Contact graphs, boundaries and a central limit theorem for CAT(0) cubical complexes (Q6594733)

Let \(X\) be a finite-dimensional CAT(0) cubical complex, \(G\) a discrete countable group acting by cubical automorphisms on \(X\) and let \(\mu\) be an admissible probability measure on \(G\). A random walk on \(G\) is a sequence of independent identically distributed random variables \((g_{i})_{i \geq 1}\) of law \(\mu\) and let \(Z_{n}=g_{1}g_{2} \dots g_{n}\). Fix a point \(o \in X\) (an origin). The authors [Comment. Math. Helv. 93, No. 2, 291--333 (2018; Zbl 1494.20059)] studied the behavior of the sequence \(\{Z_{n}.o\}\) and they proved that if the action of \(G\) on \(X\) is non-elementary, then \(Z_{n}.o\) converges almost surely to some regular point in the Roller boundary \(\eta_{Z_{n}} =\partial_{\mathrm{reg}}X\). Furthermore, there is a \(\lambda > 0\) such that \(\lim_{n\rightarrow \infty}\frac{1}{n} d(Z_{n}.o,o)=\lambda\), where \(d\) is the combinatorial distance on \(X\). In the paper under review, they prove the following central limit theorem:\N\NTheorem 1.1: Assume that the action of \(G\) on \(X\) is nonelementary. Assume that \(\mu\) has a finite second moment, that is,\N\[\N\int_{G} \big ( d(g.o,o) \big ) \mathrm{d}\mu(g)< +\infty.\N\]\NThen there exists \(\sigma > 0\) such that \(n^{-\frac{1}{2}}\big ( d(Z_{n}.o,o)-n\lambda \big )\) converges in distribution to a Gaussian law of variance \(\sigma^{2}\).\N\NLet \(\partial\, \mathscr{C}X\) the boundary of the contact graph \(\mathscr{C}X\). Another result in this paper is Theorem 1.2: Let \(X\) be a finite-dimensional CAT(0) cubical complex. There exists an \(\Aut(X)\)-equivariant homeomorphism between \(\partial_{\mathrm{reg}}X\) and \(\partial\,\mathscr{C}X\).