TY - JOUR

T1 - Random generators of the symmetric group: Diameter, mixing time and spectral gap

AU - Helfgott, H.A.

AU - Seress, Akos

AU - Zuk, A.

PY - 2015

Y1 - 2015

N2 - © 2014 Elsevier Inc. Let g, h be a random pair of generators of G=Sym(n) or G=Alt(n). We show that, with probability tending to 1 as n→∞, (a) the diameter of G with respect to S={g, h, g-1, h-1} is at most O(n2(logn)c), and (b) the mixing time of G with respect to S is at most O(n3(logn)c). (Both c and the implied constants are absolute.)These bounds are far lower than the strongest worst-case bounds known (in Helfgott-Seress, 2013); they roughly match the worst known examples. We also give an improved, though still non-constant, bound on the spectral gap.Our results rest on a combination of the algorithm in (Babai-Beals-Seress, 2004) and the fact that the action of a pair of random permutations is almost certain to act as an expander on ℓ-tuples, where ℓ is an arbitrary constant (Friedman et al., 1998).

AB - © 2014 Elsevier Inc. Let g, h be a random pair of generators of G=Sym(n) or G=Alt(n). We show that, with probability tending to 1 as n→∞, (a) the diameter of G with respect to S={g, h, g-1, h-1} is at most O(n2(logn)c), and (b) the mixing time of G with respect to S is at most O(n3(logn)c). (Both c and the implied constants are absolute.)These bounds are far lower than the strongest worst-case bounds known (in Helfgott-Seress, 2013); they roughly match the worst known examples. We also give an improved, though still non-constant, bound on the spectral gap.Our results rest on a combination of the algorithm in (Babai-Beals-Seress, 2004) and the fact that the action of a pair of random permutations is almost certain to act as an expander on ℓ-tuples, where ℓ is an arbitrary constant (Friedman et al., 1998).

U2 - 10.1016/j.jalgebra.2014.08.033

DO - 10.1016/j.jalgebra.2014.08.033

M3 - Article

VL - 421

SP - 349

EP - 368

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -