Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Mittagsseminar Talk Information

Date and Time: Tuesday, July 27, 2004, 12:15 pm

Duration: This information is not available in the database

Location: This information is not available in the database

Speaker: Volker Kaibel (TU Berlin)

Low-dimensional faces of random 0/1-polytopes

Let P be a random 0/1-polytope in R^d with n(d) vertices, and denote by φ_k(P) the k-face density of P, i.e., the quotient of the number of k-dimensional faces of P and \binom{n(d)}{k+1}. For each k >= 2, we establish the existence of a sharp threshold for the k-face density and determine the values of the threshold numbers τ_k such that, for all ε > 0,
E(φ_k(P)) =

• 1- o(1), if n(d)<= 2^(τ_k-ε)d for all d
• o(1), if n(d)>= 2^(τ_k+ε)d for all d

holds for the expected value of φ_k(P). The threshold for k=1 has recently been determined by K. and Remshagen (2003).

In particular, these results indicate that the high face densities often encountered in polyhedral combinatorics (e.g., for the cut-polytopes of complete graphs) should be considered more as a phenomenon of the general geometry of 0/1-polytopes than as a feature of the special combinatorics of the underlying problems.

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