On drift and entropy growth for random walks on groups

Erschler, Anna

doi:10.1214/aop/1055425775

Cited by 35 publications

(34 citation statements)

References 9 publications

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“…In the case of solvable groups the situation is much less known. Erschler showed in [17] that a group quasi-isometric to a solvable group may be not virtually solvable. Thus, the class of virtually solvable groups is not closed under quasi-isometry.…”

Section: The Geodesic Length Problem (Glp)mentioning

confidence: 99%

The word and geodesic problems in free solvable groups

Myasnikov

Roman'kov

Ushakov

et al. 2010

Trans. Amer. Math. Soc.

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Abstract. We study the computational complexity of the Word Problem (WP) in free solvable groups S r,d , where r ≥ 2 is the rank and d ≥ 2 is the solvability class of the group. It is known that the Magnus embedding of S r,d into matrices provides a polynomial time decision algorithm for WP in a fixed group S r,d . Unfortunately, the degree of the polynomial grows together with d, so the uniform algorithm is not polynomial in d. In this paper we show that WP has time complexity O(rn log 2 n) in S r,2 , and O(n 3 rd) in S r,d for d ≥ 3. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in S r,2 is NP -complete. We prove also that one can compute Fox derivatives of elements from S r,d in time O(n 3 rd); in particular, one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as on relatively new geometric ideas; in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.

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Section: The Geodesic Length Problem (Glp)mentioning

confidence: 99%

The word and geodesic problems in free solvable groups

Myasnikov

Roman'kov

Ushakov

et al. 2010

Trans. Amer. Math. Soc.

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“…For any symmetric finitely supported random walk on a group G, there exists K ą 0 such that Lpnq ď K a n log vpnq`logpnq for all n, see [14,Lemma 7.(ii)].…”

Section: 3mentioning

confidence: 99%

Poisson–Furstenberg boundary and growth of groups

Bartholdi

Erschler

2016

Probab. Theory Relat. Fields

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Abstract. We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this criterion is useful to establish exponential word growth of groups. We construct groups of exponential growth such that all finitely supported (not necessarily symmetric, possibly degenerate) random walks on these groups have trivial boundary. This gives a negative answer to a question of Kaimanovich and Vershik.

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“…The matrix elements of the Fourier transform of a function on a lattice motion group can be defined as: (10) Using the properties of the delta function, (10) can be rewritten as (11) where k is drawn from the reciprocal lattice (see the appendix). By setting , (11) becomes (12) Equation (12) can be identified as a d-dimensional space DFT-transform, where d is the dimension of the lattice.…”

Section: Analysis: Fourier Transformmentioning

confidence: 99%

“…The topic of random walks on lattices and on finite groups has been of interest to mathematicians for at least half a century [15,24,36], and remains of interest in recent times [9,10]. Often the main mathematical goals in studying random walks center around limiting behaviors of ensembles of walks as their length becomes large [28,41,43].…”

Section: Introductionmentioning

confidence: 99%

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

2010

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This paper presents an efficient group-theoretic approach for computing the statistics of non-reversal random walks (NRRW) on lattices. These framed walks evolve on proper crystallographic space groups. In a previous paper we introduced a convolution method for computing the statistics of NRRWs in which the convolution product is defined relative to the space-group operation. Here we use the corresponding concept of the fast Fourier transform for functions on crystallographic space groups together with a non-Abelian version of the convolution theorem. We develop the theory behind this technique and present numerical results for two-dimensional and three-dimensional lattices (square, cubic and diamond). In order to verify our results, the statistics of the end-to-end distance and the probability of ring closure are calculated and compared with results obtained in the literature for the random walks for which closed-form expressions exist.

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On drift and entropy growth for random walks on groups

Cited by 35 publications

References 9 publications

The word and geodesic problems in free solvable groups

The word and geodesic problems in free solvable groups

Poisson–Furstenberg boundary and growth of groups

Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

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