AbstractWe consider a random walk Wn on the locally free group (or equivalently a signed random heap) with m generators subject to periodic boundary conditions. Let #T(Wn) denote the number of removable elements, which determines the heap's growth rate. We prove that limn→∞𝔼(#T(Wn))/m ≤ 0.32893 for m ≥ 4. This result disproves a conjecture (due to Vershik, Nechaev and Bikbov [Comm Math Phys 212 (2000), 469–501]) that the limit tends to 1/3 as m → ∞. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005