• Bernoulli
  • Volume 22, Number 2 (2016), 653-680.

The number of accessible paths in the hypercube

Julien Berestycki, Éric Brunet, and Zhan Shi

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Motivated by an evolutionary biology question, we study the following problem: we consider the hypercube $\{0,1\}^{L}$ where each node carries an independent random variable uniformly distributed on $[0,1]$, except $(1,1,\ldots,1)$ which carries the value $1$ and $(0,0,\ldots,0)$ which carries the value $x\in[0,1]$. We study the number $\Theta$ of paths from vertex $(0,0,\ldots,0)$ to the opposite vertex $(1,1,\ldots,1)$ along which the values on the nodes form an increasing sequence. We show that if the value on $(0,0,\ldots,0)$ is set to $x=X/L$ then $\Theta/L$ converges in law as $L\to\infty$ to $\mathrm{e}^{-X}$ times the product of two standard independent exponential variables.

As a first step in the analysis, we study the same question when the graph is that of a tree where the root has arity $L$, each node at level 1 has arity $L-1$, …, and the nodes at level $L-1$ have only one offspring which are the leaves of the tree (all the leaves are assigned the value 1, the root the value $x\in[0,1]$).

Article information

Bernoulli, Volume 22, Number 2 (2016), 653-680.

Received: November 2013
First available in Project Euclid: 9 November 2015

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branching processes evolutionary biology percolation trees


Berestycki, Julien; Brunet, Éric; Shi, Zhan. The number of accessible paths in the hypercube. Bernoulli 22 (2016), no. 2, 653--680. doi:10.3150/14-BEJ641.

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  • [1] Aldous, D. and Steele, J.M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Berlin: Springer.
  • [2] Altenberg, L. (1997). NK fitness landscapes. In Handbook of Evolutionary Computation (T. Bäck, D.B. Fogel and Z. Michalewicz, eds.) B2.7:5–B2.7:10. New York: Oxford Univ. Press.
  • [3] Berestycki, J., Brunet, E. and Shi, Z. (2014). Accessibility percolation with backsteps. Preprint. Available at arXiv:1401.6894.
  • [4] Carneiro, M. and Hartl, D.L. (2010). Adaptive landscapes and protein evolution. Proc. Natl. Acad. Sci. USA 107, Suppl 1 1747–1751.
  • [5] Chen, X. (2014). Increasing paths on $N$-ary trees. Preprint. Available at arXiv:1403.0843.
  • [6] Franke, J., Klözer, A., de Visser, J.A.G.M. and Krug, J. (2011). Evolutionary accessibility of mutational pathways. PLoS Comput. Biol. 7 e1002134, 9.
  • [7] Gillespie, J.H. (1983). A simple stochastic gene substitution model. Theor. Popul. Biol. 23 202–215.
  • [8] Hegarty, P. and Martinsson, A. (2014). On the existence of accessible paths in various models of fitness landscapes. Ann. Appl. Probab. 24 1375–1395.
  • [9] Kauffman, S. and Levin, S. (1987). Towards a general theory of adaptive walks on rugged landscapes. J. Theoret. Biol. 128 11–45.
  • [10] Kingman, J.F.C. (1978). A simple model for the balance between selection and mutation. J. Appl. Probab. 15 1–12.
  • [11] Klozner, A. (2008). NK fitness landscapes. Diplomarbeit Universität zu Köln.
  • [12] Lalley, S.P. and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15 1052–1061.
  • [13] Nowak, S. and Krug, J. (2013). Accessibility percolation on $n$-trees. Europhys. Lett. 101 66004.
  • [14] On-line Encyclopedia of Integer Sequences. Available at
  • [15] Roberts, M.I. and Zhao, L.Z. (2013). Increasing paths in regular trees. Electron. Commun. Probab. 18 1–10.
  • [16] Weinreich, D.M., Delaney, N.F., DePristo, M.A. and Hartl, D.M. (2006). Darwinian evolution can follow only very few mutational paths to fitter proteins. Science 312 111–114.
  • [17] Weinreich, D.M., Watson, R.A. and Chao, L. (2005). Perspective: Sign epistasis and genetic constraints on evolutionary trajectories. Evolution 59 1165–1174.