2000/06/07 by P. L. Krapivsky, Satya N. Majumdar
Physics and Astronomy · #cond-mat.stat-mech
paper · pdf · doi:10.1103/physrevlett.85.5492
published as Phys. Rev. Lett., 85 (2000) 5492 · 4 pages Revtex
arxiv created 2000/06/07 · arxiv updated 2009/11/30
We study a random bisection problem where an initial interval of length x is cut into two random fragments at the first stage, then each of these two fragments is cut further, etc. We compute the probability Pn(x) that at the n-th stage, each of the 2n fragments is shorter than 1. We show that Pn(x) approaches a traveling wave form, and the front position xn increases as xn∼ nβρn for large n. We compute exactly the exponents ρ=1.261076... and β=0.453025.... as roots of transcendental equations. We also solve the m-section problem where each interval is broken into m fragments. In particular, the generalized exponents grow as ρm≈ m/(ln m) and βm≈ 3/(2ln m) in the large m limit. Our approach establishes an intriguing connection between extreme value statistics and traveling wave propagation in the context of the fragmentation problem.