow propagation by acting as a bottleneck: all its edges are saturated. In the special case where G({0})>1-pc, we prove a law of large numbers for the size of minimal cutsets associated with the maximal flow in a flat cylinder, where its top and bottom correspond respectively to the source and the sink
Complete list of metadata

https://tel.archives-ouvertes.fr/tel-03175532
Contributor : ABES STAR :  Contact
Submitted on : Saturday, March 20, 2021 - 12:51:08 PM
Last modification on : Friday, August 5, 2022 - 3:00:08 PM
Long-term archiving on: : Monday, June 21, 2021 - 6:14:20 PM

File

DEMBIN_Barbara_va.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-03175532, version 1

Citation

Barbara Dembin. Percolation and first passage percolation : isoperimetric, time and flow constants. Mathematical Physics [math-ph]. Université Paris Cité, 2020. English. ⟨NNT : 2020UNIP7023⟩. ⟨tel-03175532⟩

Share

Metrics

Record views

161

Files downloads

135