The action of a few permutations onr-tuples is quickly transitive

Abstract

ABSTRACT:We prove that for every r and dG 2 there is a C such that for most choices of d permutations , , . . . , of S , the following holds: for any two r-tuples of distinct 1 2 d n Ä 4 elements in 1, . . . , n , there is a product of less than C log n of the s which map the first i r-tuple to the second. Although we came across this problem while studying a rather unrelated cryptographic problem, it belongs to a general context of which random Cayley graph quotients of S are good expanders. ᮊ

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

“…2.1]. We thank B. Tsaban for pointing this out to us; an earlier version of the present paper contained a proof close to the one that can be found in [13]. (The proof in the earlier version was inspired by the proof for the case = 1 given in [7].…”

“…(Since S = S −1 , the spectrum is real.) What was proven in [13] is that, for every , there is a δ > 0 such that the probability that λ 0 − λ 1 ≥ δ (for the largest eigenvalues λ 0 , λ 1 of the Schreier graph Γ (G → X, S), where X is the set of all -tuples of distinct elements of {1, 2, . .…”

“…We also give an improved, though still non-constant, bound on the spectral gap. Our results rest on a combination of the algorithm in (BabaiBeals-Seress, 2004) and the fact that the action of a pair of random permutations is almost certain to act as an expander on -tuples, where is an arbitrary constant (Friedman et al, 1998).…”

On the Interplay Between Černý and Babai’s Conjectures

“…The diameter of P (A), denoted diam P (A), is the maximum among the lengths of shortest (directed) paths between any two vertices. We study the behavior of diam P (A) in terms of n. The problem comes from analysis of certain aspects of Markov chains and group theory [17], but our interest in it is mainly motivated by its importance for the theory of synchronizing automata. Indeed, every synchronizing automaton A must have a letter a, say, whose action merges a pair of states.…”

“…Relying on a group-theoretic result by Dixon [12], Cameron [8] observed that an automaton formed by two permutation letters taken uniformly at random and an arbitrary non-permutation letter is synchronizing with high probability. We give an extension by using another non-trivial group-theoretical result, namely, the following theorem by Friedman et al [17]:…”