On the joint behaviour of speed and entropy of random walks on groups

2015/09/01 by Amir, Gideon
#05C81 #20E08 #20F65 #60B15 #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.1509.00256

Abstract

For every 3/4≤ δ, β< 1 satisfying δ≤ β< (1+δ)/(2) we construct a finitely generated group Γ and a (symmetric, finitely supported) random walk X_n on Γ so that its expected distance from its starting point satisfies E|X_n|\asymp n^β and its entropy satisfies H(X_n)\asymp n^δ. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions f,h up to a constant factor as long as the functions satisfy the relation n^(3)/(4)≤ h(n)≤ f(n)≤ √nh(n)/log (n+1)≤ n^γ for some γ<1.

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