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2 results

  • Non-existence of bi-infinite geodesics in the exponential corner growth model

  • Part of
  • Márton Balázs, Ofer Busani, Timo Seppäläinen
  • Journal:
    Forum of Mathematics, Sigma / Volume 8 / 2020
    Published online by Cambridge University Press:
    16 November 2020, e46
    • Article
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  • This paper gives a self-contained proof of the non-existence of nontrivial bi-infinite geodesics in directed planar last-passage percolation with exponential weights. The techniques used are couplings, coarse graining, and control of geodesics through planarity and estimates derived from increment-stationary versions of the last-passage percolation process.

  • Growth of a Population of Bacteria in a Dynamical Hostile Environment

  • Part of
  • Olivier Garet, Régine Marchand
  • Journal:
    Advances in Applied Probability / Volume 46 / Issue 3 / September 2014
    Published online by Cambridge University Press:
    22 February 2016, pp. 661-686
    Print publication:
    September 2014
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  • We study the growth of a population of bacteria in a dynamical hostile environment corresponding to the immune system of the colonized organism. The immune cells evolve as subcritical open clusters of oriented percolation and are perpetually reinforced by an immigration process, while the bacteria try to grow as a supercritical oriented percolation in the remaining empty space. We prove that the population of bacteria grows linearly when it survives. From this perspective, we build general tools to study dependent oriented percolation models issued from renormalization processes.