Accessibility Percolation with Crossing Valleys on n-ary Trees

  • Frank Duque
  • Alejandro Roldán-CorreaEmail author
  • Leon A. Valencia


In this paper, we study a variation of the accessibility percolation model. This work is also motivated by evolutionary biology and evolutionary computation. Consider a tree whose vertices are labeled with random numbers. We study the probability of having a monotone subsequence of a path from the root to a leaf, where any k consecutive vertices in the path contain at least one vertex of the subsequence. An n-ary tree, with height h, is a tree whose vertices at distance at most \(h-1\) to the root have n children. For the case of n-ary trees, we prove that, as h tends to infinity, the probability of having such subsequence: tends to 1, if n grows significantly faster than \(\root k \of {h/(ek)}\); and tends to 0, if n grows significantly slower than \(\root k \of {h/(ek)}\).


Percolation Dynamics of evolution Fitness landscape 

Mathematics Subject Classification

60K35 60C05 92D15 



The authors are thankful to Ricardo Restrepo and Daya K. Nagar for helpful discussions. Thanks are due to the anonymous referees for their careful reading, criticism and suggestions which helped us to considerably improve the paper. Research Partially supported by FORDECYT 265667 (Mexico). Research Partially supported by Universidad de Antioquia (Colombia).


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Matemáticas UNAM and Instituto de Física UASLPMexicoMexico
  2. 2.Instituto de MatemáticasUniversidad de AntioquiaMedellínColombia

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