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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 Z^d, d≥2. In a first part, we study isoperimetric properties of the infinite cluster Cp of percolation of parameter p>p_c. Conditioning on the event that 0 belongs to C_p, the anchored isoperimetric constant φ_p(n) corresponds to the infimum over all connected subgraph of C_p containing 0 of size at most n^d, of the boundary size to volume ratio. We prove that n φ_p (n) converges when n goes to infinity towards a deterministic constant φ_p, 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 φ_p in p for p>p_c. 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 µ_G(x) that corresponds to the inverse of the asymptotic propagation speed in the direction x. We study the regularity properties of the µ_G in G. In particular, we study how the graph distance in C_p 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.
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Submitted on : Sunday, September 6, 2020 - 10:01:33 AM
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Barbara Dembin. Percolation and first passage percolation: isoperimetric, time and flow constants. Probability [math.PR]. Université de Paris, 2020. English. ⟨tel-02931357⟩

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