Random walks on finite rank solvable groups.
type: Item
instance of: scholarly article
title: Random walks on finite rank solvable groups.
zbMATH Open document ID: 1057.20026
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2004-07-07T00:00:00Z
review text: Let \(\Gamma\) be a finitely generated group. Let \(p\colon\Gamma\times\Gamma\to[0,1]\) be a symmetric Markov kernel which is left-invariant, irreducible and whose support lies within bounded distance (with respect to a word metric on \(\Gamma\)) from the diagonal. We also denote by \(p\) the operator on the Hilbert space \(l^2(\Gamma)\) defined on \(f\in l^2(\Gamma)\) by the formula \(pf(x)=\sum_yp(x,y)f(y)\) and we denote by \(p^n\) the composition of \(p\) with itself \(n\) times. Let \(\delta_x\in l^2(\Gamma)\) be the characteristic function of the point \(x\). The scalar product \(p_t(x,y)=\langle p^t\delta_y,\delta_x\rangle\) can be interpreted as the probability for the random walk on \(\Gamma\) defined by \(p\) to go from \(x\) to \(y\) in time \(t\). In particular \[ \| p^t\delta_e\|_2^2=\langle p^t\delta_e,p^t\delta_e\rangle=\langle p^{2t}\delta_e,\delta_e\rangle=p_{2t}(e,e) \] corresponds to the probability of return to the origin \(e\) after \(2t\) steps. If \(f,g\) are two non-negative functions defined on positive numbers, then it will be written \(f\precsim g\) if there exist constants \(a,b>0\), such that for \(x\) large enough, \(f(x)\leq ag(bx)\). If the symmetric relation also holds, then it will be written \(f\sim g\). When a function is defined only on the integers, it is extended to the positive real axis by linear interpolation. It will be used the same name as for the original function. When using this convention for the function \(t\mapsto p_{2t}(e,e)\), the value of \(p_t(e,e)\) at an odd integer \(t\) must be interpreted as \(\tfrac12(p_{t-1}(e,e)+p_{t+1}(e,e))\). The first example of a finitely generated group with a heat decay equivalent to \(\exp(-t^{1/3})\) was given by \textit{N. Th. Varopoulos}. He showed that the heat decay on the wreath product \((\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}\) is equivalent to the expectation \(E[2^{-R_t}]\) where \(R_t\) is the random variable which counts the number of visited sites during time \(t\) for…
reviewed by: Alireza Abdollahi
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 20F69
Mathematics Subject Classification ID: 20P05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 60B15
zbMATH DE Number: 2078244
zbMATH Keywords: random walks
zbMATH Keywords: heat kernel decay
zbMATH Keywords: asymptotic invariants of infinite groups
zbMATH Keywords: Prüfer rank
zbMATH Keywords: solvable groups
zbMATH Keywords: finite symmetric generating sets
zbMATH Keywords: finitely generated groups
zbMATH Keywords: word metrics
author: Christophe Pittet
author: Laurent Saloff-Coste
MaRDI profile type: Publication
full work available at URL: s10097-003-0054-4
OpenAlex ID: W1974112462
DOI: 10.1007/S10097-003-0054-4
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description: scientific article; zbMATH DE number 2078244