Ergodic Schrodinger operators on the Bethe lattice and a modified Thouless formula

Abstract

The main result of this paper is a modified Thouless formula relating the density of states for ergodic Schrodinger operators on the Bethe lattice to the Lyapunov exponent. The modified Thouless formula consists of a Thouless-like term, involving the density of states, and a remainder term. The remainder term vanishes when the connectivity $κ$ equals one, yielding the usual Thouless formula for ergodic Schrodinger operators on $\mathbb{Z}$. We prove the remainder term is nontrivial for $κ\geq 2$. We also discuss the automorphism group of the Bethe lattice and its relation to ergodic Schrodinger operators. In particular, we clarify the use of the multiparameter noncommutative ergodic theorem in evaluating the limit of Green's functions along certain paths.

Figures (5)

• Figure 1: Basic geometry of the Bethe Lattice: the figure shows a finite section of the Bethe lattice for connectivity $\kappa = 2$. Distinguishing one vertex as the root (shown in the center) results in a radial structure (coordination spheres or levels) about the root, where each coordination sphere consists of vertices at a fixed distance (=number of edges) from the root. The arrows display the natural partial order emanating from the root (see Section \ref{['subsec:bethe1']} for a precise definition); in particular, while the root is connected to $\kappa + 1$ forward neighbors, each other vertex has precisely $\kappa$ vertices in the forward direction. The labeling of the vertices is explained in Section \ref{['subsec:bethe1']}; see also Figure \ref{['fig_bethe-lattice_vertex-labeling']}.
• Figure 2: Vertex labeling of the Bethe lattice for $\kappa = 2$: the information about each vertex at level $\ell \geqslant 1$ is uniquely encoded by the $\ell+1$-tuple $(0,a_1, \dots, a_\ell)$ where $a_1 \in \{0,1,2\}$ (corresponding to the $\kappa + 1 = 3$ forward neighbors of the root) and $a_j \in \{0,1\}$, for $2 \leqslant j \leqslant \ell$ (corresponding to the $\kappa = 2$ forward neighbors of vertices except the root). To keep the figure concise, we abbreviate this labeling by only showing the last entry $a_\ell$ of the tuple next to the respective vertex. For instance, the three vertices in the first coordination sphere $(\ell = 1)$ are labeled in the figure as $0,1,2$ to abbreviate the ordered pairs $(0,0)$, $(0,1)$, and $(0,2)$.
• Figure 3: Illustration of the map $\tau_1$, defined in (\ref{['eq_transl1']}): the graph automorphism $\tau_1$ acts as a level translation by shifting the root $(0)$ up by one one level to the vertex $(0,0)$; the remaining vertices respond accordingly (i.e., so that edges are preserved). The two panels of the figure represent a before (left panel) and after (right panel) picture of the action of $\tau_1$, where the transformation of the root $(0)$ (shown as a pentagon) and the vertex $(0,2)$ (shown as a diamond) are especially highlighted.
• Figure 4: Illustration of the map $\tau_2$, defined in (\ref{['eq:transl2']}): the action of the graph automorphism $\tau_2$ on a vertex $x$ at level $\ell \geqslant 1$ can be understood as the result of $\ell$ consecutive rotations (or cyclic permutations) where each vertex along the unique path connecting the root to $x$ is rotated; this rotation at each level is indicated in the left panel by dotted arrows. The side by side display of the left and right panel presents a before and after picture in which we illustrate the result of the action of $\tau_2$ for the three vertices $(0,2)$ (diamond), $(0,2,0)$ (pentagon), and $(0,2,1)$ (star).
• Figure 5: Geometric perspective of the generalized shift $\tau_x$, defined in (\ref{['eq_generalized-shift-vertex']}): we illustrate the action of $\tau_x$ for $x = (0,2,0)$ on three vertices $z_1= (0,0)$ (solid pentagon), $z_2=(0,1)$ (solid diamond), and $z_3 = (0,2)$ (star). The results of $\tau_x(z_j)$, for $1 \leqslant j \leqslant 3$, are shown by the respective unshaded symbols. The figure illustrates that $\tau_x(z_j)$ can be obtained geometrically by appropriately attaching (as indicated by the labeling) the unique path $\gamma_{z_j}$, connecting the root to $z_j$, to the vertex $x$. In this sense, $\tau_x$ can be viewed as shifting the root to the vertex $x$, which is shown by the arrow in the figure.

Theorems & Definitions (18)

• Remark 2.1
• Remark 2.2
• Proposition 2.1
• Remark 2.3
• proof
• Lemma 2.1
• Theorem 2.4: RW-expansion of the resolvent
• proof
• Theorem 2.5: SAW-expansion of the resolvent
• Remark 2.6