Electronic Journal of Probability

Detecting tampering in a random hypercube

Ross Pinsky

Full-text: Open access


Consider the random hypercube $H_2^n(p_n)$ obtained from the hypercube $H_2^n$ by deleting any given edge with probabilty $1 -p_n$, independently of all the other edges. A diameter path in $H_2^n$ is a longest geodesic path   in $H_2^n$. Consider the following two ways of tampering with the random graph $H_2^n(p_n)$: (i) choose a diameter path at random and adjoin all of its edges to $H_2^n(p_n)$; (ii)  choose a diameter path at random from among those that start at $0=(0,\cdots, 0)$, and adjoin all of its edges to $H_2^n(p_n)$. We study the question of whether these tamperings are detectable asymptotically as $n\to\infty$.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 28, 12 pp.

Accepted: 18 February 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability

random graph random hypercube total variation norm detection

This work is licensed under a Creative Commons Attribution 3.0 License.


Pinsky, Ross. Detecting tampering in a random hypercube. Electron. J. Probab. 18 (2013), paper no. 28, 12 pp. doi:10.1214/EJP.v18-2290. https://projecteuclid.org/euclid.ejp/1465064253

Export citation


  • Baik, Jinho; Deift, Percy; Johansson, Kurt. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178.
  • Berger, N. and Peres, Y. Detecting the Trail of a Random Walker in a Random Scenery, preprint, arXiv:1210.0314
  • Bollobás, Béla. Modern graph theory. Graduate Texts in Mathematics, 184. Springer-Verlag, New York, 1998. xiv+394 pp. ISBN: 0-387-98488-7
  • Harris, Matthew; Keane, Michael. Random coin tossing. Probab. Theory Related Fields 109 (1997), no. 1, 27–37.
  • Janson, Svante. The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph. Combin. Probab. Comput. 3 (1994), no. 1, 97–126.
  • Komlós, János; Szemerédi, Endre. Limit distribution for the existence of Hamiltonian cycles in a random graph. Discrete Math. 43 (1983), no. 1, 55–63.
  • Levin, David A.; Pemantle, Robin; Peres, Yuval. A phase transition in random coin tossing. Ann. Probab. 29 (2001), no. 4, 1637–1669.
  • Logan, B. F.; Shepp, L. A. A variational problem for random Young tableaux. Advances in Math. 26 (1977), no. 2, 206–222.
  • Pinsky, Ross G. Law of large numbers for increasing subsequences of random permutations. Random Structures Algorithms 29 (2006), no. 3, 277–295.
  • Pinsky, Ross G. When the law of large numbers fails for increasing subsequences of random permutations. Ann. Probab. 35 (2007), no. 2, 758–772.
  • Vershik, A. M.; Kerov, S. V. Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group. (Russian) Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25–36, 96.