DOI: 10.1515/anona-2025-0198 ISSN: 2191-950X
Finite-volume eigenfunction correlators for the one-dimensional Anderson model: a numerical study
Alexander Bao Chen, Emily Mei, Neev Shaw, Yiding Max Tang, Sebastian Vargas Loaiza, Xueyin Wang Abstract
For the one-dimensional discrete Anderson model, we numerically study the finite-volume eigenfunction correlator
Q
ω
,
N
(
k
)
=
∑
j
=
−
N
N
ψ
ω
,
N
,
j
(
0
)
ψ
ω
,
N
,
j
(
k
)
,
$${Q}_{\omega ,N}\left(k\right)=\sum _{j=-N}^{N}\left\vert {\psi }_{\omega ,N,j}\left(0\right)\right\vert \left\vert {\psi }_{\omega ,N,j}\left(k\right)\right\vert ,$$
of the Dirichlet restriction to
[
−
N
,
N
]
∩
Z
$\left[-N,N\right]\cap \mathbb{Z}$
, which dominates the site-to-site transition amplitude uniformly in time. We prove a concentration estimate, uniform in
k
, for its Monte Carlo approximation. Comparing centered Bernoulli and continuous uniform single-site laws at matched variance
σ
2
, we find that their estimated finite-volume decay rates exhibit closely parallel dependence on
σ
2
, although their values are not identical at fixed variance. We also quantify the dependence of the estimated decay rate on
N
and on
σ
2
over the parameter ranges considered.