The large connectivity limit of the Anderson model on tree graphs (Q2924879)

Let \(\mathcal{T}_K\) be an infinite regular tree, where \(K\geq 2\) is the degree. The paper under review deals with the Anderson model on \(\mathcal{T}_K\), that is the study of the following operator acting on \(\ell^2(\mathcal{T}_K)\) NEWLINE\[NEWLINEH_{K,t} = -t\;T_K + V,NEWLINE\]NEWLINE where \(T_K\) is the adjacency matrix of the tree, \(V = (V_x)_{x\in \mathcal{T}_K}\) is a sequence of independent and identically distributed random variables (random potential) and \(t>0\) is interpreted as a hopping strength. According to the value of \(t\), the operator can be in a localized or extended phase. The main result of this paper (Theorem 1) provides upper and lower bounds on the critical hopping strength (note that the \(I\) in the statement of Theorem 1 is presumably the energy range \([E \pm \delta_E]\)). These bounds match in the large connectivity limit (\(K\to\infty\), Corollary 1), confirming thereby early predictions of \textit{R. Abou-Chacra} and \textit{P. W. Anderson} and \textit{D. J. Thouless} [``A self-consistent theory of localization'', J. Phys. C. 6, No. 10, 1734 (1973; \url{doi:10.1088/0022-3719/6/10/009})], who first investigated the localization issue on such graphs. This result is derived under certain limitations on the disorder distribution which, from the author's point of view, are essentially technical. The proof is contained in sections 2 and 3. It makes use in particular of the free energy function and relies on a recent progress made by \textit{M. Aizenmann} and \textit{S. Warzel} [J. Eur. Math. Soc. (JEMS) 15, No. 4, 1167--1222 (2013; Zbl 1267.47064)], who provided a criterion for the presence of extended states. Section 4 contains a numerical confirmation.