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

Babai’s conjecture for high-rank classical groups with random generators

“…We analyze this action by again using the language of trajectories and coincidences, and the trace method: we bound a high moment of the second eigenvalue by bounding the trace of the corresponding power of the adjacency matrix, interpretting the latter in terms of closed trajectories. This is analogous to a result for the symmetric group due to Friedman, Joux, Roichman, Stern, and Tillich [15], building on earlier work of Broder-Shamir [8]. However, in the case of classical groups there are some extra combinatorial complications that do not arise for symmetric groups.…”

“…In fact, we will prove that the Schreier graph of the action of G on O(1)-tuples of vectors is almost surely a union of expander graphs. (The analogous result for the symmetric group is a result of Friedman, Joux, Roichman, Stern, and Tillich [15], and was essential in [26].) Theorem 1.3 Let G = Cl n (q), and let x 1 , .…”

Fast Synchronization of Inhomogenous Random Automata

“…Let us highlight the most important differences between the model we investigate and the usual model of independent uniform random mappings. The independence of the two random permutations guarantees that our automaton is strongly connected with high probability [7], moreover generates S n or A n with high probability [5], neither of which holds for an automaton with three random mappings. The idea of using permutations is motivated by the work of Nicaud and Berlinkov [1].…”

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, . .…”