Harmonic maps on amenable groups and a diffusive lower bound for random walks (Q378807)

Consider an infinite connected \(d\)-regular graph \(G\), having a transitive automorphism group \(\Gamma\), which is moreover amenable, in the sense that there exists a sequence \((S_j)\) of finite subsets of \(G\) such that \(|\partial S_j|= \sigma(|S_j|)\).NEWLINENEWLINENEWLINE Let \((X_t)\) denote the symmetric simple random walk on \(G\). The authors prove the following, which extends a result by A. Erschler (valid for \(G\) being moreover a Cayley graph).NEWLINENEWLINENEWLINE Theorem: (i) \(\mathbb{E}(d(X_0,X_t)^2)\geq t/d\) for any \(t\in\mathbb{N}\);NEWLINENEWLINENEWLINE (ii) \(\mathbb{E}(d(X_0,X_t))\geq C\sqrt{t/d}\) for a universal \(C>0\) and for \(t\geq d\);NEWLINENEWLINENEWLINE (iii) \({1\over t} \sum^t_{s=1} \mathbb{P}(d(X_0, X_s)\leq \varepsilon\sqrt{t/d})\leq C'\varepsilon\) for a universal \(C'\) and any \(\varepsilon> 0\) and \(t\geq\max\{d,\varepsilon^{-2}\}\).NEWLINENEWLINENEWLINE They also prove a version for finite graphs which holds up to the relaxation time of \((X_t)\).NEWLINENEWLINENEWLINE Their proof is based on the existence of a non-constant equivariant harmonic map \(\Psi\) on \(G\): they indeed establish the following. NEWLINENEWLINENEWLINE Theorem: There exist a Hilbert space \(H\) on which \(\Gamma\) acts by isometries and a non-constant \(\Psi:G\to H\) such that:NEWLINENEWLINENEWLINE (i) \(\gamma\cdot\Psi(g)= \Psi(\gamma\cdot g)\) for any \(\gamma\in\Gamma\) and \(g\in G\);NEWLINENEWLINENEWLINE (ii) \(P\psi=\Psi\), where \(P\) is the transition matrix of \((X_t)\).