Consider the Schrödinger operator $-\triangle +\lambda V$ with non-negative iid random potential $V$ of strength $\lambda >0$. We prove existence and uniqueness of the associated landscape function on the whole space, and show that its correlations decay exponentially. As a main ingredient we establish the (annealed and quenched) exponential decay of the Green function of $-\triangle +\lambda V$ using Agmon’s positivity method, rank-one perturbation in dimensions $d\ge 3$, and first-passage percolation in dimensions $d=1,2$.
Soit $-\triangle +\lambda V$ l’opérateur de Schrödinger avec un potentiel aléatoire $V$ positif (à valeurs indépendantes et identiquement distribuées) d’intensité $\lambda >0$. Nous démontrons l’existence et l’unicité de la fonction de paysage associée sur tout l’espace et établissons la décroissance exponentielle de ses corrélations. L’ingrédient principal est la décroissance exponentielle (trajectorielle et en moyenne) de la fonction de Green de $-\triangle +\lambda V$ que nous montrons en combinant la méthode de positivité d’Agmon avec un argument de perturbation de rang un en dimensions $d\ge 3$ et des résultats de percolation de premier passage en dimensions $d=1,2$.
Accepté le :
Publié le :
Guy David 1 ; Antoine Gloria 2 ; Svitlana Mayboroda 3
CC-BY 4.0
@article{AFST_2025_6_34_2_315_0,
author = {Guy David and Antoine Gloria and Svitlana Mayboroda},
title = {The landscape function on $\mathbb{R}^d$},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {315--337},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {2},
year = {2025},
doi = {10.5802/afst.1814},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1814/}
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Guy David; Antoine Gloria; Svitlana Mayboroda. The landscape function on $\mathbb{R}^d$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 2, pp. 315-337. doi : 10.5802/afst.1814. https://afst.centre-mersenne.org/articles/10.5802/afst.1814/
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