Noise sensitivity of random walks on groups

A random walk on a group is noise sensitive if resampling every step independantly with a small probability results in an almost independant output. We precisely define two notions: $\ell^1$-noise sensitivity and entropy noise sensitivity. Groups with one of these properties are necessarily Liouville. Homomorphisms to free abelian groups provide an obstruction to $\ell^1$-noise sensitivity. We also provide examples of $\ell^1$ and entropy noise sensitive random walks. Noise sensitivity raises many open questions which are described at the end of the paper.

[1]  Tianyi Zheng,et al.  Speed of random walks, isoperimetry and compression of finitely generated groups , 2015, Annals of Mathematics.

[2]  Jérémie Brieussel.  Growth behaviors in the range er , 2020 .

[3]  O. Tamuz,et al.  Choquet-Deny groups and the infinite conjugacy class property , 2018, Annals of Mathematics.

[4]  Gil Kalai,et al.  Three Puzzles on Mathematics, Computation, and Games , 2018, Proceedings of the International Congress of Mathematicians (ICM 2018).

[5]  Volodymyr Nekrashevych,et al.  Palindromic subshifts and simple periodic groups of intermediate growth , 2016, 1601.01033.

[6]  G. Tiozzo.  Sublinear deviation between geodesics and sample paths , 2012, 1210.7352.

[7]  Christophe Garban,et al.  Noise Sensitivity of Boolean Functions and Percolation , 2011, 1102.5761.

[8]  Nicol'as Matte Bon.  Subshifts with slow complexity and simple groups with the Liouville property , 2014, 1402.2234.

[9]  Nicol'as Matte Bon,et al.  The Liouville property for groups acting on rooted trees , 2013, 1307.5652.

[10]  Jérémie Brieussel.  Behaviors of entropy on finitely generated groups , 2011, 1110.5099.

[11]  Omer Angel,et al.  Amenability of linear-activity automaton groups , 2013 .

[12]  Speed exponents of random walks on groups , 2012, 1203.6226.

[13]  L. Bartholdi,et al.  Growth of permutational extensions , 2010, 1011.5266.

[14]  L. Bartholdi,et al.  Groups of given intermediate word growth , 2011, 1110.3650.

[15]  Jérémie Brieussel.  Growth behaviors in the range $$e^{r^\alpha }$$erα , 2011, 1107.1632.

[16]  Laurent Bartholdi,et al.  On amenability of automata groups , 2008, 0802.2837.

[17]  Jérémie Brieussel.  Amenability and non-uniform growth of some directed automorphism groups of a rooted tree , 2009 .

[18]  Oded Schramm,et al.  The Fourier spectrum of critical percolation , 2008, 0803.3750.

[19]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[20]  Volodymyr Nekrashevych,et al.  Self-Similar Groups , 2005, 2304.11232.

[21]  Laurent Saloff-Coste,et al.  Random Walks on Finite Groups , 2004 .

[22]  L. Bartholdi,et al.  ON THE WORD AND PERIOD GROWTH OF SOME GROUPS OF TREE AUTOMORPHISMS , 2000, math/0005113.

[23]  R. I. Grigorchuk,et al.  Just Infinite Branch Groups , 2000 .

[24]  I. Benjamini,et al.  Noise sensitivity of Boolean functions and applications to percolation , 1998, math/9811157.

[25]  V. Kaimanovich.  The Poisson formula for groups with hyperbolic properties , 1998, math/9802132.

[26]  D. McDonald,et al.  An elementary proof of the local central limit theorem , 1995 .

[27]  Pál Révész,et al.  Random walk in random and non-random environments , 1990 .

[28]  R. Grigorchuk.  ON THE GROWTH DEGREES OF $ p$-GROUPS AND TORSION-FREE GROUPS , 1986 .

[29]  R. Grigorchuk.  Degrees of Growth of Finitely Generated Groups, and the Theory of Invariant Means , 1985 .

[30]  Vadim A. Kaimanovich,et al.  Random Walks on Discrete Groups: Boundary and Entropy , 1983 .

[31]  S. V. Aleshin.  Finite automata and Burnside's problem for periodic groups , 1972 .